numeric.c

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/*-------------------------------------------------------------------------
 *
 * numeric.c
 *    An exact numeric data type for the Postgres database system
 *
 * Original coding 1998, Jan Wieck.  Heavily revised 2003, Tom Lane.
 *
 * Many of the algorithmic ideas are borrowed from David M. Smith's "FM"
 * multiple-precision math library, most recently published as Algorithm
 * 786: Multiple-Precision Complex Arithmetic and Functions, ACM
 * Transactions on Mathematical Software, Vol. 24, No. 4, December 1998,
 * pages 359-367.
 *
 * Copyright (c) 1998-2025, PostgreSQL Global Development Group
 *
 * IDENTIFICATION
 *    src/backend/utils/adt/numeric.c
 *
 *-------------------------------------------------------------------------
 */
 
#include "postgres.h"
 
#include <ctype.h>
#include <float.h>
#include <limits.h>
#include <math.h>
 
#include "common/hashfn.h"
#include "common/int.h"
#include "funcapi.h"
#include "lib/hyperloglog.h"
#include "libpq/pqformat.h"
#include "miscadmin.h"
#include "nodes/nodeFuncs.h"
#include "nodes/supportnodes.h"
#include "optimizer/optimizer.h"
#include "utils/array.h"
#include "utils/builtins.h"
#include "utils/float.h"
#include "utils/guc.h"
#include "utils/numeric.h"
#include "utils/pg_lsn.h"
#include "utils/sortsupport.h"
 
/* ----------
 * Uncomment the following to enable compilation of dump_numeric()
 * and dump_var() and to get a dump of any result produced by make_result().
 * ----------
#define NUMERIC_DEBUG
 */
 
 
/* ----------
 * Local data types
 *
 * Numeric values are represented in a base-NBASE floating point format.
 * Each "digit" ranges from 0 to NBASE-1.  The type NumericDigit is signed
 * and wide enough to store a digit.  We assume that NBASE*NBASE can fit in
 * an int.  Although the purely calculational routines could handle any even
 * NBASE that's less than sqrt(INT_MAX), in practice we are only interested
 * in NBASE a power of ten, so that I/O conversions and decimal rounding
 * are easy.  Also, it's actually more efficient if NBASE is rather less than
 * sqrt(INT_MAX), so that there is "headroom" for mul_var and div_var to
 * postpone processing carries.
 *
 * Values of NBASE other than 10000 are considered of historical interest only
 * and are no longer supported in any sense; no mechanism exists for the client
 * to discover the base, so every client supporting binary mode expects the
 * base-10000 format.  If you plan to change this, also note the numeric
 * abbreviation code, which assumes NBASE=10000.
 * ----------
 */
 
#if 0
#define NBASE       10
#define HALF_NBASE  5
#define DEC_DIGITS  1           /* decimal digits per NBASE digit */
#define MUL_GUARD_DIGITS    4   /* these are measured in NBASE digits */
#define DIV_GUARD_DIGITS    8
 
typedef signed char NumericDigit;
#endif
 
#if 0
#define NBASE       100
#define HALF_NBASE  50
#define DEC_DIGITS  2           /* decimal digits per NBASE digit */
#define MUL_GUARD_DIGITS    3   /* these are measured in NBASE digits */
#define DIV_GUARD_DIGITS    6
 
typedef signed char NumericDigit;
#endif
 
#if 1
#define NBASE       10000
#define HALF_NBASE  5000
#define DEC_DIGITS  4           /* decimal digits per NBASE digit */
#define MUL_GUARD_DIGITS    2   /* these are measured in NBASE digits */
#define DIV_GUARD_DIGITS    4
 
typedef int16 NumericDigit;
#endif
 
#define NBASE_SQR   (NBASE * NBASE)
 
/*
 * The Numeric type as stored on disk.
 *
 * If the high bits of the first word of a NumericChoice (n_header, or
 * n_short.n_header, or n_long.n_sign_dscale) are NUMERIC_SHORT, then the
 * numeric follows the NumericShort format; if they are NUMERIC_POS or
 * NUMERIC_NEG, it follows the NumericLong format. If they are NUMERIC_SPECIAL,
 * the value is a NaN or Infinity.  We currently always store SPECIAL values
 * using just two bytes (i.e. only n_header), but previous releases used only
 * the NumericLong format, so we might find 4-byte NaNs (though not infinities)
 * on disk if a database has been migrated using pg_upgrade.  In either case,
 * the low-order bits of a special value's header are reserved and currently
 * should always be set to zero.
 *
 * In the NumericShort format, the remaining 14 bits of the header word
 * (n_short.n_header) are allocated as follows: 1 for sign (positive or
 * negative), 6 for dynamic scale, and 7 for weight.  In practice, most
 * commonly-encountered values can be represented this way.
 *
 * In the NumericLong format, the remaining 14 bits of the header word
 * (n_long.n_sign_dscale) represent the display scale; and the weight is
 * stored separately in n_weight.
 *
 * NOTE: by convention, values in the packed form have been stripped of
 * all leading and trailing zero digits (where a "digit" is of base NBASE).
 * In particular, if the value is zero, there will be no digits at all!
 * The weight is arbitrary in that case, but we normally set it to zero.
 */
 
struct NumericShort
{
    uint16      n_header;       /* Sign + display scale + weight */
    NumericDigit n_data[FLEXIBLE_ARRAY_MEMBER]; /* Digits */
};
 
struct NumericLong
{
    uint16      n_sign_dscale;  /* Sign + display scale */
    int16       n_weight;       /* Weight of 1st digit  */
    NumericDigit n_data[FLEXIBLE_ARRAY_MEMBER]; /* Digits */
};
 
union NumericChoice
{
    uint16      n_header;       /* Header word */
    struct NumericLong n_long;  /* Long form (4-byte header) */
    struct NumericShort n_short;    /* Short form (2-byte header) */
};
 
struct NumericData
{
    int32       vl_len_;        /* varlena header (do not touch directly!) */
    union NumericChoice choice; /* choice of format */
};
 
 
/*
 * Interpretation of high bits.
 */
 
#define NUMERIC_SIGN_MASK   0xC000
#define NUMERIC_POS         0x0000
#define NUMERIC_NEG         0x4000
#define NUMERIC_SHORT       0x8000
#define NUMERIC_SPECIAL     0xC000
 
#define NUMERIC_FLAGBITS(n) ((n)->choice.n_header & NUMERIC_SIGN_MASK)
#define NUMERIC_IS_SHORT(n)     (NUMERIC_FLAGBITS(n) == NUMERIC_SHORT)
#define NUMERIC_IS_SPECIAL(n)   (NUMERIC_FLAGBITS(n) == NUMERIC_SPECIAL)
 
#define NUMERIC_HDRSZ   (VARHDRSZ + sizeof(uint16) + sizeof(int16))
#define NUMERIC_HDRSZ_SHORT (VARHDRSZ + sizeof(uint16))
 
/*
 * If the flag bits are NUMERIC_SHORT or NUMERIC_SPECIAL, we want the short
 * header; otherwise, we want the long one.  Instead of testing against each
 * value, we can just look at the high bit, for a slight efficiency gain.
 */
#define NUMERIC_HEADER_IS_SHORT(n)  (((n)->choice.n_header & 0x8000) != 0)
#define NUMERIC_HEADER_SIZE(n) \
    (VARHDRSZ + sizeof(uint16) + \
     (NUMERIC_HEADER_IS_SHORT(n) ? 0 : sizeof(int16)))
 
/*
 * Definitions for special values (NaN, positive infinity, negative infinity).
 *
 * The two bits after the NUMERIC_SPECIAL bits are 00 for NaN, 01 for positive
 * infinity, 11 for negative infinity.  (This makes the sign bit match where
 * it is in a short-format value, though we make no use of that at present.)
 * We could mask off the remaining bits before testing the active bits, but
 * currently those bits must be zeroes, so masking would just add cycles.
 */
#define NUMERIC_EXT_SIGN_MASK   0xF000  /* high bits plus NaN/Inf flag bits */
#define NUMERIC_NAN             0xC000
#define NUMERIC_PINF            0xD000
#define NUMERIC_NINF            0xF000
#define NUMERIC_INF_SIGN_MASK   0x2000
 
#define NUMERIC_EXT_FLAGBITS(n) ((n)->choice.n_header & NUMERIC_EXT_SIGN_MASK)
#define NUMERIC_IS_NAN(n)       ((n)->choice.n_header == NUMERIC_NAN)
#define NUMERIC_IS_PINF(n)      ((n)->choice.n_header == NUMERIC_PINF)
#define NUMERIC_IS_NINF(n)      ((n)->choice.n_header == NUMERIC_NINF)
#define NUMERIC_IS_INF(n) \
    (((n)->choice.n_header & ~NUMERIC_INF_SIGN_MASK) == NUMERIC_PINF)
 
/*
 * Short format definitions.
 */
 
#define NUMERIC_SHORT_SIGN_MASK         0x2000
#define NUMERIC_SHORT_DSCALE_MASK       0x1F80
#define NUMERIC_SHORT_DSCALE_SHIFT      7
#define NUMERIC_SHORT_DSCALE_MAX        \
    (NUMERIC_SHORT_DSCALE_MASK >> NUMERIC_SHORT_DSCALE_SHIFT)
#define NUMERIC_SHORT_WEIGHT_SIGN_MASK  0x0040
#define NUMERIC_SHORT_WEIGHT_MASK       0x003F
#define NUMERIC_SHORT_WEIGHT_MAX        NUMERIC_SHORT_WEIGHT_MASK
#define NUMERIC_SHORT_WEIGHT_MIN        (-(NUMERIC_SHORT_WEIGHT_MASK+1))
 
/*
 * Extract sign, display scale, weight.  These macros extract field values
 * suitable for the NumericVar format from the Numeric (on-disk) format.
 *
 * Note that we don't trouble to ensure that dscale and weight read as zero
 * for an infinity; however, that doesn't matter since we never convert
 * "special" numerics to NumericVar form.  Only the constants defined below
 * (const_nan, etc) ever represent a non-finite value as a NumericVar.
 */
 
#define NUMERIC_DSCALE_MASK         0x3FFF
#define NUMERIC_DSCALE_MAX          NUMERIC_DSCALE_MASK
 
#define NUMERIC_SIGN(n) \
    (NUMERIC_IS_SHORT(n) ? \
        (((n)->choice.n_short.n_header & NUMERIC_SHORT_SIGN_MASK) ? \
         NUMERIC_NEG : NUMERIC_POS) : \
        (NUMERIC_IS_SPECIAL(n) ? \
         NUMERIC_EXT_FLAGBITS(n) : NUMERIC_FLAGBITS(n)))
#define NUMERIC_DSCALE(n)   (NUMERIC_HEADER_IS_SHORT((n)) ? \
    ((n)->choice.n_short.n_header & NUMERIC_SHORT_DSCALE_MASK) \
        >> NUMERIC_SHORT_DSCALE_SHIFT \
    : ((n)->choice.n_long.n_sign_dscale & NUMERIC_DSCALE_MASK))
#define NUMERIC_WEIGHT(n)   (NUMERIC_HEADER_IS_SHORT((n)) ? \
    (((n)->choice.n_short.n_header & NUMERIC_SHORT_WEIGHT_SIGN_MASK ? \
        ~NUMERIC_SHORT_WEIGHT_MASK : 0) \
     | ((n)->choice.n_short.n_header & NUMERIC_SHORT_WEIGHT_MASK)) \
    : ((n)->choice.n_long.n_weight))
 
/*
 * Maximum weight of a stored Numeric value (based on the use of int16 for the
 * weight in NumericLong).  Note that intermediate values held in NumericVar
 * and NumericSumAccum variables may have much larger weights.
 */
#define NUMERIC_WEIGHT_MAX          PG_INT16_MAX
 
/* ----------
 * NumericVar is the format we use for arithmetic.  The digit-array part
 * is the same as the NumericData storage format, but the header is more
 * complex.
 *
 * The value represented by a NumericVar is determined by the sign, weight,
 * ndigits, and digits[] array.  If it is a "special" value (NaN or Inf)
 * then only the sign field matters; ndigits should be zero, and the weight
 * and dscale fields are ignored.
 *
 * Note: the first digit of a NumericVar's value is assumed to be multiplied
 * by NBASE ** weight.  Another way to say it is that there are weight+1
 * digits before the decimal point.  It is possible to have weight < 0.
 *
 * buf points at the physical start of the palloc'd digit buffer for the
 * NumericVar.  digits points at the first digit in actual use (the one
 * with the specified weight).  We normally leave an unused digit or two
 * (preset to zeroes) between buf and digits, so that there is room to store
 * a carry out of the top digit without reallocating space.  We just need to
 * decrement digits (and increment weight) to make room for the carry digit.
 * (There is no such extra space in a numeric value stored in the database,
 * only in a NumericVar in memory.)
 *
 * If buf is NULL then the digit buffer isn't actually palloc'd and should
 * not be freed --- see the constants below for an example.
 *
 * dscale, or display scale, is the nominal precision expressed as number
 * of digits after the decimal point (it must always be >= 0 at present).
 * dscale may be more than the number of physically stored fractional digits,
 * implying that we have suppressed storage of significant trailing zeroes.
 * It should never be less than the number of stored digits, since that would
 * imply hiding digits that are present.  NOTE that dscale is always expressed
 * in *decimal* digits, and so it may correspond to a fractional number of
 * base-NBASE digits --- divide by DEC_DIGITS to convert to NBASE digits.
 *
 * rscale, or result scale, is the target precision for a computation.
 * Like dscale it is expressed as number of *decimal* digits after the decimal
 * point, and is always >= 0 at present.
 * Note that rscale is not stored in variables --- it's figured on-the-fly
 * from the dscales of the inputs.
 *
 * While we consistently use "weight" to refer to the base-NBASE weight of
 * a numeric value, it is convenient in some scale-related calculations to
 * make use of the base-10 weight (ie, the approximate log10 of the value).
 * To avoid confusion, such a decimal-units weight is called a "dweight".
 *
 * NB: All the variable-level functions are written in a style that makes it
 * possible to give one and the same variable as argument and destination.
 * This is feasible because the digit buffer is separate from the variable.
 * ----------
 */
typedef struct NumericVar
{
    int         ndigits;        /* # of digits in digits[] - can be 0! */
    int         weight;         /* weight of first digit */
    int         sign;           /* NUMERIC_POS, _NEG, _NAN, _PINF, or _NINF */
    int         dscale;         /* display scale */
    NumericDigit *buf;          /* start of palloc'd space for digits[] */
    NumericDigit *digits;       /* base-NBASE digits */
} NumericVar;
 
 
/* ----------
 * Data for generate_series
 * ----------
 */
typedef struct
{
    NumericVar  current;
    NumericVar  stop;
    NumericVar  step;
} generate_series_numeric_fctx;
 
 
/* ----------
 * Sort support.
 * ----------
 */
typedef struct
{
    void       *buf;            /* buffer for short varlenas */
    int64       input_count;    /* number of non-null values seen */
    bool        estimating;     /* true if estimating cardinality */
 
    hyperLogLogState abbr_card; /* cardinality estimator */
} NumericSortSupport;
 
 
/* ----------
 * Fast sum accumulator.
 *
 * NumericSumAccum is used to implement SUM(), and other standard aggregates
 * that track the sum of input values.  It uses 32-bit integers to store the
 * digits, instead of the normal 16-bit integers (with NBASE=10000).  This
 * way, we can safely accumulate up to NBASE - 1 values without propagating
 * carry, before risking overflow of any of the digits.  'num_uncarried'
 * tracks how many values have been accumulated without propagating carry.
 *
 * Positive and negative values are accumulated separately, in 'pos_digits'
 * and 'neg_digits'.  This is simpler and faster than deciding whether to add
 * or subtract from the current value, for each new value (see sub_var() for
 * the logic we avoid by doing this).  Both buffers are of same size, and
 * have the same weight and scale.  In accum_sum_final(), the positive and
 * negative sums are added together to produce the final result.
 *
 * When a new value has a larger ndigits or weight than the accumulator
 * currently does, the accumulator is enlarged to accommodate the new value.
 * We normally have one zero digit reserved for carry propagation, and that
 * is indicated by the 'have_carry_space' flag.  When accum_sum_carry() uses
 * up the reserved digit, it clears the 'have_carry_space' flag.  The next
 * call to accum_sum_add() will enlarge the buffer, to make room for the
 * extra digit, and set the flag again.
 *
 * To initialize a new accumulator, simply reset all fields to zeros.
 *
 * The accumulator does not handle NaNs.
 * ----------
 */
typedef struct NumericSumAccum
{
    int         ndigits;
    int         weight;
    int         dscale;
    int         num_uncarried;
    bool        have_carry_space;
    int32      *pos_digits;
    int32      *neg_digits;
} NumericSumAccum;
 
 
/*
 * We define our own macros for packing and unpacking abbreviated-key
 * representations for numeric values in order to avoid depending on
 * USE_FLOAT8_BYVAL.  The type of abbreviation we use is based only on
 * the size of a datum, not the argument-passing convention for float8.
 *
 * The range of abbreviations for finite values is from +PG_INT64/32_MAX
 * to -PG_INT64/32_MAX.  NaN has the abbreviation PG_INT64/32_MIN, and we
 * define the sort ordering to make that work out properly (see further
 * comments below).  PINF and NINF share the abbreviations of the largest
 * and smallest finite abbreviation classes.
 */
#define NUMERIC_ABBREV_BITS (SIZEOF_DATUM * BITS_PER_BYTE)
#if SIZEOF_DATUM == 8
#define NumericAbbrevGetDatum(X) ((Datum) (X))
#define DatumGetNumericAbbrev(X) ((int64) (X))
#define NUMERIC_ABBREV_NAN       NumericAbbrevGetDatum(PG_INT64_MIN)
#define NUMERIC_ABBREV_PINF      NumericAbbrevGetDatum(-PG_INT64_MAX)
#define NUMERIC_ABBREV_NINF      NumericAbbrevGetDatum(PG_INT64_MAX)
#else
#define NumericAbbrevGetDatum(X) ((Datum) (X))
#define DatumGetNumericAbbrev(X) ((int32) (X))
#define NUMERIC_ABBREV_NAN       NumericAbbrevGetDatum(PG_INT32_MIN)
#define NUMERIC_ABBREV_PINF      NumericAbbrevGetDatum(-PG_INT32_MAX)
#define NUMERIC_ABBREV_NINF      NumericAbbrevGetDatum(PG_INT32_MAX)
#endif
 
 
/* ----------
 * Some preinitialized constants
 * ----------
 */
static const NumericDigit const_zero_data[1] = {0};
static const NumericVar const_zero =
{0, 0, NUMERIC_POS, 0, NULL, (NumericDigit *) const_zero_data};
 
static const NumericDigit const_one_data[1] = {1};
static const NumericVar const_one =
{1, 0, NUMERIC_POS, 0, NULL, (NumericDigit *) const_one_data};
 
static const NumericVar const_minus_one =
{1, 0, NUMERIC_NEG, 0, NULL, (NumericDigit *) const_one_data};
 
static const NumericDigit const_two_data[1] = {2};
static const NumericVar const_two =
{1, 0, NUMERIC_POS, 0, NULL, (NumericDigit *) const_two_data};
 
#if DEC_DIGITS == 4
static const NumericDigit const_zero_point_nine_data[1] = {9000};
#elif DEC_DIGITS == 2
static const NumericDigit const_zero_point_nine_data[1] = {90};
#elif DEC_DIGITS == 1
static const NumericDigit const_zero_point_nine_data[1] = {9};
#endif
static const NumericVar const_zero_point_nine =
{1, -1, NUMERIC_POS, 1, NULL, (NumericDigit *) const_zero_point_nine_data};
 
#if DEC_DIGITS == 4
static const NumericDigit const_one_point_one_data[2] = {1, 1000};
#elif DEC_DIGITS == 2
static const NumericDigit const_one_point_one_data[2] = {1, 10};
#elif DEC_DIGITS == 1
static const NumericDigit const_one_point_one_data[2] = {1, 1};
#endif
static const NumericVar const_one_point_one =
{2, 0, NUMERIC_POS, 1, NULL, (NumericDigit *) const_one_point_one_data};
 
static const NumericVar const_nan =
{0, 0, NUMERIC_NAN, 0, NULL, NULL};
 
static const NumericVar const_pinf =
{0, 0, NUMERIC_PINF, 0, NULL, NULL};
 
static const NumericVar const_ninf =
{0, 0, NUMERIC_NINF, 0, NULL, NULL};
 
#if DEC_DIGITS == 4
static const int round_powers[4] = {0, 1000, 100, 10};
#endif
 
 
/* ----------
 * Local functions
 * ----------
 */
 
#ifdef NUMERIC_DEBUG
static void dump_numeric(const char *str, Numeric num);
static void dump_var(const char *str, NumericVar *var);
#else
#define dump_numeric(s,n)
#define dump_var(s,v)
#endif
 
#define digitbuf_alloc(ndigits)  \
    ((NumericDigit *) palloc((ndigits) * sizeof(NumericDigit)))
#define digitbuf_free(buf)  \
    do { \
         if ((buf) != NULL) \
             pfree(buf); \
    } while (0)
 
#define init_var(v)     memset(v, 0, sizeof(NumericVar))
 
#define NUMERIC_DIGITS(num) (NUMERIC_HEADER_IS_SHORT(num) ? \
    (num)->choice.n_short.n_data : (num)->choice.n_long.n_data)
#define NUMERIC_NDIGITS(num) \
    ((VARSIZE(num) - NUMERIC_HEADER_SIZE(num)) / sizeof(NumericDigit))
#define NUMERIC_CAN_BE_SHORT(scale,weight) \
    ((scale) <= NUMERIC_SHORT_DSCALE_MAX && \
    (weight) <= NUMERIC_SHORT_WEIGHT_MAX && \
    (weight) >= NUMERIC_SHORT_WEIGHT_MIN)
 
static void alloc_var(NumericVar *var, int ndigits);
static void free_var(NumericVar *var);
static void zero_var(NumericVar *var);
 
static bool set_var_from_str(const char *str, const char *cp,
                             NumericVar *dest, const char **endptr,
                             Node *escontext);
static bool set_var_from_non_decimal_integer_str(const char *str,
                                                 const char *cp, int sign,
                                                 int base, NumericVar *dest,
                                                 const char **endptr,
                                                 Node *escontext);
static void set_var_from_num(Numeric num, NumericVar *dest);
static void init_var_from_num(Numeric num, NumericVar *dest);
static void set_var_from_var(const NumericVar *value, NumericVar *dest);
static char *get_str_from_var(const NumericVar *var);
static char *get_str_from_var_sci(const NumericVar *var, int rscale);
 
static void numericvar_serialize(StringInfo buf, const NumericVar *var);
static void numericvar_deserialize(StringInfo buf, NumericVar *var);
 
static Numeric duplicate_numeric(Numeric num);
static Numeric make_result(const NumericVar *var);
static Numeric make_result_opt_error(const NumericVar *var, bool *have_error);
 
static bool apply_typmod(NumericVar *var, int32 typmod, Node *escontext);
static bool apply_typmod_special(Numeric num, int32 typmod, Node *escontext);
 
static bool numericvar_to_int32(const NumericVar *var, int32 *result);
static bool numericvar_to_int64(const NumericVar *var, int64 *result);
static void int64_to_numericvar(int64 val, NumericVar *var);
static bool numericvar_to_uint64(const NumericVar *var, uint64 *result);
#ifdef HAVE_INT128
static bool numericvar_to_int128(const NumericVar *var, int128 *result);
static void int128_to_numericvar(int128 val, NumericVar *var);
#endif
static double numericvar_to_double_no_overflow(const NumericVar *var);
 
static Datum numeric_abbrev_convert(Datum original_datum, SortSupport ssup);
static bool numeric_abbrev_abort(int memtupcount, SortSupport ssup);
static int  numeric_fast_cmp(Datum x, Datum y, SortSupport ssup);
static int  numeric_cmp_abbrev(Datum x, Datum y, SortSupport ssup);
 
static Datum numeric_abbrev_convert_var(const NumericVar *var,
                                        NumericSortSupport *nss);
 
static int  cmp_numerics(Numeric num1, Numeric num2);
static int  cmp_var(const NumericVar *var1, const NumericVar *var2);
static int  cmp_var_common(const NumericDigit *var1digits, int var1ndigits,
                           int var1weight, int var1sign,
                           const NumericDigit *var2digits, int var2ndigits,
                           int var2weight, int var2sign);
static void add_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void sub_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void mul_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result,
                    int rscale);
static void mul_var_short(const NumericVar *var1, const NumericVar *var2,
                          NumericVar *result);
static void div_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result, int rscale, bool round, bool exact);
static void div_var_int(const NumericVar *var, int ival, int ival_weight,
                        NumericVar *result, int rscale, bool round);
#ifdef HAVE_INT128
static void div_var_int64(const NumericVar *var, int64 ival, int ival_weight,
                          NumericVar *result, int rscale, bool round);
#endif
static int  select_div_scale(const NumericVar *var1, const NumericVar *var2);
static void mod_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void div_mod_var(const NumericVar *var1, const NumericVar *var2,
                        NumericVar *quot, NumericVar *rem);
static void ceil_var(const NumericVar *var, NumericVar *result);
static void floor_var(const NumericVar *var, NumericVar *result);
 
static void gcd_var(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void sqrt_var(const NumericVar *arg, NumericVar *result, int rscale);
static void exp_var(const NumericVar *arg, NumericVar *result, int rscale);
static int  estimate_ln_dweight(const NumericVar *var);
static void ln_var(const NumericVar *arg, NumericVar *result, int rscale);
static void log_var(const NumericVar *base, const NumericVar *num,
                    NumericVar *result);
static void power_var(const NumericVar *base, const NumericVar *exp,
                      NumericVar *result);
static void power_var_int(const NumericVar *base, int exp, int exp_dscale,
                          NumericVar *result);
static void power_ten_int(int exp, NumericVar *result);
static void random_var(pg_prng_state *state, const NumericVar *rmin,
                       const NumericVar *rmax, NumericVar *result);
 
static int  cmp_abs(const NumericVar *var1, const NumericVar *var2);
static int  cmp_abs_common(const NumericDigit *var1digits, int var1ndigits,
                           int var1weight,
                           const NumericDigit *var2digits, int var2ndigits,
                           int var2weight);
static void add_abs(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void sub_abs(const NumericVar *var1, const NumericVar *var2,
                    NumericVar *result);
static void round_var(NumericVar *var, int rscale);
static void trunc_var(NumericVar *var, int rscale);
static void strip_var(NumericVar *var);
static void compute_bucket(Numeric operand, Numeric bound1, Numeric bound2,
                           const NumericVar *count_var,
                           NumericVar *result_var);
 
static void accum_sum_add(NumericSumAccum *accum, const NumericVar *val);
static void accum_sum_rescale(NumericSumAccum *accum, const NumericVar *val);
static void accum_sum_carry(NumericSumAccum *accum);
static void accum_sum_reset(NumericSumAccum *accum);
static void accum_sum_final(NumericSumAccum *accum, NumericVar *result);
static void accum_sum_copy(NumericSumAccum *dst, NumericSumAccum *src);
static void accum_sum_combine(NumericSumAccum *accum, NumericSumAccum *accum2);
 
 
/* ----------------------------------------------------------------------
 *
 * Input-, output- and rounding-functions
 *
 * ----------------------------------------------------------------------
 */
 
 
/*
 * numeric_in() -
 *
 *  Input function for numeric data type
 */
Datum
numeric_in(PG_FUNCTION_ARGS)
{
    char       *str = PG_GETARG_CSTRING(0);
#ifdef NOT_USED
    Oid         typelem = PG_GETARG_OID(1);
#endif
    int32       typmod = PG_GETARG_INT32(2);
    Node       *escontext = fcinfo->context;
    Numeric     res;
    const char *cp;
    const char *numstart;
    int         sign;
 
    /* Skip leading spaces */
    cp = str;
    while (*cp)
    {
        if (!isspace((unsigned char) *cp))
            break;
        cp++;
    }
 
    /*
     * Process the number's sign. This duplicates logic in set_var_from_str(),
     * but it's worth doing here, since it simplifies the handling of
     * infinities and non-decimal integers.
     */
    numstart = cp;
    sign = NUMERIC_POS;
 
    if (*cp == '+')
        cp++;
    else if (*cp == '-')
    {
        sign = NUMERIC_NEG;
        cp++;
    }
 
    /*
     * Check for NaN and infinities.  We recognize the same strings allowed by
     * float8in().
     *
     * Since all other legal inputs have a digit or a decimal point after the
     * sign, we need only check for NaN/infinity if that's not the case.
     */
    if (!isdigit((unsigned char) *cp) && *cp != '.')
    {
        /*
         * The number must be NaN or infinity; anything else can only be a
         * syntax error. Note that NaN mustn't have a sign.
         */
        if (pg_strncasecmp(numstart, "NaN", 3) == 0)
        {
            res = make_result(&const_nan);
            cp = numstart + 3;
        }
        else if (pg_strncasecmp(cp, "Infinity", 8) == 0)
        {
            res = make_result(sign == NUMERIC_POS ? &const_pinf : &const_ninf);
            cp += 8;
        }
        else if (pg_strncasecmp(cp, "inf", 3) == 0)
        {
            res = make_result(sign == NUMERIC_POS ? &const_pinf : &const_ninf);
            cp += 3;
        }
        else
            goto invalid_syntax;
 
        /*
         * Check for trailing junk; there should be nothing left but spaces.
         *
         * We intentionally do this check before applying the typmod because
         * we would like to throw any trailing-junk syntax error before any
         * semantic error resulting from apply_typmod_special().
         */
        while (*cp)
        {
            if (!isspace((unsigned char) *cp))
                goto invalid_syntax;
            cp++;
        }
 
        if (!apply_typmod_special(res, typmod, escontext))
            PG_RETURN_NULL();
    }
    else
    {
        /*
         * We have a normal numeric value, which may be a non-decimal integer
         * or a regular decimal number.
         */
        NumericVar  value;
        int         base;
        bool        have_error;
 
        init_var(&value);
 
        /*
         * Determine the number's base by looking for a non-decimal prefix
         * indicator ("0x", "0o", or "0b").
         */
        if (cp[0] == '0')
        {
            switch (cp[1])
            {
                case 'x':
                case 'X':
                    base = 16;
                    break;
                case 'o':
                case 'O':
                    base = 8;
                    break;
                case 'b':
                case 'B':
                    base = 2;
                    break;
                default:
                    base = 10;
            }
        }
        else
            base = 10;
 
        /* Parse the rest of the number and apply the sign */
        if (base == 10)
        {
            if (!set_var_from_str(str, cp, &value, &cp, escontext))
                PG_RETURN_NULL();
            value.sign = sign;
        }
        else
        {
            if (!set_var_from_non_decimal_integer_str(str, cp + 2, sign, base,
                                                      &value, &cp, escontext))
                PG_RETURN_NULL();
        }
 
        /*
         * Should be nothing left but spaces. As above, throw any typmod error
         * after finishing syntax check.
         */
        while (*cp)
        {
            if (!isspace((unsigned char) *cp))
                goto invalid_syntax;
            cp++;
        }
 
        if (!apply_typmod(&value, typmod, escontext))
            PG_RETURN_NULL();
 
        res = make_result_opt_error(&value, &have_error);
 
        if (have_error)
            ereturn(escontext, (Datum) 0,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("value overflows numeric format")));
 
        free_var(&value);
    }
 
    PG_RETURN_NUMERIC(res);
 
invalid_syntax:
    ereturn(escontext, (Datum) 0,
            (errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
             errmsg("invalid input syntax for type %s: \"%s\"",
                    "numeric", str)));
}
 
 
/*
 * numeric_out() -
 *
 *  Output function for numeric data type
 */
Datum
numeric_out(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  x;
    char       *str;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            PG_RETURN_CSTRING(pstrdup("Infinity"));
        else if (NUMERIC_IS_NINF(num))
            PG_RETURN_CSTRING(pstrdup("-Infinity"));
        else
            PG_RETURN_CSTRING(pstrdup("NaN"));
    }
 
    /*
     * Get the number in the variable format.
     */
    init_var_from_num(num, &x);
 
    str = get_str_from_var(&x);
 
    PG_RETURN_CSTRING(str);
}
 
/*
 * numeric_is_nan() -
 *
 *  Is Numeric value a NaN?
 */
bool
numeric_is_nan(Numeric num)
{
    return NUMERIC_IS_NAN(num);
}
 
/*
 * numeric_is_inf() -
 *
 *  Is Numeric value an infinity?
 */
bool
numeric_is_inf(Numeric num)
{
    return NUMERIC_IS_INF(num);
}
 
/*
 * numeric_is_integral() -
 *
 *  Is Numeric value integral?
 */
static bool
numeric_is_integral(Numeric num)
{
    NumericVar  arg;
 
    /* Reject NaN, but infinities are considered integral */
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_NAN(num))
            return false;
        return true;
    }
 
    /* Integral if there are no digits to the right of the decimal point */
    init_var_from_num(num, &arg);
 
    return (arg.ndigits == 0 || arg.ndigits <= arg.weight + 1);
}
 
/*
 * make_numeric_typmod() -
 *
 *  Pack numeric precision and scale values into a typmod.  The upper 16 bits
 *  are used for the precision (though actually not all these bits are needed,
 *  since the maximum allowed precision is 1000).  The lower 16 bits are for
 *  the scale, but since the scale is constrained to the range [-1000, 1000],
 *  we use just the lower 11 of those 16 bits, and leave the remaining 5 bits
 *  unset, for possible future use.
 *
 *  For purely historical reasons VARHDRSZ is then added to the result, thus
 *  the unused space in the upper 16 bits is not all as freely available as it
 *  might seem.  (We can't let the result overflow to a negative int32, as
 *  other parts of the system would interpret that as not-a-valid-typmod.)
 */
static inline int32
make_numeric_typmod(int precision, int scale)
{
    return ((precision << 16) | (scale & 0x7ff)) + VARHDRSZ;
}
 
/*
 * Because of the offset, valid numeric typmods are at least VARHDRSZ
 */
static inline bool
is_valid_numeric_typmod(int32 typmod)
{
    return typmod >= (int32) VARHDRSZ;
}
 
/*
 * numeric_typmod_precision() -
 *
 *  Extract the precision from a numeric typmod --- see make_numeric_typmod().
 */
static inline int
numeric_typmod_precision(int32 typmod)
{
    return ((typmod - VARHDRSZ) >> 16) & 0xffff;
}
 
/*
 * numeric_typmod_scale() -
 *
 *  Extract the scale from a numeric typmod --- see make_numeric_typmod().
 *
 *  Note that the scale may be negative, so we must do sign extension when
 *  unpacking it.  We do this using the bit hack (x^1024)-1024, which sign
 *  extends an 11-bit two's complement number x.
 */
static inline int
numeric_typmod_scale(int32 typmod)
{
    return (((typmod - VARHDRSZ) & 0x7ff) ^ 1024) - 1024;
}
 
/*
 * numeric_maximum_size() -
 *
 *  Maximum size of a numeric with given typmod, or -1 if unlimited/unknown.
 */
int32
numeric_maximum_size(int32 typmod)
{
    int         precision;
    int         numeric_digits;
 
    if (!is_valid_numeric_typmod(typmod))
        return -1;
 
    /* precision (ie, max # of digits) is in upper bits of typmod */
    precision = numeric_typmod_precision(typmod);
 
    /*
     * This formula computes the maximum number of NumericDigits we could need
     * in order to store the specified number of decimal digits. Because the
     * weight is stored as a number of NumericDigits rather than a number of
     * decimal digits, it's possible that the first NumericDigit will contain
     * only a single decimal digit.  Thus, the first two decimal digits can
     * require two NumericDigits to store, but it isn't until we reach
     * DEC_DIGITS + 2 decimal digits that we potentially need a third
     * NumericDigit.
     */
    numeric_digits = (precision + 2 * (DEC_DIGITS - 1)) / DEC_DIGITS;
 
    /*
     * In most cases, the size of a numeric will be smaller than the value
     * computed below, because the varlena header will typically get toasted
     * down to a single byte before being stored on disk, and it may also be
     * possible to use a short numeric header.  But our job here is to compute
     * the worst case.
     */
    return NUMERIC_HDRSZ + (numeric_digits * sizeof(NumericDigit));
}
 
/*
 * numeric_out_sci() -
 *
 *  Output function for numeric data type in scientific notation.
 */
char *
numeric_out_sci(Numeric num, int scale)
{
    NumericVar  x;
    char       *str;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            return pstrdup("Infinity");
        else if (NUMERIC_IS_NINF(num))
            return pstrdup("-Infinity");
        else
            return pstrdup("NaN");
    }
 
    init_var_from_num(num, &x);
 
    str = get_str_from_var_sci(&x, scale);
 
    return str;
}
 
/*
 * numeric_normalize() -
 *
 *  Output function for numeric data type, suppressing insignificant trailing
 *  zeroes and then any trailing decimal point.  The intent of this is to
 *  produce strings that are equal if and only if the input numeric values
 *  compare equal.
 */
char *
numeric_normalize(Numeric num)
{
    NumericVar  x;
    char       *str;
    int         last;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            return pstrdup("Infinity");
        else if (NUMERIC_IS_NINF(num))
            return pstrdup("-Infinity");
        else
            return pstrdup("NaN");
    }
 
    init_var_from_num(num, &x);
 
    str = get_str_from_var(&x);
 
    /* If there's no decimal point, there's certainly nothing to remove. */
    if (strchr(str, '.') != NULL)
    {
        /*
         * Back up over trailing fractional zeroes.  Since there is a decimal
         * point, this loop will terminate safely.
         */
        last = strlen(str) - 1;
        while (str[last] == '0')
            last--;
 
        /* We want to get rid of the decimal point too, if it's now last. */
        if (str[last] == '.')
            last--;
 
        /* Delete whatever we backed up over. */
        str[last + 1] = '\0';
    }
 
    return str;
}
 
/*
 *      numeric_recv            - converts external binary format to numeric
 *
 * External format is a sequence of int16's:
 * ndigits, weight, sign, dscale, NumericDigits.
 */
Datum
numeric_recv(PG_FUNCTION_ARGS)
{
    StringInfo  buf = (StringInfo) PG_GETARG_POINTER(0);
 
#ifdef NOT_USED
    Oid         typelem = PG_GETARG_OID(1);
#endif
    int32       typmod = PG_GETARG_INT32(2);
    NumericVar  value;
    Numeric     res;
    int         len,
                i;
 
    init_var(&value);
 
    len = (uint16) pq_getmsgint(buf, sizeof(uint16));
 
    alloc_var(&value, len);
 
    value.weight = (int16) pq_getmsgint(buf, sizeof(int16));
    /* we allow any int16 for weight --- OK? */
 
    value.sign = (uint16) pq_getmsgint(buf, sizeof(uint16));
    if (!(value.sign == NUMERIC_POS ||
          value.sign == NUMERIC_NEG ||
          value.sign == NUMERIC_NAN ||
          value.sign == NUMERIC_PINF ||
          value.sign == NUMERIC_NINF))
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
                 errmsg("invalid sign in external \"numeric\" value")));
 
    value.dscale = (uint16) pq_getmsgint(buf, sizeof(uint16));
    if ((value.dscale & NUMERIC_DSCALE_MASK) != value.dscale)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
                 errmsg("invalid scale in external \"numeric\" value")));
 
    for (i = 0; i < len; i++)
    {
        NumericDigit d = pq_getmsgint(buf, sizeof(NumericDigit));
 
        if (d < 0 || d >= NBASE)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
                     errmsg("invalid digit in external \"numeric\" value")));
        value.digits[i] = d;
    }
 
    /*
     * If the given dscale would hide any digits, truncate those digits away.
     * We could alternatively throw an error, but that would take a bunch of
     * extra code (about as much as trunc_var involves), and it might cause
     * client compatibility issues.  Be careful not to apply trunc_var to
     * special values, as it could do the wrong thing; we don't need it
     * anyway, since make_result will ignore all but the sign field.
     *
     * After doing that, be sure to check the typmod restriction.
     */
    if (value.sign == NUMERIC_POS ||
        value.sign == NUMERIC_NEG)
    {
        trunc_var(&value, value.dscale);
 
        (void) apply_typmod(&value, typmod, NULL);
 
        res = make_result(&value);
    }
    else
    {
        /* apply_typmod_special wants us to make the Numeric first */
        res = make_result(&value);
 
        (void) apply_typmod_special(res, typmod, NULL);
    }
 
    free_var(&value);
 
    PG_RETURN_NUMERIC(res);
}
 
/*
 *      numeric_send            - converts numeric to binary format
 */
Datum
numeric_send(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  x;
    StringInfoData buf;
    int         i;
 
    init_var_from_num(num, &x);
 
    pq_begintypsend(&buf);
 
    pq_sendint16(&buf, x.ndigits);
    pq_sendint16(&buf, x.weight);
    pq_sendint16(&buf, x.sign);
    pq_sendint16(&buf, x.dscale);
    for (i = 0; i < x.ndigits; i++)
        pq_sendint16(&buf, x.digits[i]);
 
    PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}
 
 
/*
 * numeric_support()
 *
 * Planner support function for the numeric() length coercion function.
 *
 * Flatten calls that solely represent increases in allowable precision.
 * Scale changes mutate every datum, so they are unoptimizable.  Some values,
 * e.g. 1E-1001, can only fit into an unconstrained numeric, so a change from
 * an unconstrained numeric to any constrained numeric is also unoptimizable.
 */
Datum
numeric_support(PG_FUNCTION_ARGS)
{
    Node       *rawreq = (Node *) PG_GETARG_POINTER(0);
    Node       *ret = NULL;
 
    if (IsA(rawreq, SupportRequestSimplify))
    {
        SupportRequestSimplify *req = (SupportRequestSimplify *) rawreq;
        FuncExpr   *expr = req->fcall;
        Node       *typmod;
 
        Assert(list_length(expr->args) >= 2);
 
        typmod = (Node *) lsecond(expr->args);
 
        if (IsA(typmod, Const) && !((Const *) typmod)->constisnull)
        {
            Node       *source = (Node *) linitial(expr->args);
            int32       old_typmod = exprTypmod(source);
            int32       new_typmod = DatumGetInt32(((Const *) typmod)->constvalue);
            int32       old_scale = numeric_typmod_scale(old_typmod);
            int32       new_scale = numeric_typmod_scale(new_typmod);
            int32       old_precision = numeric_typmod_precision(old_typmod);
            int32       new_precision = numeric_typmod_precision(new_typmod);
 
            /*
             * If new_typmod is invalid, the destination is unconstrained;
             * that's always OK.  If old_typmod is valid, the source is
             * constrained, and we're OK if the scale is unchanged and the
             * precision is not decreasing.  See further notes in function
             * header comment.
             */
            if (!is_valid_numeric_typmod(new_typmod) ||
                (is_valid_numeric_typmod(old_typmod) &&
                 new_scale == old_scale && new_precision >= old_precision))
                ret = relabel_to_typmod(source, new_typmod);
        }
    }
 
    PG_RETURN_POINTER(ret);
}
 
/*
 * numeric() -
 *
 *  This is a special function called by the Postgres database system
 *  before a value is stored in a tuple's attribute. The precision and
 *  scale of the attribute have to be applied on the value.
 */
Datum
numeric     (PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    int32       typmod = PG_GETARG_INT32(1);
    Numeric     new;
    int         precision;
    int         scale;
    int         ddigits;
    int         maxdigits;
    int         dscale;
    NumericVar  var;
 
    /*
     * Handle NaN and infinities: if apply_typmod_special doesn't complain,
     * just return a copy of the input.
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        (void) apply_typmod_special(num, typmod, NULL);
        PG_RETURN_NUMERIC(duplicate_numeric(num));
    }
 
    /*
     * If the value isn't a valid type modifier, simply return a copy of the
     * input value
     */
    if (!is_valid_numeric_typmod(typmod))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    /*
     * Get the precision and scale out of the typmod value
     */
    precision = numeric_typmod_precision(typmod);
    scale = numeric_typmod_scale(typmod);
    maxdigits = precision - scale;
 
    /* The target display scale is non-negative */
    dscale = Max(scale, 0);
 
    /*
     * If the number is certainly in bounds and due to the target scale no
     * rounding could be necessary, just make a copy of the input and modify
     * its scale fields, unless the larger scale forces us to abandon the
     * short representation.  (Note we assume the existing dscale is
     * honest...)
     */
    ddigits = (NUMERIC_WEIGHT(num) + 1) * DEC_DIGITS;
    if (ddigits <= maxdigits && scale >= NUMERIC_DSCALE(num)
        && (NUMERIC_CAN_BE_SHORT(dscale, NUMERIC_WEIGHT(num))
            || !NUMERIC_IS_SHORT(num)))
    {
        new = duplicate_numeric(num);
        if (NUMERIC_IS_SHORT(num))
            new->choice.n_short.n_header =
                (num->choice.n_short.n_header & ~NUMERIC_SHORT_DSCALE_MASK)
                | (dscale << NUMERIC_SHORT_DSCALE_SHIFT);
        else
            new->choice.n_long.n_sign_dscale = NUMERIC_SIGN(new) |
                ((uint16) dscale & NUMERIC_DSCALE_MASK);
        PG_RETURN_NUMERIC(new);
    }
 
    /*
     * We really need to fiddle with things - unpack the number into a
     * variable and let apply_typmod() do it.
     */
    init_var(&var);
 
    set_var_from_num(num, &var);
    (void) apply_typmod(&var, typmod, NULL);
    new = make_result(&var);
 
    free_var(&var);
 
    PG_RETURN_NUMERIC(new);
}
 
Datum
numerictypmodin(PG_FUNCTION_ARGS)
{
    ArrayType  *ta = PG_GETARG_ARRAYTYPE_P(0);
    int32      *tl;
    int         n;
    int32       typmod;
 
    tl = ArrayGetIntegerTypmods(ta, &n);
 
    if (n == 2)
    {
        if (tl[0] < 1 || tl[0] > NUMERIC_MAX_PRECISION)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                     errmsg("NUMERIC precision %d must be between 1 and %d",
                            tl[0], NUMERIC_MAX_PRECISION)));
        if (tl[1] < NUMERIC_MIN_SCALE || tl[1] > NUMERIC_MAX_SCALE)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                     errmsg("NUMERIC scale %d must be between %d and %d",
                            tl[1], NUMERIC_MIN_SCALE, NUMERIC_MAX_SCALE)));
        typmod = make_numeric_typmod(tl[0], tl[1]);
    }
    else if (n == 1)
    {
        if (tl[0] < 1 || tl[0] > NUMERIC_MAX_PRECISION)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                     errmsg("NUMERIC precision %d must be between 1 and %d",
                            tl[0], NUMERIC_MAX_PRECISION)));
        /* scale defaults to zero */
        typmod = make_numeric_typmod(tl[0], 0);
    }
    else
    {
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                 errmsg("invalid NUMERIC type modifier")));
        typmod = 0;             /* keep compiler quiet */
    }
 
    PG_RETURN_INT32(typmod);
}
 
Datum
numerictypmodout(PG_FUNCTION_ARGS)
{
    int32       typmod = PG_GETARG_INT32(0);
    char       *res = (char *) palloc(64);
 
    if (is_valid_numeric_typmod(typmod))
        snprintf(res, 64, "(%d,%d)",
                 numeric_typmod_precision(typmod),
                 numeric_typmod_scale(typmod));
    else
        *res = '\0';
 
    PG_RETURN_CSTRING(res);
}
 
 
/* ----------------------------------------------------------------------
 *
 * Sign manipulation, rounding and the like
 *
 * ----------------------------------------------------------------------
 */
 
Datum
numeric_abs(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
 
    /*
     * Do it the easy way directly on the packed format
     */
    res = duplicate_numeric(num);
 
    if (NUMERIC_IS_SHORT(num))
        res->choice.n_short.n_header =
            num->choice.n_short.n_header & ~NUMERIC_SHORT_SIGN_MASK;
    else if (NUMERIC_IS_SPECIAL(num))
    {
        /* This changes -Inf to Inf, and doesn't affect NaN */
        res->choice.n_short.n_header =
            num->choice.n_short.n_header & ~NUMERIC_INF_SIGN_MASK;
    }
    else
        res->choice.n_long.n_sign_dscale = NUMERIC_POS | NUMERIC_DSCALE(num);
 
    PG_RETURN_NUMERIC(res);
}
 
 
Datum
numeric_uminus(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
 
    /*
     * Do it the easy way directly on the packed format
     */
    res = duplicate_numeric(num);
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        /* Flip the sign, if it's Inf or -Inf */
        if (!NUMERIC_IS_NAN(num))
            res->choice.n_short.n_header =
                num->choice.n_short.n_header ^ NUMERIC_INF_SIGN_MASK;
    }
 
    /*
     * The packed format is known to be totally zero digit trimmed always. So
     * once we've eliminated specials, we can identify a zero by the fact that
     * there are no digits at all. Do nothing to a zero.
     */
    else if (NUMERIC_NDIGITS(num) != 0)
    {
        /* Else, flip the sign */
        if (NUMERIC_IS_SHORT(num))
            res->choice.n_short.n_header =
                num->choice.n_short.n_header ^ NUMERIC_SHORT_SIGN_MASK;
        else if (NUMERIC_SIGN(num) == NUMERIC_POS)
            res->choice.n_long.n_sign_dscale =
                NUMERIC_NEG | NUMERIC_DSCALE(num);
        else
            res->choice.n_long.n_sign_dscale =
                NUMERIC_POS | NUMERIC_DSCALE(num);
    }
 
    PG_RETURN_NUMERIC(res);
}
 
 
Datum
numeric_uplus(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
 
    PG_RETURN_NUMERIC(duplicate_numeric(num));
}
 
 
/*
 * numeric_sign_internal() -
 *
 * Returns -1 if the argument is less than 0, 0 if the argument is equal
 * to 0, and 1 if the argument is greater than zero.  Caller must have
 * taken care of the NaN case, but we can handle infinities here.
 */
static int
numeric_sign_internal(Numeric num)
{
    if (NUMERIC_IS_SPECIAL(num))
    {
        Assert(!NUMERIC_IS_NAN(num));
        /* Must be Inf or -Inf */
        if (NUMERIC_IS_PINF(num))
            return 1;
        else
            return -1;
    }
 
    /*
     * The packed format is known to be totally zero digit trimmed always. So
     * once we've eliminated specials, we can identify a zero by the fact that
     * there are no digits at all.
     */
    else if (NUMERIC_NDIGITS(num) == 0)
        return 0;
    else if (NUMERIC_SIGN(num) == NUMERIC_NEG)
        return -1;
    else
        return 1;
}
 
/*
 * numeric_sign() -
 *
 * returns -1 if the argument is less than 0, 0 if the argument is equal
 * to 0, and 1 if the argument is greater than zero.
 */
Datum
numeric_sign(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
 
    /*
     * Handle NaN (infinities can be handled normally)
     */
    if (NUMERIC_IS_NAN(num))
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    switch (numeric_sign_internal(num))
    {
        case 0:
            PG_RETURN_NUMERIC(make_result(&const_zero));
        case 1:
            PG_RETURN_NUMERIC(make_result(&const_one));
        case -1:
            PG_RETURN_NUMERIC(make_result(&const_minus_one));
    }
 
    Assert(false);
    return (Datum) 0;
}
 
 
/*
 * numeric_round() -
 *
 *  Round a value to have 'scale' digits after the decimal point.
 *  We allow negative 'scale', implying rounding before the decimal
 *  point --- Oracle interprets rounding that way.
 */
Datum
numeric_round(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    int32       scale = PG_GETARG_INT32(1);
    Numeric     res;
    NumericVar  arg;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    /*
     * Limit the scale value to avoid possible overflow in calculations.
     *
     * These limits are based on the maximum number of digits a Numeric value
     * can have before and after the decimal point, but we must allow for one
     * extra digit before the decimal point, in case the most significant
     * digit rounds up; we must check if that causes Numeric overflow.
     */
    scale = Max(scale, -(NUMERIC_WEIGHT_MAX + 1) * DEC_DIGITS - 1);
    scale = Min(scale, NUMERIC_DSCALE_MAX);
 
    /*
     * Unpack the argument and round it at the proper digit position
     */
    init_var(&arg);
    set_var_from_num(num, &arg);
 
    round_var(&arg, scale);
 
    /* We don't allow negative output dscale */
    if (scale < 0)
        arg.dscale = 0;
 
    /*
     * Return the rounded result
     */
    res = make_result(&arg);
 
    free_var(&arg);
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_trunc() -
 *
 *  Truncate a value to have 'scale' digits after the decimal point.
 *  We allow negative 'scale', implying a truncation before the decimal
 *  point --- Oracle interprets truncation that way.
 */
Datum
numeric_trunc(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    int32       scale = PG_GETARG_INT32(1);
    Numeric     res;
    NumericVar  arg;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    /*
     * Limit the scale value to avoid possible overflow in calculations.
     *
     * These limits are based on the maximum number of digits a Numeric value
     * can have before and after the decimal point.
     */
    scale = Max(scale, -(NUMERIC_WEIGHT_MAX + 1) * DEC_DIGITS);
    scale = Min(scale, NUMERIC_DSCALE_MAX);
 
    /*
     * Unpack the argument and truncate it at the proper digit position
     */
    init_var(&arg);
    set_var_from_num(num, &arg);
 
    trunc_var(&arg, scale);
 
    /* We don't allow negative output dscale */
    if (scale < 0)
        arg.dscale = 0;
 
    /*
     * Return the truncated result
     */
    res = make_result(&arg);
 
    free_var(&arg);
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_ceil() -
 *
 *  Return the smallest integer greater than or equal to the argument
 */
Datum
numeric_ceil(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  result;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    init_var_from_num(num, &result);
    ceil_var(&result, &result);
 
    res = make_result(&result);
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_floor() -
 *
 *  Return the largest integer equal to or less than the argument
 */
Datum
numeric_floor(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  result;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    init_var_from_num(num, &result);
    floor_var(&result, &result);
 
    res = make_result(&result);
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * generate_series_numeric() -
 *
 *  Generate series of numeric.
 */
Datum
generate_series_numeric(PG_FUNCTION_ARGS)
{
    return generate_series_step_numeric(fcinfo);
}
 
Datum
generate_series_step_numeric(PG_FUNCTION_ARGS)
{
    generate_series_numeric_fctx *fctx;
    FuncCallContext *funcctx;
    MemoryContext oldcontext;
 
    if (SRF_IS_FIRSTCALL())
    {
        Numeric     start_num = PG_GETARG_NUMERIC(0);
        Numeric     stop_num = PG_GETARG_NUMERIC(1);
        NumericVar  steploc = const_one;
 
        /* Reject NaN and infinities in start and stop values */
        if (NUMERIC_IS_SPECIAL(start_num))
        {
            if (NUMERIC_IS_NAN(start_num))
                ereport(ERROR,
                        (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                         errmsg("start value cannot be NaN")));
            else
                ereport(ERROR,
                        (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                         errmsg("start value cannot be infinity")));
        }
        if (NUMERIC_IS_SPECIAL(stop_num))
        {
            if (NUMERIC_IS_NAN(stop_num))
                ereport(ERROR,
                        (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                         errmsg("stop value cannot be NaN")));
            else
                ereport(ERROR,
                        (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                         errmsg("stop value cannot be infinity")));
        }
 
        /* see if we were given an explicit step size */
        if (PG_NARGS() == 3)
        {
            Numeric     step_num = PG_GETARG_NUMERIC(2);
 
            if (NUMERIC_IS_SPECIAL(step_num))
            {
                if (NUMERIC_IS_NAN(step_num))
                    ereport(ERROR,
                            (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                             errmsg("step size cannot be NaN")));
                else
                    ereport(ERROR,
                            (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                             errmsg("step size cannot be infinity")));
            }
 
            init_var_from_num(step_num, &steploc);
 
            if (cmp_var(&steploc, &const_zero) == 0)
                ereport(ERROR,
                        (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                         errmsg("step size cannot equal zero")));
        }
 
        /* create a function context for cross-call persistence */
        funcctx = SRF_FIRSTCALL_INIT();
 
        /*
         * Switch to memory context appropriate for multiple function calls.
         */
        oldcontext = MemoryContextSwitchTo(funcctx->multi_call_memory_ctx);
 
        /* allocate memory for user context */
        fctx = (generate_series_numeric_fctx *)
            palloc(sizeof(generate_series_numeric_fctx));
 
        /*
         * Use fctx to keep state from call to call. Seed current with the
         * original start value. We must copy the start_num and stop_num
         * values rather than pointing to them, since we may have detoasted
         * them in the per-call context.
         */
        init_var(&fctx->current);
        init_var(&fctx->stop);
        init_var(&fctx->step);
 
        set_var_from_num(start_num, &fctx->current);
        set_var_from_num(stop_num, &fctx->stop);
        set_var_from_var(&steploc, &fctx->step);
 
        funcctx->user_fctx = fctx;
        MemoryContextSwitchTo(oldcontext);
    }
 
    /* stuff done on every call of the function */
    funcctx = SRF_PERCALL_SETUP();
 
    /*
     * Get the saved state and use current state as the result of this
     * iteration.
     */
    fctx = funcctx->user_fctx;
 
    if ((fctx->step.sign == NUMERIC_POS &&
         cmp_var(&fctx->current, &fctx->stop) <= 0) ||
        (fctx->step.sign == NUMERIC_NEG &&
         cmp_var(&fctx->current, &fctx->stop) >= 0))
    {
        Numeric     result = make_result(&fctx->current);
 
        /* switch to memory context appropriate for iteration calculation */
        oldcontext = MemoryContextSwitchTo(funcctx->multi_call_memory_ctx);
 
        /* increment current in preparation for next iteration */
        add_var(&fctx->current, &fctx->step, &fctx->current);
        MemoryContextSwitchTo(oldcontext);
 
        /* do when there is more left to send */
        SRF_RETURN_NEXT(funcctx, NumericGetDatum(result));
    }
    else
        /* do when there is no more left */
        SRF_RETURN_DONE(funcctx);
}
 
/*
 * Planner support function for generate_series(numeric, numeric [, numeric])
 */
Datum
generate_series_numeric_support(PG_FUNCTION_ARGS)
{
    Node       *rawreq = (Node *) PG_GETARG_POINTER(0);
    Node       *ret = NULL;
 
    if (IsA(rawreq, SupportRequestRows))
    {
        /* Try to estimate the number of rows returned */
        SupportRequestRows *req = (SupportRequestRows *) rawreq;
 
        if (is_funcclause(req->node))   /* be paranoid */
        {
            List       *args = ((FuncExpr *) req->node)->args;
            Node       *arg1,
                       *arg2,
                       *arg3;
 
            /* We can use estimated argument values here */
            arg1 = estimate_expression_value(req->root, linitial(args));
            arg2 = estimate_expression_value(req->root, lsecond(args));
            if (list_length(args) >= 3)
                arg3 = estimate_expression_value(req->root, lthird(args));
            else
                arg3 = NULL;
 
            /*
             * If any argument is constant NULL, we can safely assume that
             * zero rows are returned.  Otherwise, if they're all non-NULL
             * constants, we can calculate the number of rows that will be
             * returned.
             */
            if ((IsA(arg1, Const) &&
                 ((Const *) arg1)->constisnull) ||
                (IsA(arg2, Const) &&
                 ((Const *) arg2)->constisnull) ||
                (arg3 != NULL && IsA(arg3, Const) &&
                 ((Const *) arg3)->constisnull))
            {
                req->rows = 0;
                ret = (Node *) req;
            }
            else if (IsA(arg1, Const) &&
                     IsA(arg2, Const) &&
                     (arg3 == NULL || IsA(arg3, Const)))
            {
                Numeric     start_num;
                Numeric     stop_num;
                NumericVar  step = const_one;
 
                /*
                 * If any argument is NaN or infinity, generate_series() will
                 * error out, so we needn't produce an estimate.
                 */
                start_num = DatumGetNumeric(((Const *) arg1)->constvalue);
                stop_num = DatumGetNumeric(((Const *) arg2)->constvalue);
 
                if (NUMERIC_IS_SPECIAL(start_num) ||
                    NUMERIC_IS_SPECIAL(stop_num))
                    PG_RETURN_POINTER(NULL);
 
                if (arg3)
                {
                    Numeric     step_num;
 
                    step_num = DatumGetNumeric(((Const *) arg3)->constvalue);
 
                    if (NUMERIC_IS_SPECIAL(step_num))
                        PG_RETURN_POINTER(NULL);
 
                    init_var_from_num(step_num, &step);
                }
 
                /*
                 * The number of rows that will be returned is given by
                 * floor((stop - start) / step) + 1, if the sign of step
                 * matches the sign of stop - start.  Otherwise, no rows will
                 * be returned.
                 */
                if (cmp_var(&step, &const_zero) != 0)
                {
                    NumericVar  start;
                    NumericVar  stop;
                    NumericVar  res;
 
                    init_var_from_num(start_num, &start);
                    init_var_from_num(stop_num, &stop);
 
                    init_var(&res);
                    sub_var(&stop, &start, &res);
 
                    if (step.sign != res.sign)
                    {
                        /* no rows will be returned */
                        req->rows = 0;
                        ret = (Node *) req;
                    }
                    else
                    {
                        if (arg3)
                            div_var(&res, &step, &res, 0, false, false);
                        else
                            trunc_var(&res, 0); /* step = 1 */
 
                        req->rows = numericvar_to_double_no_overflow(&res) + 1;
                        ret = (Node *) req;
                    }
 
                    free_var(&res);
                }
            }
        }
    }
 
    PG_RETURN_POINTER(ret);
}
 
 
/*
 * Implements the numeric version of the width_bucket() function
 * defined by SQL2003. See also width_bucket_float8().
 *
 * 'bound1' and 'bound2' are the lower and upper bounds of the
 * histogram's range, respectively. 'count' is the number of buckets
 * in the histogram. width_bucket() returns an integer indicating the
 * bucket number that 'operand' belongs to in an equiwidth histogram
 * with the specified characteristics. An operand smaller than the
 * lower bound is assigned to bucket 0. An operand greater than the
 * upper bound is assigned to an additional bucket (with number
 * count+1). We don't allow "NaN" for any of the numeric arguments.
 */
Datum
width_bucket_numeric(PG_FUNCTION_ARGS)
{
    Numeric     operand = PG_GETARG_NUMERIC(0);
    Numeric     bound1 = PG_GETARG_NUMERIC(1);
    Numeric     bound2 = PG_GETARG_NUMERIC(2);
    int32       count = PG_GETARG_INT32(3);
    NumericVar  count_var;
    NumericVar  result_var;
    int32       result;
 
    if (count <= 0)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_ARGUMENT_FOR_WIDTH_BUCKET_FUNCTION),
                 errmsg("count must be greater than zero")));
 
    if (NUMERIC_IS_SPECIAL(operand) ||
        NUMERIC_IS_SPECIAL(bound1) ||
        NUMERIC_IS_SPECIAL(bound2))
    {
        if (NUMERIC_IS_NAN(operand) ||
            NUMERIC_IS_NAN(bound1) ||
            NUMERIC_IS_NAN(bound2))
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_WIDTH_BUCKET_FUNCTION),
                     errmsg("operand, lower bound, and upper bound cannot be NaN")));
        /* We allow "operand" to be infinite; cmp_numerics will cope */
        if (NUMERIC_IS_INF(bound1) || NUMERIC_IS_INF(bound2))
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_WIDTH_BUCKET_FUNCTION),
                     errmsg("lower and upper bounds must be finite")));
    }
 
    init_var(&result_var);
    init_var(&count_var);
 
    /* Convert 'count' to a numeric, for ease of use later */
    int64_to_numericvar((int64) count, &count_var);
 
    switch (cmp_numerics(bound1, bound2))
    {
        case 0:
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_WIDTH_BUCKET_FUNCTION),
                     errmsg("lower bound cannot equal upper bound")));
            break;
 
            /* bound1 < bound2 */
        case -1:
            if (cmp_numerics(operand, bound1) < 0)
                set_var_from_var(&const_zero, &result_var);
            else if (cmp_numerics(operand, bound2) >= 0)
                add_var(&count_var, &const_one, &result_var);
            else
                compute_bucket(operand, bound1, bound2, &count_var,
                               &result_var);
            break;
 
            /* bound1 > bound2 */
        case 1:
            if (cmp_numerics(operand, bound1) > 0)
                set_var_from_var(&const_zero, &result_var);
            else if (cmp_numerics(operand, bound2) <= 0)
                add_var(&count_var, &const_one, &result_var);
            else
                compute_bucket(operand, bound1, bound2, &count_var,
                               &result_var);
            break;
    }
 
    /* if result exceeds the range of a legal int4, we ereport here */
    if (!numericvar_to_int32(&result_var, &result))
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("integer out of range")));
 
    free_var(&count_var);
    free_var(&result_var);
 
    PG_RETURN_INT32(result);
}
 
/*
 * 'operand' is inside the bucket range, so determine the correct
 * bucket for it to go in. The calculations performed by this function
 * are derived directly from the SQL2003 spec. Note however that we
 * multiply by count before dividing, to avoid unnecessary roundoff error.
 */
static void
compute_bucket(Numeric operand, Numeric bound1, Numeric bound2,
               const NumericVar *count_var, NumericVar *result_var)
{
    NumericVar  bound1_var;
    NumericVar  bound2_var;
    NumericVar  operand_var;
 
    init_var_from_num(bound1, &bound1_var);
    init_var_from_num(bound2, &bound2_var);
    init_var_from_num(operand, &operand_var);
 
    /*
     * Per spec, bound1 is inclusive and bound2 is exclusive, and so we have
     * bound1 <= operand < bound2 or bound1 >= operand > bound2.  Either way,
     * the result is ((operand - bound1) * count) / (bound2 - bound1) + 1,
     * where the quotient is computed using floor division (i.e., division to
     * zero decimal places with truncation), which guarantees that the result
     * is in the range [1, count].  Reversing the bounds doesn't affect the
     * computation, because the signs cancel out when dividing.
     */
    sub_var(&operand_var, &bound1_var, &operand_var);
    sub_var(&bound2_var, &bound1_var, &bound2_var);
 
    mul_var(&operand_var, count_var, &operand_var,
            operand_var.dscale + count_var->dscale);
    div_var(&operand_var, &bound2_var, result_var, 0, false, true);
    add_var(result_var, &const_one, result_var);
 
    free_var(&bound1_var);
    free_var(&bound2_var);
    free_var(&operand_var);
}
 
/* ----------------------------------------------------------------------
 *
 * Comparison functions
 *
 * Note: btree indexes need these routines not to leak memory; therefore,
 * be careful to free working copies of toasted datums.  Most places don't
 * need to be so careful.
 *
 * Sort support:
 *
 * We implement the sortsupport strategy routine in order to get the benefit of
 * abbreviation. The ordinary numeric comparison can be quite slow as a result
 * of palloc/pfree cycles (due to detoasting packed values for alignment);
 * while this could be worked on itself, the abbreviation strategy gives more
 * speedup in many common cases.
 *
 * Two different representations are used for the abbreviated form, one in
 * int32 and one in int64, whichever fits into a by-value Datum.  In both cases
 * the representation is negated relative to the original value, because we use
 * the largest negative value for NaN, which sorts higher than other values. We
 * convert the absolute value of the numeric to a 31-bit or 63-bit positive
 * value, and then negate it if the original number was positive.
 *
 * We abort the abbreviation process if the abbreviation cardinality is below
 * 0.01% of the row count (1 per 10k non-null rows).  The actual break-even
 * point is somewhat below that, perhaps 1 per 30k (at 1 per 100k there's a
 * very small penalty), but we don't want to build up too many abbreviated
 * values before first testing for abort, so we take the slightly pessimistic
 * number.  We make no attempt to estimate the cardinality of the real values,
 * since it plays no part in the cost model here (if the abbreviation is equal,
 * the cost of comparing equal and unequal underlying values is comparable).
 * We discontinue even checking for abort (saving us the hashing overhead) if
 * the estimated cardinality gets to 100k; that would be enough to support many
 * billions of rows while doing no worse than breaking even.
 *
 * ----------------------------------------------------------------------
 */
 
/*
 * Sort support strategy routine.
 */
Datum
numeric_sortsupport(PG_FUNCTION_ARGS)
{
    SortSupport ssup = (SortSupport) PG_GETARG_POINTER(0);
 
    ssup->comparator = numeric_fast_cmp;
 
    if (ssup->abbreviate)
    {
        NumericSortSupport *nss;
        MemoryContext oldcontext = MemoryContextSwitchTo(ssup->ssup_cxt);
 
        nss = palloc(sizeof(NumericSortSupport));
 
        /*
         * palloc a buffer for handling unaligned packed values in addition to
         * the support struct
         */
        nss->buf = palloc(VARATT_SHORT_MAX + VARHDRSZ + 1);
 
        nss->input_count = 0;
        nss->estimating = true;
        initHyperLogLog(&nss->abbr_card, 10);
 
        ssup->ssup_extra = nss;
 
        ssup->abbrev_full_comparator = ssup->comparator;
        ssup->comparator = numeric_cmp_abbrev;
        ssup->abbrev_converter = numeric_abbrev_convert;
        ssup->abbrev_abort = numeric_abbrev_abort;
 
        MemoryContextSwitchTo(oldcontext);
    }
 
    PG_RETURN_VOID();
}
 
/*
 * Abbreviate a numeric datum, handling NaNs and detoasting
 * (must not leak memory!)
 */
static Datum
numeric_abbrev_convert(Datum original_datum, SortSupport ssup)
{
    NumericSortSupport *nss = ssup->ssup_extra;
    void       *original_varatt = PG_DETOAST_DATUM_PACKED(original_datum);
    Numeric     value;
    Datum       result;
 
    nss->input_count += 1;
 
    /*
     * This is to handle packed datums without needing a palloc/pfree cycle;
     * we keep and reuse a buffer large enough to handle any short datum.
     */
    if (VARATT_IS_SHORT(original_varatt))
    {
        void       *buf = nss->buf;
        Size        sz = VARSIZE_SHORT(original_varatt) - VARHDRSZ_SHORT;
 
        Assert(sz <= VARATT_SHORT_MAX - VARHDRSZ_SHORT);
 
        SET_VARSIZE(buf, VARHDRSZ + sz);
        memcpy(VARDATA(buf), VARDATA_SHORT(original_varatt), sz);
 
        value = (Numeric) buf;
    }
    else
        value = (Numeric) original_varatt;
 
    if (NUMERIC_IS_SPECIAL(value))
    {
        if (NUMERIC_IS_PINF(value))
            result = NUMERIC_ABBREV_PINF;
        else if (NUMERIC_IS_NINF(value))
            result = NUMERIC_ABBREV_NINF;
        else
            result = NUMERIC_ABBREV_NAN;
    }
    else
    {
        NumericVar  var;
 
        init_var_from_num(value, &var);
 
        result = numeric_abbrev_convert_var(&var, nss);
    }
 
    /* should happen only for external/compressed toasts */
    if ((Pointer) original_varatt != DatumGetPointer(original_datum))
        pfree(original_varatt);
 
    return result;
}
 
/*
 * Consider whether to abort abbreviation.
 *
 * We pay no attention to the cardinality of the non-abbreviated data. There is
 * no reason to do so: unlike text, we have no fast check for equal values, so
 * we pay the full overhead whenever the abbreviations are equal regardless of
 * whether the underlying values are also equal.
 */
static bool
numeric_abbrev_abort(int memtupcount, SortSupport ssup)
{
    NumericSortSupport *nss = ssup->ssup_extra;
    double      abbr_card;
 
    if (memtupcount < 10000 || nss->input_count < 10000 || !nss->estimating)
        return false;
 
    abbr_card = estimateHyperLogLog(&nss->abbr_card);
 
    /*
     * If we have >100k distinct values, then even if we were sorting many
     * billion rows we'd likely still break even, and the penalty of undoing
     * that many rows of abbrevs would probably not be worth it. Stop even
     * counting at that point.
     */
    if (abbr_card > 100000.0)
    {
        if (trace_sort)
            elog(LOG,
                 "numeric_abbrev: estimation ends at cardinality %f"
                 " after " INT64_FORMAT " values (%d rows)",
                 abbr_card, nss->input_count, memtupcount);
        nss->estimating = false;
        return false;
    }
 
    /*
     * Target minimum cardinality is 1 per ~10k of non-null inputs.  (The
     * break even point is somewhere between one per 100k rows, where
     * abbreviation has a very slight penalty, and 1 per 10k where it wins by
     * a measurable percentage.)  We use the relatively pessimistic 10k
     * threshold, and add a 0.5 row fudge factor, because it allows us to
     * abort earlier on genuinely pathological data where we've had exactly
     * one abbreviated value in the first 10k (non-null) rows.
     */
    if (abbr_card < nss->input_count / 10000.0 + 0.5)
    {
        if (trace_sort)
            elog(LOG,
                 "numeric_abbrev: aborting abbreviation at cardinality %f"
                 " below threshold %f after " INT64_FORMAT " values (%d rows)",
                 abbr_card, nss->input_count / 10000.0 + 0.5,
                 nss->input_count, memtupcount);
        return true;
    }
 
    if (trace_sort)
        elog(LOG,
             "numeric_abbrev: cardinality %f"
             " after " INT64_FORMAT " values (%d rows)",
             abbr_card, nss->input_count, memtupcount);
 
    return false;
}
 
/*
 * Non-fmgr interface to the comparison routine to allow sortsupport to elide
 * the fmgr call.  The saving here is small given how slow numeric comparisons
 * are, but it is a required part of the sort support API when abbreviations
 * are performed.
 *
 * Two palloc/pfree cycles could be saved here by using persistent buffers for
 * aligning short-varlena inputs, but this has not so far been considered to
 * be worth the effort.
 */
static int
numeric_fast_cmp(Datum x, Datum y, SortSupport ssup)
{
    Numeric     nx = DatumGetNumeric(x);
    Numeric     ny = DatumGetNumeric(y);
    int         result;
 
    result = cmp_numerics(nx, ny);
 
    if ((Pointer) nx != DatumGetPointer(x))
        pfree(nx);
    if ((Pointer) ny != DatumGetPointer(y))
        pfree(ny);
 
    return result;
}
 
/*
 * Compare abbreviations of values. (Abbreviations may be equal where the true
 * values differ, but if the abbreviations differ, they must reflect the
 * ordering of the true values.)
 */
static int
numeric_cmp_abbrev(Datum x, Datum y, SortSupport ssup)
{
    /*
     * NOTE WELL: this is intentionally backwards, because the abbreviation is
     * negated relative to the original value, to handle NaN/infinity cases.
     */
    if (DatumGetNumericAbbrev(x) < DatumGetNumericAbbrev(y))
        return 1;
    if (DatumGetNumericAbbrev(x) > DatumGetNumericAbbrev(y))
        return -1;
    return 0;
}
 
/*
 * Abbreviate a NumericVar according to the available bit size.
 *
 * The 31-bit value is constructed as:
 *
 *  0 + 7bits digit weight + 24 bits digit value
 *
 * where the digit weight is in single decimal digits, not digit words, and
 * stored in excess-44 representation[1]. The 24-bit digit value is the 7 most
 * significant decimal digits of the value converted to binary. Values whose
 * weights would fall outside the representable range are rounded off to zero
 * (which is also used to represent actual zeros) or to 0x7FFFFFFF (which
 * otherwise cannot occur). Abbreviation therefore fails to gain any advantage
 * where values are outside the range 10^-44 to 10^83, which is not considered
 * to be a serious limitation, or when values are of the same magnitude and
 * equal in the first 7 decimal digits, which is considered to be an
 * unavoidable limitation given the available bits. (Stealing three more bits
 * to compare another digit would narrow the range of representable weights by
 * a factor of 8, which starts to look like a real limiting factor.)
 *
 * (The value 44 for the excess is essentially arbitrary)
 *
 * The 63-bit value is constructed as:
 *
 *  0 + 7bits weight + 4 x 14-bit packed digit words
 *
 * The weight in this case is again stored in excess-44, but this time it is
 * the original weight in digit words (i.e. powers of 10000). The first four
 * digit words of the value (if present; trailing zeros are assumed as needed)
 * are packed into 14 bits each to form the rest of the value. Again,
 * out-of-range values are rounded off to 0 or 0x7FFFFFFFFFFFFFFF. The
 * representable range in this case is 10^-176 to 10^332, which is considered
 * to be good enough for all practical purposes, and comparison of 4 words
 * means that at least 13 decimal digits are compared, which is considered to
 * be a reasonable compromise between effectiveness and efficiency in computing
 * the abbreviation.
 *
 * (The value 44 for the excess is even more arbitrary here, it was chosen just
 * to match the value used in the 31-bit case)
 *
 * [1] - Excess-k representation means that the value is offset by adding 'k'
 * and then treated as unsigned, so the smallest representable value is stored
 * with all bits zero. This allows simple comparisons to work on the composite
 * value.
 */
 
#if NUMERIC_ABBREV_BITS == 64
 
static Datum
numeric_abbrev_convert_var(const NumericVar *var, NumericSortSupport *nss)
{
    int         ndigits = var->ndigits;
    int         weight = var->weight;
    int64       result;
 
    if (ndigits == 0 || weight < -44)
    {
        result = 0;
    }
    else if (weight > 83)
    {
        result = PG_INT64_MAX;
    }
    else
    {
        result = ((int64) (weight + 44) << 56);
 
        switch (ndigits)
        {
            default:
                result |= ((int64) var->digits[3]);
                /* FALLTHROUGH */
            case 3:
                result |= ((int64) var->digits[2]) << 14;
                /* FALLTHROUGH */
            case 2:
                result |= ((int64) var->digits[1]) << 28;
                /* FALLTHROUGH */
            case 1:
                result |= ((int64) var->digits[0]) << 42;
                break;
        }
    }
 
    /* the abbrev is negated relative to the original */
    if (var->sign == NUMERIC_POS)
        result = -result;
 
    if (nss->estimating)
    {
        uint32      tmp = ((uint32) result
                           ^ (uint32) ((uint64) result >> 32));
 
        addHyperLogLog(&nss->abbr_card, DatumGetUInt32(hash_uint32(tmp)));
    }
 
    return NumericAbbrevGetDatum(result);
}
 
#endif                          /* NUMERIC_ABBREV_BITS == 64 */
 
#if NUMERIC_ABBREV_BITS == 32
 
static Datum
numeric_abbrev_convert_var(const NumericVar *var, NumericSortSupport *nss)
{
    int         ndigits = var->ndigits;
    int         weight = var->weight;
    int32       result;
 
    if (ndigits == 0 || weight < -11)
    {
        result = 0;
    }
    else if (weight > 20)
    {
        result = PG_INT32_MAX;
    }
    else
    {
        NumericDigit nxt1 = (ndigits > 1) ? var->digits[1] : 0;
 
        weight = (weight + 11) * 4;
 
        result = var->digits[0];
 
        /*
         * "result" now has 1 to 4 nonzero decimal digits. We pack in more
         * digits to make 7 in total (largest we can fit in 24 bits)
         */
 
        if (result > 999)
        {
            /* already have 4 digits, add 3 more */
            result = (result * 1000) + (nxt1 / 10);
            weight += 3;
        }
        else if (result > 99)
        {
            /* already have 3 digits, add 4 more */
            result = (result * 10000) + nxt1;
            weight += 2;
        }
        else if (result > 9)
        {
            NumericDigit nxt2 = (ndigits > 2) ? var->digits[2] : 0;
 
            /* already have 2 digits, add 5 more */
            result = (result * 100000) + (nxt1 * 10) + (nxt2 / 1000);
            weight += 1;
        }
        else
        {
            NumericDigit nxt2 = (ndigits > 2) ? var->digits[2] : 0;
 
            /* already have 1 digit, add 6 more */
            result = (result * 1000000) + (nxt1 * 100) + (nxt2 / 100);
        }
 
        result = result | (weight << 24);
    }
 
    /* the abbrev is negated relative to the original */
    if (var->sign == NUMERIC_POS)
        result = -result;
 
    if (nss->estimating)
    {
        uint32      tmp = (uint32) result;
 
        addHyperLogLog(&nss->abbr_card, DatumGetUInt32(hash_uint32(tmp)));
    }
 
    return NumericAbbrevGetDatum(result);
}
 
#endif                          /* NUMERIC_ABBREV_BITS == 32 */
 
/*
 * Ordinary (non-sortsupport) comparisons follow.
 */
 
Datum
numeric_cmp(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    int         result;
 
    result = cmp_numerics(num1, num2);
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_INT32(result);
}
 
 
Datum
numeric_eq(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) == 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
Datum
numeric_ne(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) != 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
Datum
numeric_gt(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) > 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
Datum
numeric_ge(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) >= 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
Datum
numeric_lt(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) < 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
Datum
numeric_le(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    bool        result;
 
    result = cmp_numerics(num1, num2) <= 0;
 
    PG_FREE_IF_COPY(num1, 0);
    PG_FREE_IF_COPY(num2, 1);
 
    PG_RETURN_BOOL(result);
}
 
static int
cmp_numerics(Numeric num1, Numeric num2)
{
    int         result;
 
    /*
     * We consider all NANs to be equal and larger than any non-NAN (including
     * Infinity).  This is somewhat arbitrary; the important thing is to have
     * a consistent sort order.
     */
    if (NUMERIC_IS_SPECIAL(num1))
    {
        if (NUMERIC_IS_NAN(num1))
        {
            if (NUMERIC_IS_NAN(num2))
                result = 0;     /* NAN = NAN */
            else
                result = 1;     /* NAN > non-NAN */
        }
        else if (NUMERIC_IS_PINF(num1))
        {
            if (NUMERIC_IS_NAN(num2))
                result = -1;    /* PINF < NAN */
            else if (NUMERIC_IS_PINF(num2))
                result = 0;     /* PINF = PINF */
            else
                result = 1;     /* PINF > anything else */
        }
        else                    /* num1 must be NINF */
        {
            if (NUMERIC_IS_NINF(num2))
                result = 0;     /* NINF = NINF */
            else
                result = -1;    /* NINF < anything else */
        }
    }
    else if (NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NINF(num2))
            result = 1;         /* normal > NINF */
        else
            result = -1;        /* normal < NAN or PINF */
    }
    else
    {
        result = cmp_var_common(NUMERIC_DIGITS(num1), NUMERIC_NDIGITS(num1),
                                NUMERIC_WEIGHT(num1), NUMERIC_SIGN(num1),
                                NUMERIC_DIGITS(num2), NUMERIC_NDIGITS(num2),
                                NUMERIC_WEIGHT(num2), NUMERIC_SIGN(num2));
    }
 
    return result;
}
 
/*
 * in_range support function for numeric.
 */
Datum
in_range_numeric_numeric(PG_FUNCTION_ARGS)
{
    Numeric     val = PG_GETARG_NUMERIC(0);
    Numeric     base = PG_GETARG_NUMERIC(1);
    Numeric     offset = PG_GETARG_NUMERIC(2);
    bool        sub = PG_GETARG_BOOL(3);
    bool        less = PG_GETARG_BOOL(4);
    bool        result;
 
    /*
     * Reject negative (including -Inf) or NaN offset.  Negative is per spec,
     * and NaN is because appropriate semantics for that seem non-obvious.
     */
    if (NUMERIC_IS_NAN(offset) ||
        NUMERIC_IS_NINF(offset) ||
        NUMERIC_SIGN(offset) == NUMERIC_NEG)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_PRECEDING_OR_FOLLOWING_SIZE),
                 errmsg("invalid preceding or following size in window function")));
 
    /*
     * Deal with cases where val and/or base is NaN, following the rule that
     * NaN sorts after non-NaN (cf cmp_numerics).  The offset cannot affect
     * the conclusion.
     */
    if (NUMERIC_IS_NAN(val))
    {
        if (NUMERIC_IS_NAN(base))
            result = true;      /* NAN = NAN */
        else
            result = !less;     /* NAN > non-NAN */
    }
    else if (NUMERIC_IS_NAN(base))
    {
        result = less;          /* non-NAN < NAN */
    }
 
    /*
     * Deal with infinite offset (necessarily +Inf, at this point).
     */
    else if (NUMERIC_IS_SPECIAL(offset))
    {
        Assert(NUMERIC_IS_PINF(offset));
        if (sub ? NUMERIC_IS_PINF(base) : NUMERIC_IS_NINF(base))
        {
            /*
             * base +/- offset would produce NaN, so return true for any val
             * (see in_range_float8_float8() for reasoning).
             */
            result = true;
        }
        else if (sub)
        {
            /* base - offset must be -inf */
            if (less)
                result = NUMERIC_IS_NINF(val);  /* only -inf is <= sum */
            else
                result = true;  /* any val is >= sum */
        }
        else
        {
            /* base + offset must be +inf */
            if (less)
                result = true;  /* any val is <= sum */
            else
                result = NUMERIC_IS_PINF(val);  /* only +inf is >= sum */
        }
    }
 
    /*
     * Deal with cases where val and/or base is infinite.  The offset, being
     * now known finite, cannot affect the conclusion.
     */
    else if (NUMERIC_IS_SPECIAL(val))
    {
        if (NUMERIC_IS_PINF(val))
        {
            if (NUMERIC_IS_PINF(base))
                result = true;  /* PINF = PINF */
            else
                result = !less; /* PINF > any other non-NAN */
        }
        else                    /* val must be NINF */
        {
            if (NUMERIC_IS_NINF(base))
                result = true;  /* NINF = NINF */
            else
                result = less;  /* NINF < anything else */
        }
    }
    else if (NUMERIC_IS_SPECIAL(base))
    {
        if (NUMERIC_IS_NINF(base))
            result = !less;     /* normal > NINF */
        else
            result = less;      /* normal < PINF */
    }
    else
    {
        /*
         * Otherwise go ahead and compute base +/- offset.  While it's
         * possible for this to overflow the numeric format, it's unlikely
         * enough that we don't take measures to prevent it.
         */
        NumericVar  valv;
        NumericVar  basev;
        NumericVar  offsetv;
        NumericVar  sum;
 
        init_var_from_num(val, &valv);
        init_var_from_num(base, &basev);
        init_var_from_num(offset, &offsetv);
        init_var(&sum);
 
        if (sub)
            sub_var(&basev, &offsetv, &sum);
        else
            add_var(&basev, &offsetv, &sum);
 
        if (less)
            result = (cmp_var(&valv, &sum) <= 0);
        else
            result = (cmp_var(&valv, &sum) >= 0);
 
        free_var(&sum);
    }
 
    PG_FREE_IF_COPY(val, 0);
    PG_FREE_IF_COPY(base, 1);
    PG_FREE_IF_COPY(offset, 2);
 
    PG_RETURN_BOOL(result);
}
 
Datum
hash_numeric(PG_FUNCTION_ARGS)
{
    Numeric     key = PG_GETARG_NUMERIC(0);
    Datum       digit_hash;
    Datum       result;
    int         weight;
    int         start_offset;
    int         end_offset;
    int         i;
    int         hash_len;
    NumericDigit *digits;
 
    /* If it's NaN or infinity, don't try to hash the rest of the fields */
    if (NUMERIC_IS_SPECIAL(key))
        PG_RETURN_UINT32(0);
 
    weight = NUMERIC_WEIGHT(key);
    start_offset = 0;
    end_offset = 0;
 
    /*
     * Omit any leading or trailing zeros from the input to the hash. The
     * numeric implementation *should* guarantee that leading and trailing
     * zeros are suppressed, but we're paranoid. Note that we measure the
     * starting and ending offsets in units of NumericDigits, not bytes.
     */
    digits = NUMERIC_DIGITS(key);
    for (i = 0; i < NUMERIC_NDIGITS(key); i++)
    {
        if (digits[i] != (NumericDigit) 0)
            break;
 
        start_offset++;
 
        /*
         * The weight is effectively the # of digits before the decimal point,
         * so decrement it for each leading zero we skip.
         */
        weight--;
    }
 
    /*
     * If there are no non-zero digits, then the value of the number is zero,
     * regardless of any other fields.
     */
    if (NUMERIC_NDIGITS(key) == start_offset)
        PG_RETURN_UINT32(-1);
 
    for (i = NUMERIC_NDIGITS(key) - 1; i >= 0; i--)
    {
        if (digits[i] != (NumericDigit) 0)
            break;
 
        end_offset++;
    }
 
    /* If we get here, there should be at least one non-zero digit */
    Assert(start_offset + end_offset < NUMERIC_NDIGITS(key));
 
    /*
     * Note that we don't hash on the Numeric's scale, since two numerics can
     * compare equal but have different scales. We also don't hash on the
     * sign, although we could: since a sign difference implies inequality,
     * this shouldn't affect correctness.
     */
    hash_len = NUMERIC_NDIGITS(key) - start_offset - end_offset;
    digit_hash = hash_any((unsigned char *) (NUMERIC_DIGITS(key) + start_offset),
                          hash_len * sizeof(NumericDigit));
 
    /* Mix in the weight, via XOR */
    result = digit_hash ^ weight;
 
    PG_RETURN_DATUM(result);
}
 
/*
 * Returns 64-bit value by hashing a value to a 64-bit value, with a seed.
 * Otherwise, similar to hash_numeric.
 */
Datum
hash_numeric_extended(PG_FUNCTION_ARGS)
{
    Numeric     key = PG_GETARG_NUMERIC(0);
    uint64      seed = PG_GETARG_INT64(1);
    Datum       digit_hash;
    Datum       result;
    int         weight;
    int         start_offset;
    int         end_offset;
    int         i;
    int         hash_len;
    NumericDigit *digits;
 
    /* If it's NaN or infinity, don't try to hash the rest of the fields */
    if (NUMERIC_IS_SPECIAL(key))
        PG_RETURN_UINT64(seed);
 
    weight = NUMERIC_WEIGHT(key);
    start_offset = 0;
    end_offset = 0;
 
    digits = NUMERIC_DIGITS(key);
    for (i = 0; i < NUMERIC_NDIGITS(key); i++)
    {
        if (digits[i] != (NumericDigit) 0)
            break;
 
        start_offset++;
 
        weight--;
    }
 
    if (NUMERIC_NDIGITS(key) == start_offset)
        PG_RETURN_UINT64(seed - 1);
 
    for (i = NUMERIC_NDIGITS(key) - 1; i >= 0; i--)
    {
        if (digits[i] != (NumericDigit) 0)
            break;
 
        end_offset++;
    }
 
    Assert(start_offset + end_offset < NUMERIC_NDIGITS(key));
 
    hash_len = NUMERIC_NDIGITS(key) - start_offset - end_offset;
    digit_hash = hash_any_extended((unsigned char *) (NUMERIC_DIGITS(key)
                                                      + start_offset),
                                   hash_len * sizeof(NumericDigit),
                                   seed);
 
    result = UInt64GetDatum(DatumGetUInt64(digit_hash) ^ weight);
 
    PG_RETURN_DATUM(result);
}
 
 
/* ----------------------------------------------------------------------
 *
 * Basic arithmetic functions
 *
 * ----------------------------------------------------------------------
 */
 
 
/*
 * numeric_add() -
 *
 *  Add two numerics
 */
Datum
numeric_add(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
 
    res = numeric_add_opt_error(num1, num2, NULL);
 
    PG_RETURN_NUMERIC(res);
}
 
/*
 * numeric_add_opt_error() -
 *
 *  Internal version of numeric_add().  If "*have_error" flag is provided,
 *  on error it's set to true, NULL returned.  This is helpful when caller
 *  need to handle errors by itself.
 */
Numeric
numeric_add_opt_error(Numeric num1, Numeric num2, bool *have_error)
{
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            return make_result(&const_nan);
        if (NUMERIC_IS_PINF(num1))
        {
            if (NUMERIC_IS_NINF(num2))
                return make_result(&const_nan); /* Inf + -Inf */
            else
                return make_result(&const_pinf);
        }
        if (NUMERIC_IS_NINF(num1))
        {
            if (NUMERIC_IS_PINF(num2))
                return make_result(&const_nan); /* -Inf + Inf */
            else
                return make_result(&const_ninf);
        }
        /* by here, num1 must be finite, so num2 is not */
        if (NUMERIC_IS_PINF(num2))
            return make_result(&const_pinf);
        Assert(NUMERIC_IS_NINF(num2));
        return make_result(&const_ninf);
    }
 
    /*
     * Unpack the values, let add_var() compute the result and return it.
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
    add_var(&arg1, &arg2, &result);
 
    res = make_result_opt_error(&result, have_error);
 
    free_var(&result);
 
    return res;
}
 
 
/*
 * numeric_sub() -
 *
 *  Subtract one numeric from another
 */
Datum
numeric_sub(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
 
    res = numeric_sub_opt_error(num1, num2, NULL);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_sub_opt_error() -
 *
 *  Internal version of numeric_sub().  If "*have_error" flag is provided,
 *  on error it's set to true, NULL returned.  This is helpful when caller
 *  need to handle errors by itself.
 */
Numeric
numeric_sub_opt_error(Numeric num1, Numeric num2, bool *have_error)
{
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            return make_result(&const_nan);
        if (NUMERIC_IS_PINF(num1))
        {
            if (NUMERIC_IS_PINF(num2))
                return make_result(&const_nan); /* Inf - Inf */
            else
                return make_result(&const_pinf);
        }
        if (NUMERIC_IS_NINF(num1))
        {
            if (NUMERIC_IS_NINF(num2))
                return make_result(&const_nan); /* -Inf - -Inf */
            else
                return make_result(&const_ninf);
        }
        /* by here, num1 must be finite, so num2 is not */
        if (NUMERIC_IS_PINF(num2))
            return make_result(&const_ninf);
        Assert(NUMERIC_IS_NINF(num2));
        return make_result(&const_pinf);
    }
 
    /*
     * Unpack the values, let sub_var() compute the result and return it.
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
    sub_var(&arg1, &arg2, &result);
 
    res = make_result_opt_error(&result, have_error);
 
    free_var(&result);
 
    return res;
}
 
 
/*
 * numeric_mul() -
 *
 *  Calculate the product of two numerics
 */
Datum
numeric_mul(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
 
    res = numeric_mul_opt_error(num1, num2, NULL);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_mul_opt_error() -
 *
 *  Internal version of numeric_mul().  If "*have_error" flag is provided,
 *  on error it's set to true, NULL returned.  This is helpful when caller
 *  need to handle errors by itself.
 */
Numeric
numeric_mul_opt_error(Numeric num1, Numeric num2, bool *have_error)
{
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            return make_result(&const_nan);
        if (NUMERIC_IS_PINF(num1))
        {
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    return make_result(&const_nan); /* Inf * 0 */
                case 1:
                    return make_result(&const_pinf);
                case -1:
                    return make_result(&const_ninf);
            }
            Assert(false);
        }
        if (NUMERIC_IS_NINF(num1))
        {
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    return make_result(&const_nan); /* -Inf * 0 */
                case 1:
                    return make_result(&const_ninf);
                case -1:
                    return make_result(&const_pinf);
            }
            Assert(false);
        }
        /* by here, num1 must be finite, so num2 is not */
        if (NUMERIC_IS_PINF(num2))
        {
            switch (numeric_sign_internal(num1))
            {
                case 0:
                    return make_result(&const_nan); /* 0 * Inf */
                case 1:
                    return make_result(&const_pinf);
                case -1:
                    return make_result(&const_ninf);
            }
            Assert(false);
        }
        Assert(NUMERIC_IS_NINF(num2));
        switch (numeric_sign_internal(num1))
        {
            case 0:
                return make_result(&const_nan); /* 0 * -Inf */
            case 1:
                return make_result(&const_ninf);
            case -1:
                return make_result(&const_pinf);
        }
        Assert(false);
    }
 
    /*
     * Unpack the values, let mul_var() compute the result and return it.
     * Unlike add_var() and sub_var(), mul_var() will round its result. In the
     * case of numeric_mul(), which is invoked for the * operator on numerics,
     * we request exact representation for the product (rscale = sum(dscale of
     * arg1, dscale of arg2)).  If the exact result has more digits after the
     * decimal point than can be stored in a numeric, we round it.  Rounding
     * after computing the exact result ensures that the final result is
     * correctly rounded (rounding in mul_var() using a truncated product
     * would not guarantee this).
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
    mul_var(&arg1, &arg2, &result, arg1.dscale + arg2.dscale);
 
    if (result.dscale > NUMERIC_DSCALE_MAX)
        round_var(&result, NUMERIC_DSCALE_MAX);
 
    res = make_result_opt_error(&result, have_error);
 
    free_var(&result);
 
    return res;
}
 
 
/*
 * numeric_div() -
 *
 *  Divide one numeric into another
 */
Datum
numeric_div(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
 
    res = numeric_div_opt_error(num1, num2, NULL);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_div_opt_error() -
 *
 *  Internal version of numeric_div().  If "*have_error" flag is provided,
 *  on error it's set to true, NULL returned.  This is helpful when caller
 *  need to handle errors by itself.
 */
Numeric
numeric_div_opt_error(Numeric num1, Numeric num2, bool *have_error)
{
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
    int         rscale;
 
    if (have_error)
        *have_error = false;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            return make_result(&const_nan);
        if (NUMERIC_IS_PINF(num1))
        {
            if (NUMERIC_IS_SPECIAL(num2))
                return make_result(&const_nan); /* Inf / [-]Inf */
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    if (have_error)
                    {
                        *have_error = true;
                        return NULL;
                    }
                    ereport(ERROR,
                            (errcode(ERRCODE_DIVISION_BY_ZERO),
                             errmsg("division by zero")));
                    break;
                case 1:
                    return make_result(&const_pinf);
                case -1:
                    return make_result(&const_ninf);
            }
            Assert(false);
        }
        if (NUMERIC_IS_NINF(num1))
        {
            if (NUMERIC_IS_SPECIAL(num2))
                return make_result(&const_nan); /* -Inf / [-]Inf */
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    if (have_error)
                    {
                        *have_error = true;
                        return NULL;
                    }
                    ereport(ERROR,
                            (errcode(ERRCODE_DIVISION_BY_ZERO),
                             errmsg("division by zero")));
                    break;
                case 1:
                    return make_result(&const_ninf);
                case -1:
                    return make_result(&const_pinf);
            }
            Assert(false);
        }
        /* by here, num1 must be finite, so num2 is not */
 
        /*
         * POSIX would have us return zero or minus zero if num1 is zero, and
         * otherwise throw an underflow error.  But the numeric type doesn't
         * really do underflow, so let's just return zero.
         */
        return make_result(&const_zero);
    }
 
    /*
     * Unpack the arguments
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
 
    /*
     * Select scale for division result
     */
    rscale = select_div_scale(&arg1, &arg2);
 
    /*
     * If "have_error" is provided, check for division by zero here
     */
    if (have_error && (arg2.ndigits == 0 || arg2.digits[0] == 0))
    {
        *have_error = true;
        return NULL;
    }
 
    /*
     * Do the divide and return the result
     */
    div_var(&arg1, &arg2, &result, rscale, true, true);
 
    res = make_result_opt_error(&result, have_error);
 
    free_var(&result);
 
    return res;
}
 
 
/*
 * numeric_div_trunc() -
 *
 *  Divide one numeric into another, truncating the result to an integer
 */
Datum
numeric_div_trunc(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            PG_RETURN_NUMERIC(make_result(&const_nan));
        if (NUMERIC_IS_PINF(num1))
        {
            if (NUMERIC_IS_SPECIAL(num2))
                PG_RETURN_NUMERIC(make_result(&const_nan)); /* Inf / [-]Inf */
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    ereport(ERROR,
                            (errcode(ERRCODE_DIVISION_BY_ZERO),
                             errmsg("division by zero")));
                    break;
                case 1:
                    PG_RETURN_NUMERIC(make_result(&const_pinf));
                case -1:
                    PG_RETURN_NUMERIC(make_result(&const_ninf));
            }
            Assert(false);
        }
        if (NUMERIC_IS_NINF(num1))
        {
            if (NUMERIC_IS_SPECIAL(num2))
                PG_RETURN_NUMERIC(make_result(&const_nan)); /* -Inf / [-]Inf */
            switch (numeric_sign_internal(num2))
            {
                case 0:
                    ereport(ERROR,
                            (errcode(ERRCODE_DIVISION_BY_ZERO),
                             errmsg("division by zero")));
                    break;
                case 1:
                    PG_RETURN_NUMERIC(make_result(&const_ninf));
                case -1:
                    PG_RETURN_NUMERIC(make_result(&const_pinf));
            }
            Assert(false);
        }
        /* by here, num1 must be finite, so num2 is not */
 
        /*
         * POSIX would have us return zero or minus zero if num1 is zero, and
         * otherwise throw an underflow error.  But the numeric type doesn't
         * really do underflow, so let's just return zero.
         */
        PG_RETURN_NUMERIC(make_result(&const_zero));
    }
 
    /*
     * Unpack the arguments
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
 
    /*
     * Do the divide and return the result
     */
    div_var(&arg1, &arg2, &result, 0, false, true);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_mod() -
 *
 *  Calculate the modulo of two numerics
 */
Datum
numeric_mod(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
 
    res = numeric_mod_opt_error(num1, num2, NULL);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_mod_opt_error() -
 *
 *  Internal version of numeric_mod().  If "*have_error" flag is provided,
 *  on error it's set to true, NULL returned.  This is helpful when caller
 *  need to handle errors by itself.
 */
Numeric
numeric_mod_opt_error(Numeric num1, Numeric num2, bool *have_error)
{
    Numeric     res;
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
 
    if (have_error)
        *have_error = false;
 
    /*
     * Handle NaN and infinities.  We follow POSIX fmod() on this, except that
     * POSIX treats x-is-infinite and y-is-zero identically, raising EDOM and
     * returning NaN.  We choose to throw error only for y-is-zero.
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            return make_result(&const_nan);
        if (NUMERIC_IS_INF(num1))
        {
            if (numeric_sign_internal(num2) == 0)
            {
                if (have_error)
                {
                    *have_error = true;
                    return NULL;
                }
                ereport(ERROR,
                        (errcode(ERRCODE_DIVISION_BY_ZERO),
                         errmsg("division by zero")));
            }
            /* Inf % any nonzero = NaN */
            return make_result(&const_nan);
        }
        /* num2 must be [-]Inf; result is num1 regardless of sign of num2 */
        return duplicate_numeric(num1);
    }
 
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
 
    /*
     * If "have_error" is provided, check for division by zero here
     */
    if (have_error && (arg2.ndigits == 0 || arg2.digits[0] == 0))
    {
        *have_error = true;
        return NULL;
    }
 
    mod_var(&arg1, &arg2, &result);
 
    res = make_result_opt_error(&result, NULL);
 
    free_var(&result);
 
    return res;
}
 
 
/*
 * numeric_inc() -
 *
 *  Increment a number by one
 */
Datum
numeric_inc(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  arg;
    Numeric     res;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    /*
     * Compute the result and return it
     */
    init_var_from_num(num, &arg);
 
    add_var(&arg, &const_one, &arg);
 
    res = make_result(&arg);
 
    free_var(&arg);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_smaller() -
 *
 *  Return the smaller of two numbers
 */
Datum
numeric_smaller(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
 
    /*
     * Use cmp_numerics so that this will agree with the comparison operators,
     * particularly as regards comparisons involving NaN.
     */
    if (cmp_numerics(num1, num2) < 0)
        PG_RETURN_NUMERIC(num1);
    else
        PG_RETURN_NUMERIC(num2);
}
 
 
/*
 * numeric_larger() -
 *
 *  Return the larger of two numbers
 */
Datum
numeric_larger(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
 
    /*
     * Use cmp_numerics so that this will agree with the comparison operators,
     * particularly as regards comparisons involving NaN.
     */
    if (cmp_numerics(num1, num2) > 0)
        PG_RETURN_NUMERIC(num1);
    else
        PG_RETURN_NUMERIC(num2);
}
 
 
/* ----------------------------------------------------------------------
 *
 * Advanced math functions
 *
 * ----------------------------------------------------------------------
 */
 
/*
 * numeric_gcd() -
 *
 *  Calculate the greatest common divisor of two numerics
 */
Datum
numeric_gcd(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities: we consider the result to be NaN in all such
     * cases.
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    /*
     * Unpack the arguments
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
 
    /*
     * Find the GCD and return the result
     */
    gcd_var(&arg1, &arg2, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_lcm() -
 *
 *  Calculate the least common multiple of two numerics
 */
Datum
numeric_lcm(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    Numeric     res;
 
    /*
     * Handle NaN and infinities: we consider the result to be NaN in all such
     * cases.
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    /*
     * Unpack the arguments
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    init_var(&result);
 
    /*
     * Compute the result using lcm(x, y) = abs(x / gcd(x, y) * y), returning
     * zero if either input is zero.
     *
     * Note that the division is guaranteed to be exact, returning an integer
     * result, so the LCM is an integral multiple of both x and y.  A display
     * scale of Min(x.dscale, y.dscale) would be sufficient to represent it,
     * but as with other numeric functions, we choose to return a result whose
     * display scale is no smaller than either input.
     */
    if (arg1.ndigits == 0 || arg2.ndigits == 0)
        set_var_from_var(&const_zero, &result);
    else
    {
        gcd_var(&arg1, &arg2, &result);
        div_var(&arg1, &result, &result, 0, false, true);
        mul_var(&arg2, &result, &result, arg2.dscale);
        result.sign = NUMERIC_POS;
    }
 
    result.dscale = Max(arg1.dscale, arg2.dscale);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_fac()
 *
 * Compute factorial
 */
Datum
numeric_fac(PG_FUNCTION_ARGS)
{
    int64       num = PG_GETARG_INT64(0);
    Numeric     res;
    NumericVar  fact;
    NumericVar  result;
 
    if (num < 0)
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("factorial of a negative number is undefined")));
    if (num <= 1)
    {
        res = make_result(&const_one);
        PG_RETURN_NUMERIC(res);
    }
    /* Fail immediately if the result would overflow */
    if (num > 32177)
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("value overflows numeric format")));
 
    init_var(&fact);
    init_var(&result);
 
    int64_to_numericvar(num, &result);
 
    for (num = num - 1; num > 1; num--)
    {
        /* this loop can take awhile, so allow it to be interrupted */
        CHECK_FOR_INTERRUPTS();
 
        int64_to_numericvar(num, &fact);
 
        mul_var(&result, &fact, &result, 0);
    }
 
    res = make_result(&result);
 
    free_var(&fact);
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_sqrt() -
 *
 *  Compute the square root of a numeric.
 */
Datum
numeric_sqrt(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  arg;
    NumericVar  result;
    int         sweight;
    int         rscale;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        /* error should match that in sqrt_var() */
        if (NUMERIC_IS_NINF(num))
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                     errmsg("cannot take square root of a negative number")));
        /* For NAN or PINF, just duplicate the input */
        PG_RETURN_NUMERIC(duplicate_numeric(num));
    }
 
    /*
     * Unpack the argument and determine the result scale.  We choose a scale
     * to give at least NUMERIC_MIN_SIG_DIGITS significant digits; but in any
     * case not less than the input's dscale.
     */
    init_var_from_num(num, &arg);
 
    init_var(&result);
 
    /*
     * Assume the input was normalized, so arg.weight is accurate.  The result
     * then has at least sweight = floor(arg.weight * DEC_DIGITS / 2 + 1)
     * digits before the decimal point.  When DEC_DIGITS is even, we can save
     * a few cycles, since the division is exact and there is no need to round
     * towards negative infinity.
     */
#if DEC_DIGITS == ((DEC_DIGITS / 2) * 2)
    sweight = arg.weight * DEC_DIGITS / 2 + 1;
#else
    if (arg.weight >= 0)
        sweight = arg.weight * DEC_DIGITS / 2 + 1;
    else
        sweight = 1 - (1 - arg.weight * DEC_DIGITS) / 2;
#endif
 
    rscale = NUMERIC_MIN_SIG_DIGITS - sweight;
    rscale = Max(rscale, arg.dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    /*
     * Let sqrt_var() do the calculation and return the result.
     */
    sqrt_var(&arg, &result, rscale);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_exp() -
 *
 *  Raise e to the power of x
 */
Datum
numeric_exp(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  arg;
    NumericVar  result;
    int         rscale;
    double      val;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        /* Per POSIX, exp(-Inf) is zero */
        if (NUMERIC_IS_NINF(num))
            PG_RETURN_NUMERIC(make_result(&const_zero));
        /* For NAN or PINF, just duplicate the input */
        PG_RETURN_NUMERIC(duplicate_numeric(num));
    }
 
    /*
     * Unpack the argument and determine the result scale.  We choose a scale
     * to give at least NUMERIC_MIN_SIG_DIGITS significant digits; but in any
     * case not less than the input's dscale.
     */
    init_var_from_num(num, &arg);
 
    init_var(&result);
 
    /* convert input to float8, ignoring overflow */
    val = numericvar_to_double_no_overflow(&arg);
 
    /*
     * log10(result) = num * log10(e), so this is approximately the decimal
     * weight of the result:
     */
    val *= 0.434294481903252;
 
    /* limit to something that won't cause integer overflow */
    val = Max(val, -NUMERIC_MAX_RESULT_SCALE);
    val = Min(val, NUMERIC_MAX_RESULT_SCALE);
 
    rscale = NUMERIC_MIN_SIG_DIGITS - (int) val;
    rscale = Max(rscale, arg.dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    /*
     * Let exp_var() do the calculation and return the result.
     */
    exp_var(&arg, &result, rscale);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_ln() -
 *
 *  Compute the natural logarithm of x
 */
Datum
numeric_ln(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  arg;
    NumericVar  result;
    int         ln_dweight;
    int         rscale;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_NINF(num))
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_LOG),
                     errmsg("cannot take logarithm of a negative number")));
        /* For NAN or PINF, just duplicate the input */
        PG_RETURN_NUMERIC(duplicate_numeric(num));
    }
 
    init_var_from_num(num, &arg);
    init_var(&result);
 
    /* Estimated dweight of logarithm */
    ln_dweight = estimate_ln_dweight(&arg);
 
    rscale = NUMERIC_MIN_SIG_DIGITS - ln_dweight;
    rscale = Max(rscale, arg.dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    ln_var(&arg, &result, rscale);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_log() -
 *
 *  Compute the logarithm of x in a given base
 */
Datum
numeric_log(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        int         sign1,
                    sign2;
 
        if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
            PG_RETURN_NUMERIC(make_result(&const_nan));
        /* fail on negative inputs including -Inf, as log_var would */
        sign1 = numeric_sign_internal(num1);
        sign2 = numeric_sign_internal(num2);
        if (sign1 < 0 || sign2 < 0)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_LOG),
                     errmsg("cannot take logarithm of a negative number")));
        /* fail on zero inputs, as log_var would */
        if (sign1 == 0 || sign2 == 0)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_LOG),
                     errmsg("cannot take logarithm of zero")));
        if (NUMERIC_IS_PINF(num1))
        {
            /* log(Inf, Inf) reduces to Inf/Inf, so it's NaN */
            if (NUMERIC_IS_PINF(num2))
                PG_RETURN_NUMERIC(make_result(&const_nan));
            /* log(Inf, finite-positive) is zero (we don't throw underflow) */
            PG_RETURN_NUMERIC(make_result(&const_zero));
        }
        Assert(NUMERIC_IS_PINF(num2));
        /* log(finite-positive, Inf) is Inf */
        PG_RETURN_NUMERIC(make_result(&const_pinf));
    }
 
    /*
     * Initialize things
     */
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
    init_var(&result);
 
    /*
     * Call log_var() to compute and return the result; note it handles scale
     * selection itself.
     */
    log_var(&arg1, &arg2, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
/*
 * numeric_power() -
 *
 *  Raise x to the power of y
 */
Datum
numeric_power(PG_FUNCTION_ARGS)
{
    Numeric     num1 = PG_GETARG_NUMERIC(0);
    Numeric     num2 = PG_GETARG_NUMERIC(1);
    Numeric     res;
    NumericVar  arg1;
    NumericVar  arg2;
    NumericVar  result;
    int         sign1,
                sign2;
 
    /*
     * Handle NaN and infinities
     */
    if (NUMERIC_IS_SPECIAL(num1) || NUMERIC_IS_SPECIAL(num2))
    {
        /*
         * We follow the POSIX spec for pow(3), which says that NaN ^ 0 = 1,
         * and 1 ^ NaN = 1, while all other cases with NaN inputs yield NaN
         * (with no error).
         */
        if (NUMERIC_IS_NAN(num1))
        {
            if (!NUMERIC_IS_SPECIAL(num2))
            {
                init_var_from_num(num2, &arg2);
                if (cmp_var(&arg2, &const_zero) == 0)
                    PG_RETURN_NUMERIC(make_result(&const_one));
            }
            PG_RETURN_NUMERIC(make_result(&const_nan));
        }
        if (NUMERIC_IS_NAN(num2))
        {
            if (!NUMERIC_IS_SPECIAL(num1))
            {
                init_var_from_num(num1, &arg1);
                if (cmp_var(&arg1, &const_one) == 0)
                    PG_RETURN_NUMERIC(make_result(&const_one));
            }
            PG_RETURN_NUMERIC(make_result(&const_nan));
        }
        /* At least one input is infinite, but error rules still apply */
        sign1 = numeric_sign_internal(num1);
        sign2 = numeric_sign_internal(num2);
        if (sign1 == 0 && sign2 < 0)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                     errmsg("zero raised to a negative power is undefined")));
        if (sign1 < 0 && !numeric_is_integral(num2))
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                     errmsg("a negative number raised to a non-integer power yields a complex result")));
 
        /*
         * POSIX gives this series of rules for pow(3) with infinite inputs:
         *
         * For any value of y, if x is +1, 1.0 shall be returned.
         */
        if (!NUMERIC_IS_SPECIAL(num1))
        {
            init_var_from_num(num1, &arg1);
            if (cmp_var(&arg1, &const_one) == 0)
                PG_RETURN_NUMERIC(make_result(&const_one));
        }
 
        /*
         * For any value of x, if y is [-]0, 1.0 shall be returned.
         */
        if (sign2 == 0)
            PG_RETURN_NUMERIC(make_result(&const_one));
 
        /*
         * For any odd integer value of y > 0, if x is [-]0, [-]0 shall be
         * returned.  For y > 0 and not an odd integer, if x is [-]0, +0 shall
         * be returned.  (Since we don't deal in minus zero, we need not
         * distinguish these two cases.)
         */
        if (sign1 == 0 && sign2 > 0)
            PG_RETURN_NUMERIC(make_result(&const_zero));
 
        /*
         * If x is -1, and y is [-]Inf, 1.0 shall be returned.
         *
         * For |x| < 1, if y is -Inf, +Inf shall be returned.
         *
         * For |x| > 1, if y is -Inf, +0 shall be returned.
         *
         * For |x| < 1, if y is +Inf, +0 shall be returned.
         *
         * For |x| > 1, if y is +Inf, +Inf shall be returned.
         */
        if (NUMERIC_IS_INF(num2))
        {
            bool        abs_x_gt_one;
 
            if (NUMERIC_IS_SPECIAL(num1))
                abs_x_gt_one = true;    /* x is either Inf or -Inf */
            else
            {
                init_var_from_num(num1, &arg1);
                if (cmp_var(&arg1, &const_minus_one) == 0)
                    PG_RETURN_NUMERIC(make_result(&const_one));
                arg1.sign = NUMERIC_POS;    /* now arg1 = abs(x) */
                abs_x_gt_one = (cmp_var(&arg1, &const_one) > 0);
            }
            if (abs_x_gt_one == (sign2 > 0))
                PG_RETURN_NUMERIC(make_result(&const_pinf));
            else
                PG_RETURN_NUMERIC(make_result(&const_zero));
        }
 
        /*
         * For y < 0, if x is +Inf, +0 shall be returned.
         *
         * For y > 0, if x is +Inf, +Inf shall be returned.
         */
        if (NUMERIC_IS_PINF(num1))
        {
            if (sign2 > 0)
                PG_RETURN_NUMERIC(make_result(&const_pinf));
            else
                PG_RETURN_NUMERIC(make_result(&const_zero));
        }
 
        Assert(NUMERIC_IS_NINF(num1));
 
        /*
         * For y an odd integer < 0, if x is -Inf, -0 shall be returned.  For
         * y < 0 and not an odd integer, if x is -Inf, +0 shall be returned.
         * (Again, we need not distinguish these two cases.)
         */
        if (sign2 < 0)
            PG_RETURN_NUMERIC(make_result(&const_zero));
 
        /*
         * For y an odd integer > 0, if x is -Inf, -Inf shall be returned. For
         * y > 0 and not an odd integer, if x is -Inf, +Inf shall be returned.
         */
        init_var_from_num(num2, &arg2);
        if (arg2.ndigits > 0 && arg2.ndigits == arg2.weight + 1 &&
            (arg2.digits[arg2.ndigits - 1] & 1))
            PG_RETURN_NUMERIC(make_result(&const_ninf));
        else
            PG_RETURN_NUMERIC(make_result(&const_pinf));
    }
 
    /*
     * The SQL spec requires that we emit a particular SQLSTATE error code for
     * certain error conditions.  Specifically, we don't return a
     * divide-by-zero error code for 0 ^ -1.  Raising a negative number to a
     * non-integer power must produce the same error code, but that case is
     * handled in power_var().
     */
    sign1 = numeric_sign_internal(num1);
    sign2 = numeric_sign_internal(num2);
 
    if (sign1 == 0 && sign2 < 0)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                 errmsg("zero raised to a negative power is undefined")));
 
    /*
     * Initialize things
     */
    init_var(&result);
    init_var_from_num(num1, &arg1);
    init_var_from_num(num2, &arg2);
 
    /*
     * Call power_var() to compute and return the result; note it handles
     * scale selection itself.
     */
    power_var(&arg1, &arg2, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
/*
 * numeric_scale() -
 *
 *  Returns the scale, i.e. the count of decimal digits in the fractional part
 */
Datum
numeric_scale(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
 
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NULL();
 
    PG_RETURN_INT32(NUMERIC_DSCALE(num));
}
 
/*
 * Calculate minimum scale for value.
 */
static int
get_min_scale(NumericVar *var)
{
    int         min_scale;
    int         last_digit_pos;
 
    /*
     * Ordinarily, the input value will be "stripped" so that the last
     * NumericDigit is nonzero.  But we don't want to get into an infinite
     * loop if it isn't, so explicitly find the last nonzero digit.
     */
    last_digit_pos = var->ndigits - 1;
    while (last_digit_pos >= 0 &&
           var->digits[last_digit_pos] == 0)
        last_digit_pos--;
 
    if (last_digit_pos >= 0)
    {
        /* compute min_scale assuming that last ndigit has no zeroes */
        min_scale = (last_digit_pos - var->weight) * DEC_DIGITS;
 
        /*
         * We could get a negative result if there are no digits after the
         * decimal point.  In this case the min_scale must be zero.
         */
        if (min_scale > 0)
        {
            /*
             * Reduce min_scale if trailing digit(s) in last NumericDigit are
             * zero.
             */
            NumericDigit last_digit = var->digits[last_digit_pos];
 
            while (last_digit % 10 == 0)
            {
                min_scale--;
                last_digit /= 10;
            }
        }
        else
            min_scale = 0;
    }
    else
        min_scale = 0;          /* result if input is zero */
 
    return min_scale;
}
 
/*
 * Returns minimum scale required to represent supplied value without loss.
 */
Datum
numeric_min_scale(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  arg;
    int         min_scale;
 
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NULL();
 
    init_var_from_num(num, &arg);
    min_scale = get_min_scale(&arg);
    free_var(&arg);
 
    PG_RETURN_INT32(min_scale);
}
 
/*
 * Reduce scale of numeric value to represent supplied value without loss.
 */
Datum
numeric_trim_scale(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    Numeric     res;
    NumericVar  result;
 
    if (NUMERIC_IS_SPECIAL(num))
        PG_RETURN_NUMERIC(duplicate_numeric(num));
 
    init_var_from_num(num, &result);
    result.dscale = get_min_scale(&result);
    res = make_result(&result);
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
/*
 * Return a random numeric value in the range [rmin, rmax].
 */
Numeric
random_numeric(pg_prng_state *state, Numeric rmin, Numeric rmax)
{
    NumericVar  rmin_var;
    NumericVar  rmax_var;
    NumericVar  result;
    Numeric     res;
 
    /* Range bounds must not be NaN/infinity */
    if (NUMERIC_IS_SPECIAL(rmin))
    {
        if (NUMERIC_IS_NAN(rmin))
            ereport(ERROR,
                    errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                    errmsg("lower bound cannot be NaN"));
        else
            ereport(ERROR,
                    errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                    errmsg("lower bound cannot be infinity"));
    }
    if (NUMERIC_IS_SPECIAL(rmax))
    {
        if (NUMERIC_IS_NAN(rmax))
            ereport(ERROR,
                    errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                    errmsg("upper bound cannot be NaN"));
        else
            ereport(ERROR,
                    errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                    errmsg("upper bound cannot be infinity"));
    }
 
    /* Return a random value in the range [rmin, rmax] */
    init_var_from_num(rmin, &rmin_var);
    init_var_from_num(rmax, &rmax_var);
 
    init_var(&result);
 
    random_var(state, &rmin_var, &rmax_var, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    return res;
}
 
 
/* ----------------------------------------------------------------------
 *
 * Type conversion functions
 *
 * ----------------------------------------------------------------------
 */
 
Numeric
int64_to_numeric(int64 val)
{
    Numeric     res;
    NumericVar  result;
 
    init_var(&result);
 
    int64_to_numericvar(val, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    return res;
}
 
/*
 * Convert val1/(10**log10val2) to numeric.  This is much faster than normal
 * numeric division.
 */
Numeric
int64_div_fast_to_numeric(int64 val1, int log10val2)
{
    Numeric     res;
    NumericVar  result;
    int         rscale;
    int         w;
    int         m;
 
    init_var(&result);
 
    /* result scale */
    rscale = log10val2 < 0 ? 0 : log10val2;
 
    /* how much to decrease the weight by */
    w = log10val2 / DEC_DIGITS;
    /* how much is left to divide by */
    m = log10val2 % DEC_DIGITS;
    if (m < 0)
    {
        m += DEC_DIGITS;
        w--;
    }
 
    /*
     * If there is anything left to divide by (10^m with 0 < m < DEC_DIGITS),
     * multiply the dividend by 10^(DEC_DIGITS - m), and shift the weight by
     * one more.
     */
    if (m > 0)
    {
#if DEC_DIGITS == 4
        static const int pow10[] = {1, 10, 100, 1000};
#elif DEC_DIGITS == 2
        static const int pow10[] = {1, 10};
#elif DEC_DIGITS == 1
        static const int pow10[] = {1};
#else
#error unsupported NBASE
#endif
        int64       factor = pow10[DEC_DIGITS - m];
        int64       new_val1;
 
        StaticAssertDecl(lengthof(pow10) == DEC_DIGITS, "mismatch with DEC_DIGITS");
 
        if (unlikely(pg_mul_s64_overflow(val1, factor, &new_val1)))
        {
#ifdef HAVE_INT128
            /* do the multiplication using 128-bit integers */
            int128      tmp;
 
            tmp = (int128) val1 * (int128) factor;
 
            int128_to_numericvar(tmp, &result);
#else
            /* do the multiplication using numerics */
            NumericVar  tmp;
 
            init_var(&tmp);
 
            int64_to_numericvar(val1, &result);
            int64_to_numericvar(factor, &tmp);
            mul_var(&result, &tmp, &result, 0);
 
            free_var(&tmp);
#endif
        }
        else
            int64_to_numericvar(new_val1, &result);
 
        w++;
    }
    else
        int64_to_numericvar(val1, &result);
 
    result.weight -= w;
    result.dscale = rscale;
 
    res = make_result(&result);
 
    free_var(&result);
 
    return res;
}
 
Datum
int4_numeric(PG_FUNCTION_ARGS)
{
    int32       val = PG_GETARG_INT32(0);
 
    PG_RETURN_NUMERIC(int64_to_numeric(val));
}
 
int32
numeric_int4_opt_error(Numeric num, bool *have_error)
{
    NumericVar  x;
    int32       result;
 
    if (have_error)
        *have_error = false;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (have_error)
        {
            *have_error = true;
            return 0;
        }
        else
        {
            if (NUMERIC_IS_NAN(num))
                ereport(ERROR,
                        (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                         errmsg("cannot convert NaN to %s", "integer")));
            else
                ereport(ERROR,
                        (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                         errmsg("cannot convert infinity to %s", "integer")));
        }
    }
 
    /* Convert to variable format, then convert to int4 */
    init_var_from_num(num, &x);
 
    if (!numericvar_to_int32(&x, &result))
    {
        if (have_error)
        {
            *have_error = true;
            return 0;
        }
        else
        {
            ereport(ERROR,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("integer out of range")));
        }
    }
 
    return result;
}
 
Datum
numeric_int4(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
 
    PG_RETURN_INT32(numeric_int4_opt_error(num, NULL));
}
 
/*
 * Given a NumericVar, convert it to an int32. If the NumericVar
 * exceeds the range of an int32, false is returned, otherwise true is returned.
 * The input NumericVar is *not* free'd.
 */
static bool
numericvar_to_int32(const NumericVar *var, int32 *result)
{
    int64       val;
 
    if (!numericvar_to_int64(var, &val))
        return false;
 
    if (unlikely(val < PG_INT32_MIN) || unlikely(val > PG_INT32_MAX))
        return false;
 
    /* Down-convert to int4 */
    *result = (int32) val;
 
    return true;
}
 
Datum
int8_numeric(PG_FUNCTION_ARGS)
{
    int64       val = PG_GETARG_INT64(0);
 
    PG_RETURN_NUMERIC(int64_to_numeric(val));
}
 
int64
numeric_int8_opt_error(Numeric num, bool *have_error)
{
    NumericVar  x;
    int64       result;
 
    if (have_error)
        *have_error = false;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (have_error)
        {
            *have_error = true;
            return 0;
        }
        else
        {
            if (NUMERIC_IS_NAN(num))
                ereport(ERROR,
                        (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                         errmsg("cannot convert NaN to %s", "bigint")));
            else
                ereport(ERROR,
                        (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                         errmsg("cannot convert infinity to %s", "bigint")));
        }
    }
 
    /* Convert to variable format, then convert to int8 */
    init_var_from_num(num, &x);
 
    if (!numericvar_to_int64(&x, &result))
    {
        if (have_error)
        {
            *have_error = true;
            return 0;
        }
        else
        {
            ereport(ERROR,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("bigint out of range")));
        }
    }
 
    return result;
}
 
Datum
numeric_int8(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
 
    PG_RETURN_INT64(numeric_int8_opt_error(num, NULL));
}
 
 
Datum
int2_numeric(PG_FUNCTION_ARGS)
{
    int16       val = PG_GETARG_INT16(0);
 
    PG_RETURN_NUMERIC(int64_to_numeric(val));
}
 
 
Datum
numeric_int2(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  x;
    int64       val;
    int16       result;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_NAN(num))
            ereport(ERROR,
                    (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                     errmsg("cannot convert NaN to %s", "smallint")));
        else
            ereport(ERROR,
                    (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                     errmsg("cannot convert infinity to %s", "smallint")));
    }
 
    /* Convert to variable format and thence to int8 */
    init_var_from_num(num, &x);
 
    if (!numericvar_to_int64(&x, &val))
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("smallint out of range")));
 
    if (unlikely(val < PG_INT16_MIN) || unlikely(val > PG_INT16_MAX))
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("smallint out of range")));
 
    /* Down-convert to int2 */
    result = (int16) val;
 
    PG_RETURN_INT16(result);
}
 
 
Datum
float8_numeric(PG_FUNCTION_ARGS)
{
    float8      val = PG_GETARG_FLOAT8(0);
    Numeric     res;
    NumericVar  result;
    char        buf[DBL_DIG + 100];
    const char *endptr;
 
    if (isnan(val))
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    if (isinf(val))
    {
        if (val < 0)
            PG_RETURN_NUMERIC(make_result(&const_ninf));
        else
            PG_RETURN_NUMERIC(make_result(&const_pinf));
    }
 
    snprintf(buf, sizeof(buf), "%.*g", DBL_DIG, val);
 
    init_var(&result);
 
    /* Assume we need not worry about leading/trailing spaces */
    (void) set_var_from_str(buf, buf, &result, &endptr, NULL);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
Datum
numeric_float8(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    char       *tmp;
    Datum       result;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            PG_RETURN_FLOAT8(get_float8_infinity());
        else if (NUMERIC_IS_NINF(num))
            PG_RETURN_FLOAT8(-get_float8_infinity());
        else
            PG_RETURN_FLOAT8(get_float8_nan());
    }
 
    tmp = DatumGetCString(DirectFunctionCall1(numeric_out,
                                              NumericGetDatum(num)));
 
    result = DirectFunctionCall1(float8in, CStringGetDatum(tmp));
 
    pfree(tmp);
 
    PG_RETURN_DATUM(result);
}
 
 
/*
 * Convert numeric to float8; if out of range, return +/- HUGE_VAL
 *
 * (internal helper function, not directly callable from SQL)
 */
Datum
numeric_float8_no_overflow(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    double      val;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            val = HUGE_VAL;
        else if (NUMERIC_IS_NINF(num))
            val = -HUGE_VAL;
        else
            val = get_float8_nan();
    }
    else
    {
        NumericVar  x;
 
        init_var_from_num(num, &x);
        val = numericvar_to_double_no_overflow(&x);
    }
 
    PG_RETURN_FLOAT8(val);
}
 
Datum
float4_numeric(PG_FUNCTION_ARGS)
{
    float4      val = PG_GETARG_FLOAT4(0);
    Numeric     res;
    NumericVar  result;
    char        buf[FLT_DIG + 100];
    const char *endptr;
 
    if (isnan(val))
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    if (isinf(val))
    {
        if (val < 0)
            PG_RETURN_NUMERIC(make_result(&const_ninf));
        else
            PG_RETURN_NUMERIC(make_result(&const_pinf));
    }
 
    snprintf(buf, sizeof(buf), "%.*g", FLT_DIG, val);
 
    init_var(&result);
 
    /* Assume we need not worry about leading/trailing spaces */
    (void) set_var_from_str(buf, buf, &result, &endptr, NULL);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
}
 
 
Datum
numeric_float4(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    char       *tmp;
    Datum       result;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_PINF(num))
            PG_RETURN_FLOAT4(get_float4_infinity());
        else if (NUMERIC_IS_NINF(num))
            PG_RETURN_FLOAT4(-get_float4_infinity());
        else
            PG_RETURN_FLOAT4(get_float4_nan());
    }
 
    tmp = DatumGetCString(DirectFunctionCall1(numeric_out,
                                              NumericGetDatum(num)));
 
    result = DirectFunctionCall1(float4in, CStringGetDatum(tmp));
 
    pfree(tmp);
 
    PG_RETURN_DATUM(result);
}
 
 
Datum
numeric_pg_lsn(PG_FUNCTION_ARGS)
{
    Numeric     num = PG_GETARG_NUMERIC(0);
    NumericVar  x;
    XLogRecPtr  result;
 
    if (NUMERIC_IS_SPECIAL(num))
    {
        if (NUMERIC_IS_NAN(num))
            ereport(ERROR,
                    (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                     errmsg("cannot convert NaN to %s", "pg_lsn")));
        else
            ereport(ERROR,
                    (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
                     errmsg("cannot convert infinity to %s", "pg_lsn")));
    }
 
    /* Convert to variable format and thence to pg_lsn */
    init_var_from_num(num, &x);
 
    if (!numericvar_to_uint64(&x, (uint64 *) &result))
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                 errmsg("pg_lsn out of range")));
 
    PG_RETURN_LSN(result);
}
 
 
/* ----------------------------------------------------------------------
 *
 * Aggregate functions
 *
 * The transition datatype for all these aggregates is declared as INTERNAL.
 * Actually, it's a pointer to a NumericAggState allocated in the aggregate
 * context.  The digit buffers for the NumericVars will be there too.
 *
 * On platforms which support 128-bit integers some aggregates instead use a
 * 128-bit integer based transition datatype to speed up calculations.
 *
 * ----------------------------------------------------------------------
 */
 
typedef struct NumericAggState
{
    bool        calcSumX2;      /* if true, calculate sumX2 */
    MemoryContext agg_context;  /* context we're calculating in */
    int64       N;              /* count of processed numbers */
    NumericSumAccum sumX;       /* sum of processed numbers */
    NumericSumAccum sumX2;      /* sum of squares of processed numbers */
    int         maxScale;       /* maximum scale seen so far */
    int64       maxScaleCount;  /* number of values seen with maximum scale */
    /* These counts are *not* included in N!  Use NA_TOTAL_COUNT() as needed */
    int64       NaNcount;       /* count of NaN values */
    int64       pInfcount;      /* count of +Inf values */
    int64       nInfcount;      /* count of -Inf values */
} NumericAggState;
 
#define NA_TOTAL_COUNT(na) \
    ((na)->N + (na)->NaNcount + (na)->pInfcount + (na)->nInfcount)
 
/*
 * Prepare state data for a numeric aggregate function that needs to compute
 * sum, count and optionally sum of squares of the input.
 */
static NumericAggState *
makeNumericAggState(FunctionCallInfo fcinfo, bool calcSumX2)
{
    NumericAggState *state;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    old_context = MemoryContextSwitchTo(agg_context);
 
    state = (NumericAggState *) palloc0(sizeof(NumericAggState));
    state->calcSumX2 = calcSumX2;
    state->agg_context = agg_context;
 
    MemoryContextSwitchTo(old_context);
 
    return state;
}
 
/*
 * Like makeNumericAggState(), but allocate the state in the current memory
 * context.
 */
static NumericAggState *
makeNumericAggStateCurrentContext(bool calcSumX2)
{
    NumericAggState *state;
 
    state = (NumericAggState *) palloc0(sizeof(NumericAggState));
    state->calcSumX2 = calcSumX2;
    state->agg_context = CurrentMemoryContext;
 
    return state;
}
 
/*
 * Accumulate a new input value for numeric aggregate functions.
 */
static void
do_numeric_accum(NumericAggState *state, Numeric newval)
{
    NumericVar  X;
    NumericVar  X2;
    MemoryContext old_context;
 
    /* Count NaN/infinity inputs separately from all else */
    if (NUMERIC_IS_SPECIAL(newval))
    {
        if (NUMERIC_IS_PINF(newval))
            state->pInfcount++;
        else if (NUMERIC_IS_NINF(newval))
            state->nInfcount++;
        else
            state->NaNcount++;
        return;
    }
 
    /* load processed number in short-lived context */
    init_var_from_num(newval, &X);
 
    /*
     * Track the highest input dscale that we've seen, to support inverse
     * transitions (see do_numeric_discard).
     */
    if (X.dscale > state->maxScale)
    {
        state->maxScale = X.dscale;
        state->maxScaleCount = 1;
    }
    else if (X.dscale == state->maxScale)
        state->maxScaleCount++;
 
    /* if we need X^2, calculate that in short-lived context */
    if (state->calcSumX2)
    {
        init_var(&X2);
        mul_var(&X, &X, &X2, X.dscale * 2);
    }
 
    /* The rest of this needs to work in the aggregate context */
    old_context = MemoryContextSwitchTo(state->agg_context);
 
    state->N++;
 
    /* Accumulate sums */
    accum_sum_add(&(state->sumX), &X);
 
    if (state->calcSumX2)
        accum_sum_add(&(state->sumX2), &X2);
 
    MemoryContextSwitchTo(old_context);
}
 
/*
 * Attempt to remove an input value from the aggregated state.
 *
 * If the value cannot be removed then the function will return false; the
 * possible reasons for failing are described below.
 *
 * If we aggregate the values 1.01 and 2 then the result will be 3.01.
 * If we are then asked to un-aggregate the 1.01 then we must fail as we
 * won't be able to tell what the new aggregated value's dscale should be.
 * We don't want to return 2.00 (dscale = 2), since the sum's dscale would
 * have been zero if we'd really aggregated only 2.
 *
 * Note: alternatively, we could count the number of inputs with each possible
 * dscale (up to some sane limit).  Not yet clear if it's worth the trouble.
 */
static bool
do_numeric_discard(NumericAggState *state, Numeric newval)
{
    NumericVar  X;
    NumericVar  X2;
    MemoryContext old_context;
 
    /* Count NaN/infinity inputs separately from all else */
    if (NUMERIC_IS_SPECIAL(newval))
    {
        if (NUMERIC_IS_PINF(newval))
            state->pInfcount--;
        else if (NUMERIC_IS_NINF(newval))
            state->nInfcount--;
        else
            state->NaNcount--;
        return true;
    }
 
    /* load processed number in short-lived context */
    init_var_from_num(newval, &X);
 
    /*
     * state->sumX's dscale is the maximum dscale of any of the inputs.
     * Removing the last input with that dscale would require us to recompute
     * the maximum dscale of the *remaining* inputs, which we cannot do unless
     * no more non-NaN inputs remain at all.  So we report a failure instead,
     * and force the aggregation to be redone from scratch.
     */
    if (X.dscale == state->maxScale)
    {
        if (state->maxScaleCount > 1 || state->maxScale == 0)
        {
            /*
             * Some remaining inputs have same dscale, or dscale hasn't gotten
             * above zero anyway
             */
            state->maxScaleCount--;
        }
        else if (state->N == 1)
        {
            /* No remaining non-NaN inputs at all, so reset maxScale */
            state->maxScale = 0;
            state->maxScaleCount = 0;
        }
        else
        {
            /* Correct new maxScale is uncertain, must fail */
            return false;
        }
    }
 
    /* if we need X^2, calculate that in short-lived context */
    if (state->calcSumX2)
    {
        init_var(&X2);
        mul_var(&X, &X, &X2, X.dscale * 2);
    }
 
    /* The rest of this needs to work in the aggregate context */
    old_context = MemoryContextSwitchTo(state->agg_context);
 
    if (state->N-- > 1)
    {
        /* Negate X, to subtract it from the sum */
        X.sign = (X.sign == NUMERIC_POS ? NUMERIC_NEG : NUMERIC_POS);
        accum_sum_add(&(state->sumX), &X);
 
        if (state->calcSumX2)
        {
            /* Negate X^2. X^2 is always positive */
            X2.sign = NUMERIC_NEG;
            accum_sum_add(&(state->sumX2), &X2);
        }
    }
    else
    {
        /* Zero the sums */
        Assert(state->N == 0);
 
        accum_sum_reset(&state->sumX);
        if (state->calcSumX2)
            accum_sum_reset(&state->sumX2);
    }
 
    MemoryContextSwitchTo(old_context);
 
    return true;
}
 
/*
 * Generic transition function for numeric aggregates that require sumX2.
 */
Datum
numeric_accum(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makeNumericAggState(fcinfo, true);
 
    if (!PG_ARGISNULL(1))
        do_numeric_accum(state, PG_GETARG_NUMERIC(1));
 
    PG_RETURN_POINTER(state);
}
 
/*
 * Generic combine function for numeric aggregates which require sumX2
 */
Datum
numeric_combine(PG_FUNCTION_ARGS)
{
    NumericAggState *state1;
    NumericAggState *state2;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state1 = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
    state2 = PG_ARGISNULL(1) ? NULL : (NumericAggState *) PG_GETARG_POINTER(1);
 
    if (state2 == NULL)
        PG_RETURN_POINTER(state1);
 
    /* manually copy all fields from state2 to state1 */
    if (state1 == NULL)
    {
        old_context = MemoryContextSwitchTo(agg_context);
 
        state1 = makeNumericAggStateCurrentContext(true);
        state1->N = state2->N;
        state1->NaNcount = state2->NaNcount;
        state1->pInfcount = state2->pInfcount;
        state1->nInfcount = state2->nInfcount;
        state1->maxScale = state2->maxScale;
        state1->maxScaleCount = state2->maxScaleCount;
 
        accum_sum_copy(&state1->sumX, &state2->sumX);
        accum_sum_copy(&state1->sumX2, &state2->sumX2);
 
        MemoryContextSwitchTo(old_context);
 
        PG_RETURN_POINTER(state1);
    }
 
    state1->N += state2->N;
    state1->NaNcount += state2->NaNcount;
    state1->pInfcount += state2->pInfcount;
    state1->nInfcount += state2->nInfcount;
 
    if (state2->N > 0)
    {
        /*
         * These are currently only needed for moving aggregates, but let's do
         * the right thing anyway...
         */
        if (state2->maxScale > state1->maxScale)
        {
            state1->maxScale = state2->maxScale;
            state1->maxScaleCount = state2->maxScaleCount;
        }
        else if (state2->maxScale == state1->maxScale)
            state1->maxScaleCount += state2->maxScaleCount;
 
        /* The rest of this needs to work in the aggregate context */
        old_context = MemoryContextSwitchTo(agg_context);
 
        /* Accumulate sums */
        accum_sum_combine(&state1->sumX, &state2->sumX);
        accum_sum_combine(&state1->sumX2, &state2->sumX2);
 
        MemoryContextSwitchTo(old_context);
    }
    PG_RETURN_POINTER(state1);
}
 
/*
 * Generic transition function for numeric aggregates that don't require sumX2.
 */
Datum
numeric_avg_accum(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makeNumericAggState(fcinfo, false);
 
    if (!PG_ARGISNULL(1))
        do_numeric_accum(state, PG_GETARG_NUMERIC(1));
 
    PG_RETURN_POINTER(state);
}
 
/*
 * Combine function for numeric aggregates which don't require sumX2
 */
Datum
numeric_avg_combine(PG_FUNCTION_ARGS)
{
    NumericAggState *state1;
    NumericAggState *state2;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state1 = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
    state2 = PG_ARGISNULL(1) ? NULL : (NumericAggState *) PG_GETARG_POINTER(1);
 
    if (state2 == NULL)
        PG_RETURN_POINTER(state1);
 
    /* manually copy all fields from state2 to state1 */
    if (state1 == NULL)
    {
        old_context = MemoryContextSwitchTo(agg_context);
 
        state1 = makeNumericAggStateCurrentContext(false);
        state1->N = state2->N;
        state1->NaNcount = state2->NaNcount;
        state1->pInfcount = state2->pInfcount;
        state1->nInfcount = state2->nInfcount;
        state1->maxScale = state2->maxScale;
        state1->maxScaleCount = state2->maxScaleCount;
 
        accum_sum_copy(&state1->sumX, &state2->sumX);
 
        MemoryContextSwitchTo(old_context);
 
        PG_RETURN_POINTER(state1);
    }
 
    state1->N += state2->N;
    state1->NaNcount += state2->NaNcount;
    state1->pInfcount += state2->pInfcount;
    state1->nInfcount += state2->nInfcount;
 
    if (state2->N > 0)
    {
        /*
         * These are currently only needed for moving aggregates, but let's do
         * the right thing anyway...
         */
        if (state2->maxScale > state1->maxScale)
        {
            state1->maxScale = state2->maxScale;
            state1->maxScaleCount = state2->maxScaleCount;
        }
        else if (state2->maxScale == state1->maxScale)
            state1->maxScaleCount += state2->maxScaleCount;
 
        /* The rest of this needs to work in the aggregate context */
        old_context = MemoryContextSwitchTo(agg_context);
 
        /* Accumulate sums */
        accum_sum_combine(&state1->sumX, &state2->sumX);
 
        MemoryContextSwitchTo(old_context);
    }
    PG_RETURN_POINTER(state1);
}
 
/*
 * numeric_avg_serialize
 *      Serialize NumericAggState for numeric aggregates that don't require
 *      sumX2.
 */
Datum
numeric_avg_serialize(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    StringInfoData buf;
    bytea      *result;
    NumericVar  tmp_var;
 
    /* Ensure we disallow calling when not in aggregate context */
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state = (NumericAggState *) PG_GETARG_POINTER(0);
 
    init_var(&tmp_var);
 
    pq_begintypsend(&buf);
 
    /* N */
    pq_sendint64(&buf, state->N);
 
    /* sumX */
    accum_sum_final(&state->sumX, &tmp_var);
    numericvar_serialize(&buf, &tmp_var);
 
    /* maxScale */
    pq_sendint32(&buf, state->maxScale);
 
    /* maxScaleCount */
    pq_sendint64(&buf, state->maxScaleCount);
 
    /* NaNcount */
    pq_sendint64(&buf, state->NaNcount);
 
    /* pInfcount */
    pq_sendint64(&buf, state->pInfcount);
 
    /* nInfcount */
    pq_sendint64(&buf, state->nInfcount);
 
    result = pq_endtypsend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_BYTEA_P(result);
}
 
/*
 * numeric_avg_deserialize
 *      Deserialize bytea into NumericAggState for numeric aggregates that
 *      don't require sumX2.
 */
Datum
numeric_avg_deserialize(PG_FUNCTION_ARGS)
{
    bytea      *sstate;
    NumericAggState *result;
    StringInfoData buf;
    NumericVar  tmp_var;
 
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    sstate = PG_GETARG_BYTEA_PP(0);
 
    init_var(&tmp_var);
 
    /*
     * Initialize a StringInfo so that we can "receive" it using the standard
     * recv-function infrastructure.
     */
    initReadOnlyStringInfo(&buf, VARDATA_ANY(sstate),
                           VARSIZE_ANY_EXHDR(sstate));
 
    result = makeNumericAggStateCurrentContext(false);
 
    /* N */
    result->N = pq_getmsgint64(&buf);
 
    /* sumX */
    numericvar_deserialize(&buf, &tmp_var);
    accum_sum_add(&(result->sumX), &tmp_var);
 
    /* maxScale */
    result->maxScale = pq_getmsgint(&buf, 4);
 
    /* maxScaleCount */
    result->maxScaleCount = pq_getmsgint64(&buf);
 
    /* NaNcount */
    result->NaNcount = pq_getmsgint64(&buf);
 
    /* pInfcount */
    result->pInfcount = pq_getmsgint64(&buf);
 
    /* nInfcount */
    result->nInfcount = pq_getmsgint64(&buf);
 
    pq_getmsgend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_POINTER(result);
}
 
/*
 * numeric_serialize
 *      Serialization function for NumericAggState for numeric aggregates that
 *      require sumX2.
 */
Datum
numeric_serialize(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    StringInfoData buf;
    bytea      *result;
    NumericVar  tmp_var;
 
    /* Ensure we disallow calling when not in aggregate context */
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state = (NumericAggState *) PG_GETARG_POINTER(0);
 
    init_var(&tmp_var);
 
    pq_begintypsend(&buf);
 
    /* N */
    pq_sendint64(&buf, state->N);
 
    /* sumX */
    accum_sum_final(&state->sumX, &tmp_var);
    numericvar_serialize(&buf, &tmp_var);
 
    /* sumX2 */
    accum_sum_final(&state->sumX2, &tmp_var);
    numericvar_serialize(&buf, &tmp_var);
 
    /* maxScale */
    pq_sendint32(&buf, state->maxScale);
 
    /* maxScaleCount */
    pq_sendint64(&buf, state->maxScaleCount);
 
    /* NaNcount */
    pq_sendint64(&buf, state->NaNcount);
 
    /* pInfcount */
    pq_sendint64(&buf, state->pInfcount);
 
    /* nInfcount */
    pq_sendint64(&buf, state->nInfcount);
 
    result = pq_endtypsend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_BYTEA_P(result);
}
 
/*
 * numeric_deserialize
 *      Deserialization function for NumericAggState for numeric aggregates that
 *      require sumX2.
 */
Datum
numeric_deserialize(PG_FUNCTION_ARGS)
{
    bytea      *sstate;
    NumericAggState *result;
    StringInfoData buf;
    NumericVar  tmp_var;
 
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    sstate = PG_GETARG_BYTEA_PP(0);
 
    init_var(&tmp_var);
 
    /*
     * Initialize a StringInfo so that we can "receive" it using the standard
     * recv-function infrastructure.
     */
    initReadOnlyStringInfo(&buf, VARDATA_ANY(sstate),
                           VARSIZE_ANY_EXHDR(sstate));
 
    result = makeNumericAggStateCurrentContext(false);
 
    /* N */
    result->N = pq_getmsgint64(&buf);
 
    /* sumX */
    numericvar_deserialize(&buf, &tmp_var);
    accum_sum_add(&(result->sumX), &tmp_var);
 
    /* sumX2 */
    numericvar_deserialize(&buf, &tmp_var);
    accum_sum_add(&(result->sumX2), &tmp_var);
 
    /* maxScale */
    result->maxScale = pq_getmsgint(&buf, 4);
 
    /* maxScaleCount */
    result->maxScaleCount = pq_getmsgint64(&buf);
 
    /* NaNcount */
    result->NaNcount = pq_getmsgint64(&buf);
 
    /* pInfcount */
    result->pInfcount = pq_getmsgint64(&buf);
 
    /* nInfcount */
    result->nInfcount = pq_getmsgint64(&buf);
 
    pq_getmsgend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_POINTER(result);
}
 
/*
 * Generic inverse transition function for numeric aggregates
 * (with or without requirement for X^2).
 */
Datum
numeric_accum_inv(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* Should not get here with no state */
    if (state == NULL)
        elog(ERROR, "numeric_accum_inv called with NULL state");
 
    if (!PG_ARGISNULL(1))
    {
        /* If we fail to perform the inverse transition, return NULL */
        if (!do_numeric_discard(state, PG_GETARG_NUMERIC(1)))
            PG_RETURN_NULL();
    }
 
    PG_RETURN_POINTER(state);
}
 
 
/*
 * Integer data types in general use Numeric accumulators to share code
 * and avoid risk of overflow.
 *
 * However for performance reasons optimized special-purpose accumulator
 * routines are used when possible.
 *
 * On platforms with 128-bit integer support, the 128-bit routines will be
 * used when sum(X) or sum(X*X) fit into 128-bit.
 *
 * For 16 and 32 bit inputs, the N and sum(X) fit into 64-bit so the 64-bit
 * accumulators will be used for SUM and AVG of these data types.
 */
 
#ifdef HAVE_INT128
typedef struct Int128AggState
{
    bool        calcSumX2;      /* if true, calculate sumX2 */
    int64       N;              /* count of processed numbers */
    int128      sumX;           /* sum of processed numbers */
    int128      sumX2;          /* sum of squares of processed numbers */
} Int128AggState;
 
/*
 * Prepare state data for a 128-bit aggregate function that needs to compute
 * sum, count and optionally sum of squares of the input.
 */
static Int128AggState *
makeInt128AggState(FunctionCallInfo fcinfo, bool calcSumX2)
{
    Int128AggState *state;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    old_context = MemoryContextSwitchTo(agg_context);
 
    state = (Int128AggState *) palloc0(sizeof(Int128AggState));
    state->calcSumX2 = calcSumX2;
 
    MemoryContextSwitchTo(old_context);
 
    return state;
}
 
/*
 * Like makeInt128AggState(), but allocate the state in the current memory
 * context.
 */
static Int128AggState *
makeInt128AggStateCurrentContext(bool calcSumX2)
{
    Int128AggState *state;
 
    state = (Int128AggState *) palloc0(sizeof(Int128AggState));
    state->calcSumX2 = calcSumX2;
 
    return state;
}
 
/*
 * Accumulate a new input value for 128-bit aggregate functions.
 */
static void
do_int128_accum(Int128AggState *state, int128 newval)
{
    if (state->calcSumX2)
        state->sumX2 += newval * newval;
 
    state->sumX += newval;
    state->N++;
}
 
/*
 * Remove an input value from the aggregated state.
 */
static void
do_int128_discard(Int128AggState *state, int128 newval)
{
    if (state->calcSumX2)
        state->sumX2 -= newval * newval;
 
    state->sumX -= newval;
    state->N--;
}
 
typedef Int128AggState PolyNumAggState;
#define makePolyNumAggState makeInt128AggState
#define makePolyNumAggStateCurrentContext makeInt128AggStateCurrentContext
#else
typedef NumericAggState PolyNumAggState;
#define makePolyNumAggState makeNumericAggState
#define makePolyNumAggStateCurrentContext makeNumericAggStateCurrentContext
#endif
 
Datum
int2_accum(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makePolyNumAggState(fcinfo, true);
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_accum(state, (int128) PG_GETARG_INT16(1));
#else
        do_numeric_accum(state, int64_to_numeric(PG_GETARG_INT16(1)));
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
int4_accum(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makePolyNumAggState(fcinfo, true);
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_accum(state, (int128) PG_GETARG_INT32(1));
#else
        do_numeric_accum(state, int64_to_numeric(PG_GETARG_INT32(1)));
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
int8_accum(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makeNumericAggState(fcinfo, true);
 
    if (!PG_ARGISNULL(1))
        do_numeric_accum(state, int64_to_numeric(PG_GETARG_INT64(1)));
 
    PG_RETURN_POINTER(state);
}
 
/*
 * Combine function for numeric aggregates which require sumX2
 */
Datum
numeric_poly_combine(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state1;
    PolyNumAggState *state2;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state1 = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
    state2 = PG_ARGISNULL(1) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(1);
 
    if (state2 == NULL)
        PG_RETURN_POINTER(state1);
 
    /* manually copy all fields from state2 to state1 */
    if (state1 == NULL)
    {
        old_context = MemoryContextSwitchTo(agg_context);
 
        state1 = makePolyNumAggState(fcinfo, true);
        state1->N = state2->N;
 
#ifdef HAVE_INT128
        state1->sumX = state2->sumX;
        state1->sumX2 = state2->sumX2;
#else
        accum_sum_copy(&state1->sumX, &state2->sumX);
        accum_sum_copy(&state1->sumX2, &state2->sumX2);
#endif
 
        MemoryContextSwitchTo(old_context);
 
        PG_RETURN_POINTER(state1);
    }
 
    if (state2->N > 0)
    {
        state1->N += state2->N;
 
#ifdef HAVE_INT128
        state1->sumX += state2->sumX;
        state1->sumX2 += state2->sumX2;
#else
        /* The rest of this needs to work in the aggregate context */
        old_context = MemoryContextSwitchTo(agg_context);
 
        /* Accumulate sums */
        accum_sum_combine(&state1->sumX, &state2->sumX);
        accum_sum_combine(&state1->sumX2, &state2->sumX2);
 
        MemoryContextSwitchTo(old_context);
#endif
 
    }
    PG_RETURN_POINTER(state1);
}
 
/*
 * numeric_poly_serialize
 *      Serialize PolyNumAggState into bytea for aggregate functions which
 *      require sumX2.
 */
Datum
numeric_poly_serialize(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
    StringInfoData buf;
    bytea      *result;
    NumericVar  tmp_var;
 
    /* Ensure we disallow calling when not in aggregate context */
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state = (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /*
     * If the platform supports int128 then sumX and sumX2 will be a 128 bit
     * integer type. Here we'll convert that into a numeric type so that the
     * combine state is in the same format for both int128 enabled machines
     * and machines which don't support that type. The logic here is that one
     * day we might like to send these over to another server for further
     * processing and we want a standard format to work with.
     */
 
    init_var(&tmp_var);
 
    pq_begintypsend(&buf);
 
    /* N */
    pq_sendint64(&buf, state->N);
 
    /* sumX */
#ifdef HAVE_INT128
    int128_to_numericvar(state->sumX, &tmp_var);
#else
    accum_sum_final(&state->sumX, &tmp_var);
#endif
    numericvar_serialize(&buf, &tmp_var);
 
    /* sumX2 */
#ifdef HAVE_INT128
    int128_to_numericvar(state->sumX2, &tmp_var);
#else
    accum_sum_final(&state->sumX2, &tmp_var);
#endif
    numericvar_serialize(&buf, &tmp_var);
 
    result = pq_endtypsend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_BYTEA_P(result);
}
 
/*
 * numeric_poly_deserialize
 *      Deserialize PolyNumAggState from bytea for aggregate functions which
 *      require sumX2.
 */
Datum
numeric_poly_deserialize(PG_FUNCTION_ARGS)
{
    bytea      *sstate;
    PolyNumAggState *result;
    StringInfoData buf;
    NumericVar  tmp_var;
 
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    sstate = PG_GETARG_BYTEA_PP(0);
 
    init_var(&tmp_var);
 
    /*
     * Initialize a StringInfo so that we can "receive" it using the standard
     * recv-function infrastructure.
     */
    initReadOnlyStringInfo(&buf, VARDATA_ANY(sstate),
                           VARSIZE_ANY_EXHDR(sstate));
 
    result = makePolyNumAggStateCurrentContext(false);
 
    /* N */
    result->N = pq_getmsgint64(&buf);
 
    /* sumX */
    numericvar_deserialize(&buf, &tmp_var);
#ifdef HAVE_INT128
    numericvar_to_int128(&tmp_var, &result->sumX);
#else
    accum_sum_add(&result->sumX, &tmp_var);
#endif
 
    /* sumX2 */
    numericvar_deserialize(&buf, &tmp_var);
#ifdef HAVE_INT128
    numericvar_to_int128(&tmp_var, &result->sumX2);
#else
    accum_sum_add(&result->sumX2, &tmp_var);
#endif
 
    pq_getmsgend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_POINTER(result);
}
 
/*
 * Transition function for int8 input when we don't need sumX2.
 */
Datum
int8_avg_accum(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Create the state data on the first call */
    if (state == NULL)
        state = makePolyNumAggState(fcinfo, false);
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_accum(state, (int128) PG_GETARG_INT64(1));
#else
        do_numeric_accum(state, int64_to_numeric(PG_GETARG_INT64(1)));
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
/*
 * Combine function for PolyNumAggState for aggregates which don't require
 * sumX2
 */
Datum
int8_avg_combine(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state1;
    PolyNumAggState *state2;
    MemoryContext agg_context;
    MemoryContext old_context;
 
    if (!AggCheckCallContext(fcinfo, &agg_context))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state1 = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
    state2 = PG_ARGISNULL(1) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(1);
 
    if (state2 == NULL)
        PG_RETURN_POINTER(state1);
 
    /* manually copy all fields from state2 to state1 */
    if (state1 == NULL)
    {
        old_context = MemoryContextSwitchTo(agg_context);
 
        state1 = makePolyNumAggState(fcinfo, false);
        state1->N = state2->N;
 
#ifdef HAVE_INT128
        state1->sumX = state2->sumX;
#else
        accum_sum_copy(&state1->sumX, &state2->sumX);
#endif
        MemoryContextSwitchTo(old_context);
 
        PG_RETURN_POINTER(state1);
    }
 
    if (state2->N > 0)
    {
        state1->N += state2->N;
 
#ifdef HAVE_INT128
        state1->sumX += state2->sumX;
#else
        /* The rest of this needs to work in the aggregate context */
        old_context = MemoryContextSwitchTo(agg_context);
 
        /* Accumulate sums */
        accum_sum_combine(&state1->sumX, &state2->sumX);
 
        MemoryContextSwitchTo(old_context);
#endif
 
    }
    PG_RETURN_POINTER(state1);
}
 
/*
 * int8_avg_serialize
 *      Serialize PolyNumAggState into bytea using the standard
 *      recv-function infrastructure.
 */
Datum
int8_avg_serialize(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
    StringInfoData buf;
    bytea      *result;
    NumericVar  tmp_var;
 
    /* Ensure we disallow calling when not in aggregate context */
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    state = (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /*
     * If the platform supports int128 then sumX will be a 128 integer type.
     * Here we'll convert that into a numeric type so that the combine state
     * is in the same format for both int128 enabled machines and machines
     * which don't support that type. The logic here is that one day we might
     * like to send these over to another server for further processing and we
     * want a standard format to work with.
     */
 
    init_var(&tmp_var);
 
    pq_begintypsend(&buf);
 
    /* N */
    pq_sendint64(&buf, state->N);
 
    /* sumX */
#ifdef HAVE_INT128
    int128_to_numericvar(state->sumX, &tmp_var);
#else
    accum_sum_final(&state->sumX, &tmp_var);
#endif
    numericvar_serialize(&buf, &tmp_var);
 
    result = pq_endtypsend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_BYTEA_P(result);
}
 
/*
 * int8_avg_deserialize
 *      Deserialize bytea back into PolyNumAggState.
 */
Datum
int8_avg_deserialize(PG_FUNCTION_ARGS)
{
    bytea      *sstate;
    PolyNumAggState *result;
    StringInfoData buf;
    NumericVar  tmp_var;
 
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    sstate = PG_GETARG_BYTEA_PP(0);
 
    init_var(&tmp_var);
 
    /*
     * Initialize a StringInfo so that we can "receive" it using the standard
     * recv-function infrastructure.
     */
    initReadOnlyStringInfo(&buf, VARDATA_ANY(sstate),
                           VARSIZE_ANY_EXHDR(sstate));
 
    result = makePolyNumAggStateCurrentContext(false);
 
    /* N */
    result->N = pq_getmsgint64(&buf);
 
    /* sumX */
    numericvar_deserialize(&buf, &tmp_var);
#ifdef HAVE_INT128
    numericvar_to_int128(&tmp_var, &result->sumX);
#else
    accum_sum_add(&result->sumX, &tmp_var);
#endif
 
    pq_getmsgend(&buf);
 
    free_var(&tmp_var);
 
    PG_RETURN_POINTER(result);
}
 
/*
 * Inverse transition functions to go with the above.
 */
 
Datum
int2_accum_inv(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Should not get here with no state */
    if (state == NULL)
        elog(ERROR, "int2_accum_inv called with NULL state");
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_discard(state, (int128) PG_GETARG_INT16(1));
#else
        /* Should never fail, all inputs have dscale 0 */
        if (!do_numeric_discard(state, int64_to_numeric(PG_GETARG_INT16(1))))
            elog(ERROR, "do_numeric_discard failed unexpectedly");
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
int4_accum_inv(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Should not get here with no state */
    if (state == NULL)
        elog(ERROR, "int4_accum_inv called with NULL state");
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_discard(state, (int128) PG_GETARG_INT32(1));
#else
        /* Should never fail, all inputs have dscale 0 */
        if (!do_numeric_discard(state, int64_to_numeric(PG_GETARG_INT32(1))))
            elog(ERROR, "do_numeric_discard failed unexpectedly");
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
int8_accum_inv(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* Should not get here with no state */
    if (state == NULL)
        elog(ERROR, "int8_accum_inv called with NULL state");
 
    if (!PG_ARGISNULL(1))
    {
        /* Should never fail, all inputs have dscale 0 */
        if (!do_numeric_discard(state, int64_to_numeric(PG_GETARG_INT64(1))))
            elog(ERROR, "do_numeric_discard failed unexpectedly");
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
int8_avg_accum_inv(PG_FUNCTION_ARGS)
{
    PolyNumAggState *state;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* Should not get here with no state */
    if (state == NULL)
        elog(ERROR, "int8_avg_accum_inv called with NULL state");
 
    if (!PG_ARGISNULL(1))
    {
#ifdef HAVE_INT128
        do_int128_discard(state, (int128) PG_GETARG_INT64(1));
#else
        /* Should never fail, all inputs have dscale 0 */
        if (!do_numeric_discard(state, int64_to_numeric(PG_GETARG_INT64(1))))
            elog(ERROR, "do_numeric_discard failed unexpectedly");
#endif
    }
 
    PG_RETURN_POINTER(state);
}
 
Datum
numeric_poly_sum(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    Numeric     res;
    NumericVar  result;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* If there were no non-null inputs, return NULL */
    if (state == NULL || state->N == 0)
        PG_RETURN_NULL();
 
    init_var(&result);
 
    int128_to_numericvar(state->sumX, &result);
 
    res = make_result(&result);
 
    free_var(&result);
 
    PG_RETURN_NUMERIC(res);
#else
    return numeric_sum(fcinfo);
#endif
}
 
Datum
numeric_poly_avg(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    NumericVar  result;
    Datum       countd,
                sumd;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    /* If there were no non-null inputs, return NULL */
    if (state == NULL || state->N == 0)
        PG_RETURN_NULL();
 
    init_var(&result);
 
    int128_to_numericvar(state->sumX, &result);
 
    countd = NumericGetDatum(int64_to_numeric(state->N));
    sumd = NumericGetDatum(make_result(&result));
 
    free_var(&result);
 
    PG_RETURN_DATUM(DirectFunctionCall2(numeric_div, sumd, countd));
#else
    return numeric_avg(fcinfo);
#endif
}
 
Datum
numeric_avg(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    Datum       N_datum;
    Datum       sumX_datum;
    NumericVar  sumX_var;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* If there were no non-null inputs, return NULL */
    if (state == NULL || NA_TOTAL_COUNT(state) == 0)
        PG_RETURN_NULL();
 
    if (state->NaNcount > 0)    /* there was at least one NaN input */
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    /* adding plus and minus infinities gives NaN */
    if (state->pInfcount > 0 && state->nInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_nan));
    if (state->pInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_pinf));
    if (state->nInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_ninf));
 
    N_datum = NumericGetDatum(int64_to_numeric(state->N));
 
    init_var(&sumX_var);
    accum_sum_final(&state->sumX, &sumX_var);
    sumX_datum = NumericGetDatum(make_result(&sumX_var));
    free_var(&sumX_var);
 
    PG_RETURN_DATUM(DirectFunctionCall2(numeric_div, sumX_datum, N_datum));
}
 
Datum
numeric_sum(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    NumericVar  sumX_var;
    Numeric     result;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    /* If there were no non-null inputs, return NULL */
    if (state == NULL || NA_TOTAL_COUNT(state) == 0)
        PG_RETURN_NULL();
 
    if (state->NaNcount > 0)    /* there was at least one NaN input */
        PG_RETURN_NUMERIC(make_result(&const_nan));
 
    /* adding plus and minus infinities gives NaN */
    if (state->pInfcount > 0 && state->nInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_nan));
    if (state->pInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_pinf));
    if (state->nInfcount > 0)
        PG_RETURN_NUMERIC(make_result(&const_ninf));
 
    init_var(&sumX_var);
    accum_sum_final(&state->sumX, &sumX_var);
    result = make_result(&sumX_var);
    free_var(&sumX_var);
 
    PG_RETURN_NUMERIC(result);
}
 
/*
 * Workhorse routine for the standard deviance and variance
 * aggregates. 'state' is aggregate's transition state.
 * 'variance' specifies whether we should calculate the
 * variance or the standard deviation. 'sample' indicates whether the
 * caller is interested in the sample or the population
 * variance/stddev.
 *
 * If appropriate variance statistic is undefined for the input,
 * *is_null is set to true and NULL is returned.
 */
static Numeric
numeric_stddev_internal(NumericAggState *state,
                        bool variance, bool sample,
                        bool *is_null)
{
    Numeric     res;
    NumericVar  vN,
                vsumX,
                vsumX2,
                vNminus1;
    int64       totCount;
    int         rscale;
 
    /*
     * Sample stddev and variance are undefined when N <= 1; population stddev
     * is undefined when N == 0.  Return NULL in either case (note that NaNs
     * and infinities count as normal inputs for this purpose).
     */
    if (state == NULL || (totCount = NA_TOTAL_COUNT(state)) == 0)
    {
        *is_null = true;
        return NULL;
    }
 
    if (sample && totCount <= 1)
    {
        *is_null = true;
        return NULL;
    }
 
    *is_null = false;
 
    /*
     * Deal with NaN and infinity cases.  By analogy to the behavior of the
     * float8 functions, any infinity input produces NaN output.
     */
    if (state->NaNcount > 0 || state->pInfcount > 0 || state->nInfcount > 0)
        return make_result(&const_nan);
 
    /* OK, normal calculation applies */
    init_var(&vN);
    init_var(&vsumX);
    init_var(&vsumX2);
 
    int64_to_numericvar(state->N, &vN);
    accum_sum_final(&(state->sumX), &vsumX);
    accum_sum_final(&(state->sumX2), &vsumX2);
 
    init_var(&vNminus1);
    sub_var(&vN, &const_one, &vNminus1);
 
    /* compute rscale for mul_var calls */
    rscale = vsumX.dscale * 2;
 
    mul_var(&vsumX, &vsumX, &vsumX, rscale);    /* vsumX = sumX * sumX */
    mul_var(&vN, &vsumX2, &vsumX2, rscale); /* vsumX2 = N * sumX2 */
    sub_var(&vsumX2, &vsumX, &vsumX2);  /* N * sumX2 - sumX * sumX */
 
    if (cmp_var(&vsumX2, &const_zero) <= 0)
    {
        /* Watch out for roundoff error producing a negative numerator */
        res = make_result(&const_zero);
    }
    else
    {
        if (sample)
            mul_var(&vN, &vNminus1, &vNminus1, 0);  /* N * (N - 1) */
        else
            mul_var(&vN, &vN, &vNminus1, 0);    /* N * N */
        rscale = select_div_scale(&vsumX2, &vNminus1);
        div_var(&vsumX2, &vNminus1, &vsumX, rscale, true, true);    /* variance */
        if (!variance)
            sqrt_var(&vsumX, &vsumX, rscale);   /* stddev */
 
        res = make_result(&vsumX);
    }
 
    free_var(&vNminus1);
    free_var(&vsumX);
    free_var(&vsumX2);
 
    return res;
}
 
Datum
numeric_var_samp(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_stddev_internal(state, true, true, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
}
 
Datum
numeric_stddev_samp(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_stddev_internal(state, false, true, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
}
 
Datum
numeric_var_pop(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_stddev_internal(state, true, false, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
}
 
Datum
numeric_stddev_pop(PG_FUNCTION_ARGS)
{
    NumericAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (NumericAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_stddev_internal(state, false, false, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
}
 
#ifdef HAVE_INT128
static Numeric
numeric_poly_stddev_internal(Int128AggState *state,
                             bool variance, bool sample,
                             bool *is_null)
{
    NumericAggState numstate;
    Numeric     res;
 
    /* Initialize an empty agg state */
    memset(&numstate, 0, sizeof(NumericAggState));
 
    if (state)
    {
        NumericVar  tmp_var;
 
        numstate.N = state->N;
 
        init_var(&tmp_var);
 
        int128_to_numericvar(state->sumX, &tmp_var);
        accum_sum_add(&numstate.sumX, &tmp_var);
 
        int128_to_numericvar(state->sumX2, &tmp_var);
        accum_sum_add(&numstate.sumX2, &tmp_var);
 
        free_var(&tmp_var);
    }
 
    res = numeric_stddev_internal(&numstate, variance, sample, is_null);
 
    if (numstate.sumX.ndigits > 0)
    {
        pfree(numstate.sumX.pos_digits);
        pfree(numstate.sumX.neg_digits);
    }
    if (numstate.sumX2.ndigits > 0)
    {
        pfree(numstate.sumX2.pos_digits);
        pfree(numstate.sumX2.neg_digits);
    }
 
    return res;
}
#endif
 
Datum
numeric_poly_var_samp(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_poly_stddev_internal(state, true, true, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
#else
    return numeric_var_samp(fcinfo);
#endif
}
 
Datum
numeric_poly_stddev_samp(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_poly_stddev_internal(state, false, true, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
#else
    return numeric_stddev_samp(fcinfo);
#endif
}
 
Datum
numeric_poly_var_pop(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_poly_stddev_internal(state, true, false, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
#else
    return numeric_var_pop(fcinfo);
#endif
}
 
Datum
numeric_poly_stddev_pop(PG_FUNCTION_ARGS)
{
#ifdef HAVE_INT128
    PolyNumAggState *state;
    Numeric     res;
    bool        is_null;
 
    state = PG_ARGISNULL(0) ? NULL : (PolyNumAggState *) PG_GETARG_POINTER(0);
 
    res = numeric_poly_stddev_internal(state, false, false, &is_null);
 
    if (is_null)
        PG_RETURN_NULL();
    else
        PG_RETURN_NUMERIC(res);
#else
    return numeric_stddev_pop(fcinfo);
#endif
}
 
/*
 * SUM transition functions for integer datatypes.
 *
 * To avoid overflow, we use accumulators wider than the input datatype.
 * A Numeric accumulator is needed for int8 input; for int4 and int2
 * inputs, we use int8 accumulators which should be sufficient for practical
 * purposes.  (The latter two therefore don't really belong in this file,
 * but we keep them here anyway.)
 *
 * Because SQL defines the SUM() of no values to be NULL, not zero,
 * the initial condition of the transition data value needs to be NULL. This
 * means we can't rely on ExecAgg to automatically insert the first non-null
 * data value into the transition data: it doesn't know how to do the type
 * conversion.  The upshot is that these routines have to be marked non-strict
 * and handle substitution of the first non-null input themselves.
 *
 * Note: these functions are used only in plain aggregation mode.
 * In moving-aggregate mode, we use intX_avg_accum and intX_avg_accum_inv.
 */
 
Datum
int2_sum(PG_FUNCTION_ARGS)
{
    int64       newval;
 
    if (PG_ARGISNULL(0))
    {
        /* No non-null input seen so far... */
        if (PG_ARGISNULL(1))
            PG_RETURN_NULL();   /* still no non-null */
        /* This is the first non-null input. */
        newval = (int64) PG_GETARG_INT16(1);
        PG_RETURN_INT64(newval);
    }
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to avoid palloc overhead. If not, we need to return
     * the new value of the transition variable. (If int8 is pass-by-value,
     * then of course this is useless as well as incorrect, so just ifdef it
     * out.)
     */
#ifndef USE_FLOAT8_BYVAL        /* controls int8 too */
    if (AggCheckCallContext(fcinfo, NULL))
    {
        int64      *oldsum = (int64 *) PG_GETARG_POINTER(0);
 
        /* Leave the running sum unchanged in the new input is null */
        if (!PG_ARGISNULL(1))
            *oldsum = *oldsum + (int64) PG_GETARG_INT16(1);
 
        PG_RETURN_POINTER(oldsum);
    }
    else
#endif
    {
        int64       oldsum = PG_GETARG_INT64(0);
 
        /* Leave sum unchanged if new input is null. */
        if (PG_ARGISNULL(1))
            PG_RETURN_INT64(oldsum);
 
        /* OK to do the addition. */
        newval = oldsum + (int64) PG_GETARG_INT16(1);
 
        PG_RETURN_INT64(newval);
    }
}
 
Datum
int4_sum(PG_FUNCTION_ARGS)
{
    int64       newval;
 
    if (PG_ARGISNULL(0))
    {
        /* No non-null input seen so far... */
        if (PG_ARGISNULL(1))
            PG_RETURN_NULL();   /* still no non-null */
        /* This is the first non-null input. */
        newval = (int64) PG_GETARG_INT32(1);
        PG_RETURN_INT64(newval);
    }
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to avoid palloc overhead. If not, we need to return
     * the new value of the transition variable. (If int8 is pass-by-value,
     * then of course this is useless as well as incorrect, so just ifdef it
     * out.)
     */
#ifndef USE_FLOAT8_BYVAL        /* controls int8 too */
    if (AggCheckCallContext(fcinfo, NULL))
    {
        int64      *oldsum = (int64 *) PG_GETARG_POINTER(0);
 
        /* Leave the running sum unchanged in the new input is null */
        if (!PG_ARGISNULL(1))
            *oldsum = *oldsum + (int64) PG_GETARG_INT32(1);
 
        PG_RETURN_POINTER(oldsum);
    }
    else
#endif
    {
        int64       oldsum = PG_GETARG_INT64(0);
 
        /* Leave sum unchanged if new input is null. */
        if (PG_ARGISNULL(1))
            PG_RETURN_INT64(oldsum);
 
        /* OK to do the addition. */
        newval = oldsum + (int64) PG_GETARG_INT32(1);
 
        PG_RETURN_INT64(newval);
    }
}
 
/*
 * Note: this function is obsolete, it's no longer used for SUM(int8).
 */
Datum
int8_sum(PG_FUNCTION_ARGS)
{
    Numeric     oldsum;
 
    if (PG_ARGISNULL(0))
    {
        /* No non-null input seen so far... */
        if (PG_ARGISNULL(1))
            PG_RETURN_NULL();   /* still no non-null */
        /* This is the first non-null input. */
        PG_RETURN_NUMERIC(int64_to_numeric(PG_GETARG_INT64(1)));
    }
 
    /*
     * Note that we cannot special-case the aggregate case here, as we do for
     * int2_sum and int4_sum: numeric is of variable size, so we cannot modify
     * our first parameter in-place.
     */
 
    oldsum = PG_GETARG_NUMERIC(0);
 
    /* Leave sum unchanged if new input is null. */
    if (PG_ARGISNULL(1))
        PG_RETURN_NUMERIC(oldsum);
 
    /* OK to do the addition. */
    PG_RETURN_DATUM(DirectFunctionCall2(numeric_add,
                                        NumericGetDatum(oldsum),
                                        NumericGetDatum(int64_to_numeric(PG_GETARG_INT64(1)))));
}
 
 
/*
 * Routines for avg(int2) and avg(int4).  The transition datatype
 * is a two-element int8 array, holding count and sum.
 *
 * These functions are also used for sum(int2) and sum(int4) when
 * operating in moving-aggregate mode, since for correct inverse transitions
 * we need to count the inputs.
 */
 
typedef struct Int8TransTypeData
{
    int64       count;
    int64       sum;
} Int8TransTypeData;
 
Datum
int2_avg_accum(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray;
    int16       newval = PG_GETARG_INT16(1);
    Int8TransTypeData *transdata;
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to reduce palloc overhead. Otherwise we need to make
     * a copy of it before scribbling on it.
     */
    if (AggCheckCallContext(fcinfo, NULL))
        transarray = PG_GETARG_ARRAYTYPE_P(0);
    else
        transarray = PG_GETARG_ARRAYTYPE_P_COPY(0);
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
    transdata->count++;
    transdata->sum += newval;
 
    PG_RETURN_ARRAYTYPE_P(transarray);
}
 
Datum
int4_avg_accum(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray;
    int32       newval = PG_GETARG_INT32(1);
    Int8TransTypeData *transdata;
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to reduce palloc overhead. Otherwise we need to make
     * a copy of it before scribbling on it.
     */
    if (AggCheckCallContext(fcinfo, NULL))
        transarray = PG_GETARG_ARRAYTYPE_P(0);
    else
        transarray = PG_GETARG_ARRAYTYPE_P_COPY(0);
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
    transdata->count++;
    transdata->sum += newval;
 
    PG_RETURN_ARRAYTYPE_P(transarray);
}
 
Datum
int4_avg_combine(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray1;
    ArrayType  *transarray2;
    Int8TransTypeData *state1;
    Int8TransTypeData *state2;
 
    if (!AggCheckCallContext(fcinfo, NULL))
        elog(ERROR, "aggregate function called in non-aggregate context");
 
    transarray1 = PG_GETARG_ARRAYTYPE_P(0);
    transarray2 = PG_GETARG_ARRAYTYPE_P(1);
 
    if (ARR_HASNULL(transarray1) ||
        ARR_SIZE(transarray1) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    if (ARR_HASNULL(transarray2) ||
        ARR_SIZE(transarray2) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    state1 = (Int8TransTypeData *) ARR_DATA_PTR(transarray1);
    state2 = (Int8TransTypeData *) ARR_DATA_PTR(transarray2);
 
    state1->count += state2->count;
    state1->sum += state2->sum;
 
    PG_RETURN_ARRAYTYPE_P(transarray1);
}
 
Datum
int2_avg_accum_inv(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray;
    int16       newval = PG_GETARG_INT16(1);
    Int8TransTypeData *transdata;
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to reduce palloc overhead. Otherwise we need to make
     * a copy of it before scribbling on it.
     */
    if (AggCheckCallContext(fcinfo, NULL))
        transarray = PG_GETARG_ARRAYTYPE_P(0);
    else
        transarray = PG_GETARG_ARRAYTYPE_P_COPY(0);
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
    transdata->count--;
    transdata->sum -= newval;
 
    PG_RETURN_ARRAYTYPE_P(transarray);
}
 
Datum
int4_avg_accum_inv(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray;
    int32       newval = PG_GETARG_INT32(1);
    Int8TransTypeData *transdata;
 
    /*
     * If we're invoked as an aggregate, we can cheat and modify our first
     * parameter in-place to reduce palloc overhead. Otherwise we need to make
     * a copy of it before scribbling on it.
     */
    if (AggCheckCallContext(fcinfo, NULL))
        transarray = PG_GETARG_ARRAYTYPE_P(0);
    else
        transarray = PG_GETARG_ARRAYTYPE_P_COPY(0);
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
 
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
    transdata->count--;
    transdata->sum -= newval;
 
    PG_RETURN_ARRAYTYPE_P(transarray);
}
 
Datum
int8_avg(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray = PG_GETARG_ARRAYTYPE_P(0);
    Int8TransTypeData *transdata;
    Datum       countd,
                sumd;
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
 
    /* SQL defines AVG of no values to be NULL */
    if (transdata->count == 0)
        PG_RETURN_NULL();
 
    countd = NumericGetDatum(int64_to_numeric(transdata->count));
    sumd = NumericGetDatum(int64_to_numeric(transdata->sum));
 
    PG_RETURN_DATUM(DirectFunctionCall2(numeric_div, sumd, countd));
}
 
/*
 * SUM(int2) and SUM(int4) both return int8, so we can use this
 * final function for both.
 */
Datum
int2int4_sum(PG_FUNCTION_ARGS)
{
    ArrayType  *transarray = PG_GETARG_ARRAYTYPE_P(0);
    Int8TransTypeData *transdata;
 
    if (ARR_HASNULL(transarray) ||
        ARR_SIZE(transarray) != ARR_OVERHEAD_NONULLS(1) + sizeof(Int8TransTypeData))
        elog(ERROR, "expected 2-element int8 array");
    transdata = (Int8TransTypeData *) ARR_DATA_PTR(transarray);
 
    /* SQL defines SUM of no values to be NULL */
    if (transdata->count == 0)
        PG_RETURN_NULL();
 
    PG_RETURN_DATUM(Int64GetDatumFast(transdata->sum));
}
 
 
/* ----------------------------------------------------------------------
 *
 * Debug support
 *
 * ----------------------------------------------------------------------
 */
 
#ifdef NUMERIC_DEBUG
 
/*
 * dump_numeric() - Dump a value in the db storage format for debugging
 */
static void
dump_numeric(const char *str, Numeric num)
{
    NumericDigit *digits = NUMERIC_DIGITS(num);
    int         ndigits;
    int         i;
 
    ndigits = NUMERIC_NDIGITS(num);
 
    printf("%s: NUMERIC w=%d d=%d ", str,
           NUMERIC_WEIGHT(num), NUMERIC_DSCALE(num));
    switch (NUMERIC_SIGN(num))
    {
        case NUMERIC_POS:
            printf("POS");
            break;
        case NUMERIC_NEG:
            printf("NEG");
            break;
        case NUMERIC_NAN:
            printf("NaN");
            break;
        case NUMERIC_PINF:
            printf("Infinity");
            break;
        case NUMERIC_NINF:
            printf("-Infinity");
            break;
        default:
            printf("SIGN=0x%x", NUMERIC_SIGN(num));
            break;
    }
 
    for (i = 0; i < ndigits; i++)
        printf(" %0*d", DEC_DIGITS, digits[i]);
    printf("\n");
}
 
 
/*
 * dump_var() - Dump a value in the variable format for debugging
 */
static void
dump_var(const char *str, NumericVar *var)
{
    int         i;
 
    printf("%s: VAR w=%d d=%d ", str, var->weight, var->dscale);
    switch (var->sign)
    {
        case NUMERIC_POS:
            printf("POS");
            break;
        case NUMERIC_NEG:
            printf("NEG");
            break;
        case NUMERIC_NAN:
            printf("NaN");
            break;
        case NUMERIC_PINF:
            printf("Infinity");
            break;
        case NUMERIC_NINF:
            printf("-Infinity");
            break;
        default:
            printf("SIGN=0x%x", var->sign);
            break;
    }
 
    for (i = 0; i < var->ndigits; i++)
        printf(" %0*d", DEC_DIGITS, var->digits[i]);
 
    printf("\n");
}
#endif                          /* NUMERIC_DEBUG */
 
 
/* ----------------------------------------------------------------------
 *
 * Local functions follow
 *
 * In general, these do not support "special" (NaN or infinity) inputs;
 * callers should handle those possibilities first.
 * (There are one or two exceptions, noted in their header comments.)
 *
 * ----------------------------------------------------------------------
 */
 
 
/*
 * alloc_var() -
 *
 *  Allocate a digit buffer of ndigits digits (plus a spare digit for rounding)
 */
static void
alloc_var(NumericVar *var, int ndigits)
{
    digitbuf_free(var->buf);
    var->buf = digitbuf_alloc(ndigits + 1);
    var->buf[0] = 0;            /* spare digit for rounding */
    var->digits = var->buf + 1;
    var->ndigits = ndigits;
}
 
 
/*
 * free_var() -
 *
 *  Return the digit buffer of a variable to the free pool
 */
static void
free_var(NumericVar *var)
{
    digitbuf_free(var->buf);
    var->buf = NULL;
    var->digits = NULL;
    var->sign = NUMERIC_NAN;
}
 
 
/*
 * zero_var() -
 *
 *  Set a variable to ZERO.
 *  Note: its dscale is not touched.
 */
static void
zero_var(NumericVar *var)
{
    digitbuf_free(var->buf);
    var->buf = NULL;
    var->digits = NULL;
    var->ndigits = 0;
    var->weight = 0;            /* by convention; doesn't really matter */
    var->sign = NUMERIC_POS;    /* anything but NAN... */
}
 
 
/*
 * set_var_from_str()
 *
 *  Parse a string and put the number into a variable
 *
 * This function does not handle leading or trailing spaces.  It returns
 * the end+1 position parsed into *endptr, so that caller can check for
 * trailing spaces/garbage if deemed necessary.
 *
 * cp is the place to actually start parsing; str is what to use in error
 * reports.  (Typically cp would be the same except advanced over spaces.)
 *
 * Returns true on success, false on failure (if escontext points to an
 * ErrorSaveContext; otherwise errors are thrown).
 */
static bool
set_var_from_str(const char *str, const char *cp,
                 NumericVar *dest, const char **endptr,
                 Node *escontext)
{
    bool        have_dp = false;
    int         i;
    unsigned char *decdigits;
    int         sign = NUMERIC_POS;
    int         dweight = -1;
    int         ddigits;
    int         dscale = 0;
    int         weight;
    int         ndigits;
    int         offset;
    NumericDigit *digits;
 
    /*
     * We first parse the string to extract decimal digits and determine the
     * correct decimal weight.  Then convert to NBASE representation.
     */
    switch (*cp)
    {
        case '+':
            sign = NUMERIC_POS;
            cp++;
            break;
 
        case '-':
            sign = NUMERIC_NEG;
            cp++;
            break;
    }
 
    if (*cp == '.')
    {
        have_dp = true;
        cp++;
    }
 
    if (!isdigit((unsigned char) *cp))
        goto invalid_syntax;
 
    decdigits = (unsigned char *) palloc(strlen(cp) + DEC_DIGITS * 2);
 
    /* leading padding for digit alignment later */
    memset(decdigits, 0, DEC_DIGITS);
    i = DEC_DIGITS;
 
    while (*cp)
    {
        if (isdigit((unsigned char) *cp))
        {
            decdigits[i++] = *cp++ - '0';
            if (!have_dp)
                dweight++;
            else
                dscale++;
        }
        else if (*cp == '.')
        {
            if (have_dp)
                goto invalid_syntax;
            have_dp = true;
            cp++;
            /* decimal point must not be followed by underscore */
            if (*cp == '_')
                goto invalid_syntax;
        }
        else if (*cp == '_')
        {
            /* underscore must be followed by more digits */
            cp++;
            if (!isdigit((unsigned char) *cp))
                goto invalid_syntax;
        }
        else
            break;
    }
 
    ddigits = i - DEC_DIGITS;
    /* trailing padding for digit alignment later */
    memset(decdigits + i, 0, DEC_DIGITS - 1);
 
    /* Handle exponent, if any */
    if (*cp == 'e' || *cp == 'E')
    {
        int64       exponent = 0;
        bool        neg = false;
 
        /*
         * At this point, dweight and dscale can't be more than about
         * INT_MAX/2 due to the MaxAllocSize limit on string length, so
         * constraining the exponent similarly should be enough to prevent
         * integer overflow in this function.  If the value is too large to
         * fit in storage format, make_result() will complain about it later;
         * for consistency use the same ereport errcode/text as make_result().
         */
 
        /* exponent sign */
        cp++;
        if (*cp == '+')
            cp++;
        else if (*cp == '-')
        {
            neg = true;
            cp++;
        }
 
        /* exponent digits */
        if (!isdigit((unsigned char) *cp))
            goto invalid_syntax;
 
        while (*cp)
        {
            if (isdigit((unsigned char) *cp))
            {
                exponent = exponent * 10 + (*cp++ - '0');
                if (exponent > PG_INT32_MAX / 2)
                    goto out_of_range;
            }
            else if (*cp == '_')
            {
                /* underscore must be followed by more digits */
                cp++;
                if (!isdigit((unsigned char) *cp))
                    goto invalid_syntax;
            }
            else
                break;
        }
 
        if (neg)
            exponent = -exponent;
 
        dweight += (int) exponent;
        dscale -= (int) exponent;
        if (dscale < 0)
            dscale = 0;
    }
 
    /*
     * Okay, convert pure-decimal representation to base NBASE.  First we need
     * to determine the converted weight and ndigits.  offset is the number of
     * decimal zeroes to insert before the first given digit to have a
     * correctly aligned first NBASE digit.
     */
    if (dweight >= 0)
        weight = (dweight + 1 + DEC_DIGITS - 1) / DEC_DIGITS - 1;
    else
        weight = -((-dweight - 1) / DEC_DIGITS + 1);
    offset = (weight + 1) * DEC_DIGITS - (dweight + 1);
    ndigits = (ddigits + offset + DEC_DIGITS - 1) / DEC_DIGITS;
 
    alloc_var(dest, ndigits);
    dest->sign = sign;
    dest->weight = weight;
    dest->dscale = dscale;
 
    i = DEC_DIGITS - offset;
    digits = dest->digits;
 
    while (ndigits-- > 0)
    {
#if DEC_DIGITS == 4
        *digits++ = ((decdigits[i] * 10 + decdigits[i + 1]) * 10 +
                     decdigits[i + 2]) * 10 + decdigits[i + 3];
#elif DEC_DIGITS == 2
        *digits++ = decdigits[i] * 10 + decdigits[i + 1];
#elif DEC_DIGITS == 1
        *digits++ = decdigits[i];
#else
#error unsupported NBASE
#endif
        i += DEC_DIGITS;
    }
 
    pfree(decdigits);
 
    /* Strip any leading/trailing zeroes, and normalize weight if zero */
    strip_var(dest);
 
    /* Return end+1 position for caller */
    *endptr = cp;
 
    return true;
 
out_of_range:
    ereturn(escontext, false,
            (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
             errmsg("value overflows numeric format")));
 
invalid_syntax:
    ereturn(escontext, false,
            (errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
             errmsg("invalid input syntax for type %s: \"%s\"",
                    "numeric", str)));
}
 
 
/*
 * Return the numeric value of a single hex digit.
 */
static inline int
xdigit_value(char dig)
{
    return dig >= '0' && dig <= '9' ? dig - '0' :
        dig >= 'a' && dig <= 'f' ? dig - 'a' + 10 :
        dig >= 'A' && dig <= 'F' ? dig - 'A' + 10 : -1;
}
 
/*
 * set_var_from_non_decimal_integer_str()
 *
 *  Parse a string containing a non-decimal integer
 *
 * This function does not handle leading or trailing spaces.  It returns
 * the end+1 position parsed into *endptr, so that caller can check for
 * trailing spaces/garbage if deemed necessary.
 *
 * cp is the place to actually start parsing; str is what to use in error
 * reports.  The number's sign and base prefix indicator (e.g., "0x") are
 * assumed to have already been parsed, so cp should point to the number's
 * first digit in the base specified.
 *
 * base is expected to be 2, 8 or 16.
 *
 * Returns true on success, false on failure (if escontext points to an
 * ErrorSaveContext; otherwise errors are thrown).
 */
static bool
set_var_from_non_decimal_integer_str(const char *str, const char *cp, int sign,
                                     int base, NumericVar *dest,
                                     const char **endptr, Node *escontext)
{
    const char *firstdigit = cp;
    int64       tmp;
    int64       mul;
    NumericVar  tmp_var;
 
    init_var(&tmp_var);
 
    zero_var(dest);
 
    /*
     * Process input digits in groups that fit in int64.  Here "tmp" is the
     * value of the digits in the group, and "mul" is base^n, where n is the
     * number of digits in the group.  Thus tmp < mul, and we must start a new
     * group when mul * base threatens to overflow PG_INT64_MAX.
     */
    tmp = 0;
    mul = 1;
 
    if (base == 16)
    {
        while (*cp)
        {
            if (isxdigit((unsigned char) *cp))
            {
                if (mul > PG_INT64_MAX / 16)
                {
                    /* Add the contribution from this group of digits */
                    int64_to_numericvar(mul, &tmp_var);
                    mul_var(dest, &tmp_var, dest, 0);
                    int64_to_numericvar(tmp, &tmp_var);
                    add_var(dest, &tmp_var, dest);
 
                    /* Result will overflow if weight overflows int16 */
                    if (dest->weight > NUMERIC_WEIGHT_MAX)
                        goto out_of_range;
 
                    /* Begin a new group */
                    tmp = 0;
                    mul = 1;
                }
 
                tmp = tmp * 16 + xdigit_value(*cp++);
                mul = mul * 16;
            }
            else if (*cp == '_')
            {
                /* Underscore must be followed by more digits */
                cp++;
                if (!isxdigit((unsigned char) *cp))
                    goto invalid_syntax;
            }
            else
                break;
        }
    }
    else if (base == 8)
    {
        while (*cp)
        {
            if (*cp >= '0' && *cp <= '7')
            {
                if (mul > PG_INT64_MAX / 8)
                {
                    /* Add the contribution from this group of digits */
                    int64_to_numericvar(mul, &tmp_var);
                    mul_var(dest, &tmp_var, dest, 0);
                    int64_to_numericvar(tmp, &tmp_var);
                    add_var(dest, &tmp_var, dest);
 
                    /* Result will overflow if weight overflows int16 */
                    if (dest->weight > NUMERIC_WEIGHT_MAX)
                        goto out_of_range;
 
                    /* Begin a new group */
                    tmp = 0;
                    mul = 1;
                }
 
                tmp = tmp * 8 + (*cp++ - '0');
                mul = mul * 8;
            }
            else if (*cp == '_')
            {
                /* Underscore must be followed by more digits */
                cp++;
                if (*cp < '0' || *cp > '7')
                    goto invalid_syntax;
            }
            else
                break;
        }
    }
    else if (base == 2)
    {
        while (*cp)
        {
            if (*cp >= '0' && *cp <= '1')
            {
                if (mul > PG_INT64_MAX / 2)
                {
                    /* Add the contribution from this group of digits */
                    int64_to_numericvar(mul, &tmp_var);
                    mul_var(dest, &tmp_var, dest, 0);
                    int64_to_numericvar(tmp, &tmp_var);
                    add_var(dest, &tmp_var, dest);
 
                    /* Result will overflow if weight overflows int16 */
                    if (dest->weight > NUMERIC_WEIGHT_MAX)
                        goto out_of_range;
 
                    /* Begin a new group */
                    tmp = 0;
                    mul = 1;
                }
 
                tmp = tmp * 2 + (*cp++ - '0');
                mul = mul * 2;
            }
            else if (*cp == '_')
            {
                /* Underscore must be followed by more digits */
                cp++;
                if (*cp < '0' || *cp > '1')
                    goto invalid_syntax;
            }
            else
                break;
        }
    }
    else
        /* Should never happen; treat as invalid input */
        goto invalid_syntax;
 
    /* Check that we got at least one digit */
    if (unlikely(cp == firstdigit))
        goto invalid_syntax;
 
    /* Add the contribution from the final group of digits */
    int64_to_numericvar(mul, &tmp_var);
    mul_var(dest, &tmp_var, dest, 0);
    int64_to_numericvar(tmp, &tmp_var);
    add_var(dest, &tmp_var, dest);
 
    if (dest->weight > NUMERIC_WEIGHT_MAX)
        goto out_of_range;
 
    dest->sign = sign;
 
    free_var(&tmp_var);
 
    /* Return end+1 position for caller */
    *endptr = cp;
 
    return true;
 
out_of_range:
    ereturn(escontext, false,
            (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
             errmsg("value overflows numeric format")));
 
invalid_syntax:
    ereturn(escontext, false,
            (errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
             errmsg("invalid input syntax for type %s: \"%s\"",
                    "numeric", str)));
}
 
 
/*
 * set_var_from_num() -
 *
 *  Convert the packed db format into a variable
 */
static void
set_var_from_num(Numeric num, NumericVar *dest)
{
    int         ndigits;
 
    ndigits = NUMERIC_NDIGITS(num);
 
    alloc_var(dest, ndigits);
 
    dest->weight = NUMERIC_WEIGHT(num);
    dest->sign = NUMERIC_SIGN(num);
    dest->dscale = NUMERIC_DSCALE(num);
 
    memcpy(dest->digits, NUMERIC_DIGITS(num), ndigits * sizeof(NumericDigit));
}
 
 
/*
 * init_var_from_num() -
 *
 *  Initialize a variable from packed db format. The digits array is not
 *  copied, which saves some cycles when the resulting var is not modified.
 *  Also, there's no need to call free_var(), as long as you don't assign any
 *  other value to it (with set_var_* functions, or by using the var as the
 *  destination of a function like add_var())
 *
 *  CAUTION: Do not modify the digits buffer of a var initialized with this
 *  function, e.g by calling round_var() or trunc_var(), as the changes will
 *  propagate to the original Numeric! It's OK to use it as the destination
 *  argument of one of the calculational functions, though.
 */
static void
init_var_from_num(Numeric num, NumericVar *dest)
{
    dest->ndigits = NUMERIC_NDIGITS(num);
    dest->weight = NUMERIC_WEIGHT(num);
    dest->sign = NUMERIC_SIGN(num);
    dest->dscale = NUMERIC_DSCALE(num);
    dest->digits = NUMERIC_DIGITS(num);
    dest->buf = NULL;           /* digits array is not palloc'd */
}
 
 
/*
 * set_var_from_var() -
 *
 *  Copy one variable into another
 */
static void
set_var_from_var(const NumericVar *value, NumericVar *dest)
{
    NumericDigit *newbuf;
 
    newbuf = digitbuf_alloc(value->ndigits + 1);
    newbuf[0] = 0;              /* spare digit for rounding */
    if (value->ndigits > 0)     /* else value->digits might be null */
        memcpy(newbuf + 1, value->digits,
               value->ndigits * sizeof(NumericDigit));
 
    digitbuf_free(dest->buf);
 
    memmove(dest, value, sizeof(NumericVar));
    dest->buf = newbuf;
    dest->digits = newbuf + 1;
}
 
 
/*
 * get_str_from_var() -
 *
 *  Convert a var to text representation (guts of numeric_out).
 *  The var is displayed to the number of digits indicated by its dscale.
 *  Returns a palloc'd string.
 */
static char *
get_str_from_var(const NumericVar *var)
{
    int         dscale;
    char       *str;
    char       *cp;
    char       *endcp;
    int         i;
    int         d;
    NumericDigit dig;
 
#if DEC_DIGITS > 1
    NumericDigit d1;
#endif
 
    dscale = var->dscale;
 
    /*
     * Allocate space for the result.
     *
     * i is set to the # of decimal digits before decimal point. dscale is the
     * # of decimal digits we will print after decimal point. We may generate
     * as many as DEC_DIGITS-1 excess digits at the end, and in addition we
     * need room for sign, decimal point, null terminator.
     */
    i = (var->weight + 1) * DEC_DIGITS;
    if (i <= 0)
        i = 1;
 
    str = palloc(i + dscale + DEC_DIGITS + 2);
    cp = str;
 
    /*
     * Output a dash for negative values
     */
    if (var->sign == NUMERIC_NEG)
        *cp++ = '-';
 
    /*
     * Output all digits before the decimal point
     */
    if (var->weight < 0)
    {
        d = var->weight + 1;
        *cp++ = '0';
    }
    else
    {
        for (d = 0; d <= var->weight; d++)
        {
            dig = (d < var->ndigits) ? var->digits[d] : 0;
            /* In the first digit, suppress extra leading decimal zeroes */
#if DEC_DIGITS == 4
            {
                bool        putit = (d > 0);
 
                d1 = dig / 1000;
                dig -= d1 * 1000;
                putit |= (d1 > 0);
                if (putit)
                    *cp++ = d1 + '0';
                d1 = dig / 100;
                dig -= d1 * 100;
                putit |= (d1 > 0);
                if (putit)
                    *cp++ = d1 + '0';
                d1 = dig / 10;
                dig -= d1 * 10;
                putit |= (d1 > 0);
                if (putit)
                    *cp++ = d1 + '0';
                *cp++ = dig + '0';
            }
#elif DEC_DIGITS == 2
            d1 = dig / 10;
            dig -= d1 * 10;
            if (d1 > 0 || d > 0)
                *cp++ = d1 + '0';
            *cp++ = dig + '0';
#elif DEC_DIGITS == 1
            *cp++ = dig + '0';
#else
#error unsupported NBASE
#endif
        }
    }
 
    /*
     * If requested, output a decimal point and all the digits that follow it.
     * We initially put out a multiple of DEC_DIGITS digits, then truncate if
     * needed.
     */
    if (dscale > 0)
    {
        *cp++ = '.';
        endcp = cp + dscale;
        for (i = 0; i < dscale; d++, i += DEC_DIGITS)
        {
            dig = (d >= 0 && d < var->ndigits) ? var->digits[d] : 0;
#if DEC_DIGITS == 4
            d1 = dig / 1000;
            dig -= d1 * 1000;
            *cp++ = d1 + '0';
            d1 = dig / 100;
            dig -= d1 * 100;
            *cp++ = d1 + '0';
            d1 = dig / 10;
            dig -= d1 * 10;
            *cp++ = d1 + '0';
            *cp++ = dig + '0';
#elif DEC_DIGITS == 2
            d1 = dig / 10;
            dig -= d1 * 10;
            *cp++ = d1 + '0';
            *cp++ = dig + '0';
#elif DEC_DIGITS == 1
            *cp++ = dig + '0';
#else
#error unsupported NBASE
#endif
        }
        cp = endcp;
    }
 
    /*
     * terminate the string and return it
     */
    *cp = '\0';
    return str;
}
 
/*
 * get_str_from_var_sci() -
 *
 *  Convert a var to a normalised scientific notation text representation.
 *  This function does the heavy lifting for numeric_out_sci().
 *
 *  This notation has the general form a * 10^b, where a is known as the
 *  "significand" and b is known as the "exponent".
 *
 *  Because we can't do superscript in ASCII (and because we want to copy
 *  printf's behaviour) we display the exponent using E notation, with a
 *  minimum of two exponent digits.
 *
 *  For example, the value 1234 could be output as 1.2e+03.
 *
 *  We assume that the exponent can fit into an int32.
 *
 *  rscale is the number of decimal digits desired after the decimal point in
 *  the output, negative values will be treated as meaning zero.
 *
 *  Returns a palloc'd string.
 */
static char *
get_str_from_var_sci(const NumericVar *var, int rscale)
{
    int32       exponent;
    NumericVar  tmp_var;
    size_t      len;
    char       *str;
    char       *sig_out;
 
    if (rscale < 0)
        rscale = 0;
 
    /*
     * Determine the exponent of this number in normalised form.
     *
     * This is the exponent required to represent the number with only one
     * significant digit before the decimal place.
     */
    if (var->ndigits > 0)
    {
        exponent = (var->weight + 1) * DEC_DIGITS;
 
        /*
         * Compensate for leading decimal zeroes in the first numeric digit by
         * decrementing the exponent.
         */
        exponent -= DEC_DIGITS - (int) log10(var->digits[0]);
    }
    else
    {
        /*
         * If var has no digits, then it must be zero.
         *
         * Zero doesn't technically have a meaningful exponent in normalised
         * notation, but we just display the exponent as zero for consistency
         * of output.
         */
        exponent = 0;
    }
 
    /*
     * Divide var by 10^exponent to get the significand, rounding to rscale
     * decimal digits in the process.
     */
    init_var(&tmp_var);
 
    power_ten_int(exponent, &tmp_var);
    div_var(var, &tmp_var, &tmp_var, rscale, true, true);
    sig_out = get_str_from_var(&tmp_var);
 
    free_var(&tmp_var);
 
    /*
     * Allocate space for the result.
     *
     * In addition to the significand, we need room for the exponent
     * decoration ("e"), the sign of the exponent, up to 10 digits for the
     * exponent itself, and of course the null terminator.
     */
    len = strlen(sig_out) + 13;
    str = palloc(len);
    snprintf(str, len, "%se%+03d", sig_out, exponent);
 
    pfree(sig_out);
 
    return str;
}
 
 
/*
 * numericvar_serialize - serialize NumericVar to binary format
 *
 * At variable level, no checks are performed on the weight or dscale, allowing
 * us to pass around intermediate values with higher precision than supported
 * by the numeric type.  Note: this is incompatible with numeric_send/recv(),
 * which use 16-bit integers for these fields.
 */
static void
numericvar_serialize(StringInfo buf, const NumericVar *var)
{
    int         i;
 
    pq_sendint32(buf, var->ndigits);
    pq_sendint32(buf, var->weight);
    pq_sendint32(buf, var->sign);
    pq_sendint32(buf, var->dscale);
    for (i = 0; i < var->ndigits; i++)
        pq_sendint16(buf, var->digits[i]);
}
 
/*
 * numericvar_deserialize - deserialize binary format to NumericVar
 */
static void
numericvar_deserialize(StringInfo buf, NumericVar *var)
{
    int         len,
                i;
 
    len = pq_getmsgint(buf, sizeof(int32));
 
    alloc_var(var, len);        /* sets var->ndigits */
 
    var->weight = pq_getmsgint(buf, sizeof(int32));
    var->sign = pq_getmsgint(buf, sizeof(int32));
    var->dscale = pq_getmsgint(buf, sizeof(int32));
    for (i = 0; i < len; i++)
        var->digits[i] = pq_getmsgint(buf, sizeof(int16));
}
 
 
/*
 * duplicate_numeric() - copy a packed-format Numeric
 *
 * This will handle NaN and Infinity cases.
 */
static Numeric
duplicate_numeric(Numeric num)
{
    Numeric     res;
 
    res = (Numeric) palloc(VARSIZE(num));
    memcpy(res, num, VARSIZE(num));
    return res;
}
 
/*
 * make_result_opt_error() -
 *
 *  Create the packed db numeric format in palloc()'d memory from
 *  a variable.  This will handle NaN and Infinity cases.
 *
 *  If "have_error" isn't NULL, on overflow *have_error is set to true and
 *  NULL is returned.  This is helpful when caller needs to handle errors.
 */
static Numeric
make_result_opt_error(const NumericVar *var, bool *have_error)
{
    Numeric     result;
    NumericDigit *digits = var->digits;
    int         weight = var->weight;
    int         sign = var->sign;
    int         n;
    Size        len;
 
    if (have_error)
        *have_error = false;
 
    if ((sign & NUMERIC_SIGN_MASK) == NUMERIC_SPECIAL)
    {
        /*
         * Verify valid special value.  This could be just an Assert, perhaps,
         * but it seems worthwhile to expend a few cycles to ensure that we
         * never write any nonzero reserved bits to disk.
         */
        if (!(sign == NUMERIC_NAN ||
              sign == NUMERIC_PINF ||
              sign == NUMERIC_NINF))
            elog(ERROR, "invalid numeric sign value 0x%x", sign);
 
        result = (Numeric) palloc(NUMERIC_HDRSZ_SHORT);
 
        SET_VARSIZE(result, NUMERIC_HDRSZ_SHORT);
        result->choice.n_header = sign;
        /* the header word is all we need */
 
        dump_numeric("make_result()", result);
        return result;
    }
 
    n = var->ndigits;
 
    /* truncate leading zeroes */
    while (n > 0 && *digits == 0)
    {
        digits++;
        weight--;
        n--;
    }
    /* truncate trailing zeroes */
    while (n > 0 && digits[n - 1] == 0)
        n--;
 
    /* If zero result, force to weight=0 and positive sign */
    if (n == 0)
    {
        weight = 0;
        sign = NUMERIC_POS;
    }
 
    /* Build the result */
    if (NUMERIC_CAN_BE_SHORT(var->dscale, weight))
    {
        len = NUMERIC_HDRSZ_SHORT + n * sizeof(NumericDigit);
        result = (Numeric) palloc(len);
        SET_VARSIZE(result, len);
        result->choice.n_short.n_header =
            (sign == NUMERIC_NEG ? (NUMERIC_SHORT | NUMERIC_SHORT_SIGN_MASK)
             : NUMERIC_SHORT)
            | (var->dscale << NUMERIC_SHORT_DSCALE_SHIFT)
            | (weight < 0 ? NUMERIC_SHORT_WEIGHT_SIGN_MASK : 0)
            | (weight & NUMERIC_SHORT_WEIGHT_MASK);
    }
    else
    {
        len = NUMERIC_HDRSZ + n * sizeof(NumericDigit);
        result = (Numeric) palloc(len);
        SET_VARSIZE(result, len);
        result->choice.n_long.n_sign_dscale =
            sign | (var->dscale & NUMERIC_DSCALE_MASK);
        result->choice.n_long.n_weight = weight;
    }
 
    Assert(NUMERIC_NDIGITS(result) == n);
    if (n > 0)
        memcpy(NUMERIC_DIGITS(result), digits, n * sizeof(NumericDigit));
 
    /* Check for overflow of int16 fields */
    if (NUMERIC_WEIGHT(result) != weight ||
        NUMERIC_DSCALE(result) != var->dscale)
    {
        if (have_error)
        {
            *have_error = true;
            return NULL;
        }
        else
        {
            ereport(ERROR,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("value overflows numeric format")));
        }
    }
 
    dump_numeric("make_result()", result);
    return result;
}
 
 
/*
 * make_result() -
 *
 *  An interface to make_result_opt_error() without "have_error" argument.
 */
static Numeric
make_result(const NumericVar *var)
{
    return make_result_opt_error(var, NULL);
}
 
 
/*
 * apply_typmod() -
 *
 *  Do bounds checking and rounding according to the specified typmod.
 *  Note that this is only applied to normal finite values.
 *
 * Returns true on success, false on failure (if escontext points to an
 * ErrorSaveContext; otherwise errors are thrown).
 */
static bool
apply_typmod(NumericVar *var, int32 typmod, Node *escontext)
{
    int         precision;
    int         scale;
    int         maxdigits;
    int         ddigits;
    int         i;
 
    /* Do nothing if we have an invalid typmod */
    if (!is_valid_numeric_typmod(typmod))
        return true;
 
    precision = numeric_typmod_precision(typmod);
    scale = numeric_typmod_scale(typmod);
    maxdigits = precision - scale;
 
    /* Round to target scale (and set var->dscale) */
    round_var(var, scale);
 
    /* but don't allow var->dscale to be negative */
    if (var->dscale < 0)
        var->dscale = 0;
 
    /*
     * Check for overflow - note we can't do this before rounding, because
     * rounding could raise the weight.  Also note that the var's weight could
     * be inflated by leading zeroes, which will be stripped before storage
     * but perhaps might not have been yet. In any case, we must recognize a
     * true zero, whose weight doesn't mean anything.
     */
    ddigits = (var->weight + 1) * DEC_DIGITS;
    if (ddigits > maxdigits)
    {
        /* Determine true weight; and check for all-zero result */
        for (i = 0; i < var->ndigits; i++)
        {
            NumericDigit dig = var->digits[i];
 
            if (dig)
            {
                /* Adjust for any high-order decimal zero digits */
#if DEC_DIGITS == 4
                if (dig < 10)
                    ddigits -= 3;
                else if (dig < 100)
                    ddigits -= 2;
                else if (dig < 1000)
                    ddigits -= 1;
#elif DEC_DIGITS == 2
                if (dig < 10)
                    ddigits -= 1;
#elif DEC_DIGITS == 1
                /* no adjustment */
#else
#error unsupported NBASE
#endif
                if (ddigits > maxdigits)
                    ereturn(escontext, false,
                            (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                             errmsg("numeric field overflow"),
                             errdetail("A field with precision %d, scale %d must round to an absolute value less than %s%d.",
                                       precision, scale,
                    /* Display 10^0 as 1 */
                                       maxdigits ? "10^" : "",
                                       maxdigits ? maxdigits : 1
                                       )));
                break;
            }
            ddigits -= DEC_DIGITS;
        }
    }
 
    return true;
}
 
/*
 * apply_typmod_special() -
 *
 *  Do bounds checking according to the specified typmod, for an Inf or NaN.
 *  For convenience of most callers, the value is presented in packed form.
 *
 * Returns true on success, false on failure (if escontext points to an
 * ErrorSaveContext; otherwise errors are thrown).
 */
static bool
apply_typmod_special(Numeric num, int32 typmod, Node *escontext)
{
    int         precision;
    int         scale;
 
    Assert(NUMERIC_IS_SPECIAL(num));    /* caller error if not */
 
    /*
     * NaN is allowed regardless of the typmod; that's rather dubious perhaps,
     * but it's a longstanding behavior.  Inf is rejected if we have any
     * typmod restriction, since an infinity shouldn't be claimed to fit in
     * any finite number of digits.
     */
    if (NUMERIC_IS_NAN(num))
        return true;
 
    /* Do nothing if we have a default typmod (-1) */
    if (!is_valid_numeric_typmod(typmod))
        return true;
 
    precision = numeric_typmod_precision(typmod);
    scale = numeric_typmod_scale(typmod);
 
    ereturn(escontext, false,
            (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
             errmsg("numeric field overflow"),
             errdetail("A field with precision %d, scale %d cannot hold an infinite value.",
                       precision, scale)));
}
 
 
/*
 * Convert numeric to int8, rounding if needed.
 *
 * If overflow, return false (no error is raised).  Return true if okay.
 */
static bool
numericvar_to_int64(const NumericVar *var, int64 *result)
{
    NumericDigit *digits;
    int         ndigits;
    int         weight;
    int         i;
    int64       val;
    bool        neg;
    NumericVar  rounded;
 
    /* Round to nearest integer */
    init_var(&rounded);
    set_var_from_var(var, &rounded);
    round_var(&rounded, 0);
 
    /* Check for zero input */
    strip_var(&rounded);
    ndigits = rounded.ndigits;
    if (ndigits == 0)
    {
        *result = 0;
        free_var(&rounded);
        return true;
    }
 
    /*
     * For input like 10000000000, we must treat stripped digits as real. So
     * the loop assumes there are weight+1 digits before the decimal point.
     */
    weight = rounded.weight;
    Assert(weight >= 0 && ndigits <= weight + 1);
 
    /*
     * Construct the result. To avoid issues with converting a value
     * corresponding to INT64_MIN (which can't be represented as a positive 64
     * bit two's complement integer), accumulate value as a negative number.
     */
    digits = rounded.digits;
    neg = (rounded.sign == NUMERIC_NEG);
    val = -digits[0];
    for (i = 1; i <= weight; i++)
    {
        if (unlikely(pg_mul_s64_overflow(val, NBASE, &val)))
        {
            free_var(&rounded);
            return false;
        }
 
        if (i < ndigits)
        {
            if (unlikely(pg_sub_s64_overflow(val, digits[i], &val)))
            {
                free_var(&rounded);
                return false;
            }
        }
    }
 
    free_var(&rounded);
 
    if (!neg)
    {
        if (unlikely(val == PG_INT64_MIN))
            return false;
        val = -val;
    }
    *result = val;
 
    return true;
}
 
/*
 * Convert int8 value to numeric.
 */
static void
int64_to_numericvar(int64 val, NumericVar *var)
{
    uint64      uval,
                newuval;
    NumericDigit *ptr;
    int         ndigits;
 
    /* int64 can require at most 19 decimal digits; add one for safety */
    alloc_var(var, 20 / DEC_DIGITS);
    if (val < 0)
    {
        var->sign = NUMERIC_NEG;
        uval = pg_abs_s64(val);
    }
    else
    {
        var->sign = NUMERIC_POS;
        uval = val;
    }
    var->dscale = 0;
    if (val == 0)
    {
        var->ndigits = 0;
        var->weight = 0;
        return;
    }
    ptr = var->digits + var->ndigits;
    ndigits = 0;
    do
    {
        ptr--;
        ndigits++;
        newuval = uval / NBASE;
        *ptr = uval - newuval * NBASE;
        uval = newuval;
    } while (uval);
    var->digits = ptr;
    var->ndigits = ndigits;
    var->weight = ndigits - 1;
}
 
/*
 * Convert numeric to uint64, rounding if needed.
 *
 * If overflow, return false (no error is raised).  Return true if okay.
 */
static bool
numericvar_to_uint64(const NumericVar *var, uint64 *result)
{
    NumericDigit *digits;
    int         ndigits;
    int         weight;
    int         i;
    uint64      val;
    NumericVar  rounded;
 
    /* Round to nearest integer */
    init_var(&rounded);
    set_var_from_var(var, &rounded);
    round_var(&rounded, 0);
 
    /* Check for zero input */
    strip_var(&rounded);
    ndigits = rounded.ndigits;
    if (ndigits == 0)
    {
        *result = 0;
        free_var(&rounded);
        return true;
    }
 
    /* Check for negative input */
    if (rounded.sign == NUMERIC_NEG)
    {
        free_var(&rounded);
        return false;
    }
 
    /*
     * For input like 10000000000, we must treat stripped digits as real. So
     * the loop assumes there are weight+1 digits before the decimal point.
     */
    weight = rounded.weight;
    Assert(weight >= 0 && ndigits <= weight + 1);
 
    /* Construct the result */
    digits = rounded.digits;
    val = digits[0];
    for (i = 1; i <= weight; i++)
    {
        if (unlikely(pg_mul_u64_overflow(val, NBASE, &val)))
        {
            free_var(&rounded);
            return false;
        }
 
        if (i < ndigits)
        {
            if (unlikely(pg_add_u64_overflow(val, digits[i], &val)))
            {
                free_var(&rounded);
                return false;
            }
        }
    }
 
    free_var(&rounded);
 
    *result = val;
 
    return true;
}
 
#ifdef HAVE_INT128
/*
 * Convert numeric to int128, rounding if needed.
 *
 * If overflow, return false (no error is raised).  Return true if okay.
 */
static bool
numericvar_to_int128(const NumericVar *var, int128 *result)
{
    NumericDigit *digits;
    int         ndigits;
    int         weight;
    int         i;
    int128      val,
                oldval;
    bool        neg;
    NumericVar  rounded;
 
    /* Round to nearest integer */
    init_var(&rounded);
    set_var_from_var(var, &rounded);
    round_var(&rounded, 0);
 
    /* Check for zero input */
    strip_var(&rounded);
    ndigits = rounded.ndigits;
    if (ndigits == 0)
    {
        *result = 0;
        free_var(&rounded);
        return true;
    }
 
    /*
     * For input like 10000000000, we must treat stripped digits as real. So
     * the loop assumes there are weight+1 digits before the decimal point.
     */
    weight = rounded.weight;
    Assert(weight >= 0 && ndigits <= weight + 1);
 
    /* Construct the result */
    digits = rounded.digits;
    neg = (rounded.sign == NUMERIC_NEG);
    val = digits[0];
    for (i = 1; i <= weight; i++)
    {
        oldval = val;
        val *= NBASE;
        if (i < ndigits)
            val += digits[i];
 
        /*
         * The overflow check is a bit tricky because we want to accept
         * INT128_MIN, which will overflow the positive accumulator.  We can
         * detect this case easily though because INT128_MIN is the only
         * nonzero value for which -val == val (on a two's complement machine,
         * anyway).
         */
        if ((val / NBASE) != oldval)    /* possible overflow? */
        {
            if (!neg || (-val) != val || val == 0 || oldval < 0)
            {
                free_var(&rounded);
                return false;
            }
        }
    }
 
    free_var(&rounded);
 
    *result = neg ? -val : val;
    return true;
}
 
/*
 * Convert 128 bit integer to numeric.
 */
static void
int128_to_numericvar(int128 val, NumericVar *var)
{
    uint128     uval,
                newuval;
    NumericDigit *ptr;
    int         ndigits;
 
    /* int128 can require at most 39 decimal digits; add one for safety */
    alloc_var(var, 40 / DEC_DIGITS);
    if (val < 0)
    {
        var->sign = NUMERIC_NEG;
        uval = -val;
    }
    else
    {
        var->sign = NUMERIC_POS;
        uval = val;
    }
    var->dscale = 0;
    if (val == 0)
    {
        var->ndigits = 0;
        var->weight = 0;
        return;
    }
    ptr = var->digits + var->ndigits;
    ndigits = 0;
    do
    {
        ptr--;
        ndigits++;
        newuval = uval / NBASE;
        *ptr = uval - newuval * NBASE;
        uval = newuval;
    } while (uval);
    var->digits = ptr;
    var->ndigits = ndigits;
    var->weight = ndigits - 1;
}
#endif
 
/*
 * Convert a NumericVar to float8; if out of range, return +/- HUGE_VAL
 */
static double
numericvar_to_double_no_overflow(const NumericVar *var)
{
    char       *tmp;
    double      val;
    char       *endptr;
 
    tmp = get_str_from_var(var);
 
    /* unlike float8in, we ignore ERANGE from strtod */
    val = strtod(tmp, &endptr);
    if (*endptr != '\0')
    {
        /* shouldn't happen ... */
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
                 errmsg("invalid input syntax for type %s: \"%s\"",
                        "double precision", tmp)));
    }
 
    pfree(tmp);
 
    return val;
}
 
 
/*
 * cmp_var() -
 *
 *  Compare two values on variable level.  We assume zeroes have been
 *  truncated to no digits.
 */
static int
cmp_var(const NumericVar *var1, const NumericVar *var2)
{
    return cmp_var_common(var1->digits, var1->ndigits,
                          var1->weight, var1->sign,
                          var2->digits, var2->ndigits,
                          var2->weight, var2->sign);
}
 
/*
 * cmp_var_common() -
 *
 *  Main routine of cmp_var(). This function can be used by both
 *  NumericVar and Numeric.
 */
static int
cmp_var_common(const NumericDigit *var1digits, int var1ndigits,
               int var1weight, int var1sign,
               const NumericDigit *var2digits, int var2ndigits,
               int var2weight, int var2sign)
{
    if (var1ndigits == 0)
    {
        if (var2ndigits == 0)
            return 0;
        if (var2sign == NUMERIC_NEG)
            return 1;
        return -1;
    }
    if (var2ndigits == 0)
    {
        if (var1sign == NUMERIC_POS)
            return 1;
        return -1;
    }
 
    if (var1sign == NUMERIC_POS)
    {
        if (var2sign == NUMERIC_NEG)
            return 1;
        return cmp_abs_common(var1digits, var1ndigits, var1weight,
                              var2digits, var2ndigits, var2weight);
    }
 
    if (var2sign == NUMERIC_POS)
        return -1;
 
    return cmp_abs_common(var2digits, var2ndigits, var2weight,
                          var1digits, var1ndigits, var1weight);
}
 
 
/*
 * add_var() -
 *
 *  Full version of add functionality on variable level (handling signs).
 *  result might point to one of the operands too without danger.
 */
static void
add_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    /*
     * Decide on the signs of the two variables what to do
     */
    if (var1->sign == NUMERIC_POS)
    {
        if (var2->sign == NUMERIC_POS)
        {
            /*
             * Both are positive result = +(ABS(var1) + ABS(var2))
             */
            add_abs(var1, var2, result);
            result->sign = NUMERIC_POS;
        }
        else
        {
            /*
             * var1 is positive, var2 is negative Must compare absolute values
             */
            switch (cmp_abs(var1, var2))
            {
                case 0:
                    /* ----------
                     * ABS(var1) == ABS(var2)
                     * result = ZERO
                     * ----------
                     */
                    zero_var(result);
                    result->dscale = Max(var1->dscale, var2->dscale);
                    break;
 
                case 1:
                    /* ----------
                     * ABS(var1) > ABS(var2)
                     * result = +(ABS(var1) - ABS(var2))
                     * ----------
                     */
                    sub_abs(var1, var2, result);
                    result->sign = NUMERIC_POS;
                    break;
 
                case -1:
                    /* ----------
                     * ABS(var1) < ABS(var2)
                     * result = -(ABS(var2) - ABS(var1))
                     * ----------
                     */
                    sub_abs(var2, var1, result);
                    result->sign = NUMERIC_NEG;
                    break;
            }
        }
    }
    else
    {
        if (var2->sign == NUMERIC_POS)
        {
            /* ----------
             * var1 is negative, var2 is positive
             * Must compare absolute values
             * ----------
             */
            switch (cmp_abs(var1, var2))
            {
                case 0:
                    /* ----------
                     * ABS(var1) == ABS(var2)
                     * result = ZERO
                     * ----------
                     */
                    zero_var(result);
                    result->dscale = Max(var1->dscale, var2->dscale);
                    break;
 
                case 1:
                    /* ----------
                     * ABS(var1) > ABS(var2)
                     * result = -(ABS(var1) - ABS(var2))
                     * ----------
                     */
                    sub_abs(var1, var2, result);
                    result->sign = NUMERIC_NEG;
                    break;
 
                case -1:
                    /* ----------
                     * ABS(var1) < ABS(var2)
                     * result = +(ABS(var2) - ABS(var1))
                     * ----------
                     */
                    sub_abs(var2, var1, result);
                    result->sign = NUMERIC_POS;
                    break;
            }
        }
        else
        {
            /* ----------
             * Both are negative
             * result = -(ABS(var1) + ABS(var2))
             * ----------
             */
            add_abs(var1, var2, result);
            result->sign = NUMERIC_NEG;
        }
    }
}
 
 
/*
 * sub_var() -
 *
 *  Full version of sub functionality on variable level (handling signs).
 *  result might point to one of the operands too without danger.
 */
static void
sub_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    /*
     * Decide on the signs of the two variables what to do
     */
    if (var1->sign == NUMERIC_POS)
    {
        if (var2->sign == NUMERIC_NEG)
        {
            /* ----------
             * var1 is positive, var2 is negative
             * result = +(ABS(var1) + ABS(var2))
             * ----------
             */
            add_abs(var1, var2, result);
            result->sign = NUMERIC_POS;
        }
        else
        {
            /* ----------
             * Both are positive
             * Must compare absolute values
             * ----------
             */
            switch (cmp_abs(var1, var2))
            {
                case 0:
                    /* ----------
                     * ABS(var1) == ABS(var2)
                     * result = ZERO
                     * ----------
                     */
                    zero_var(result);
                    result->dscale = Max(var1->dscale, var2->dscale);
                    break;
 
                case 1:
                    /* ----------
                     * ABS(var1) > ABS(var2)
                     * result = +(ABS(var1) - ABS(var2))
                     * ----------
                     */
                    sub_abs(var1, var2, result);
                    result->sign = NUMERIC_POS;
                    break;
 
                case -1:
                    /* ----------
                     * ABS(var1) < ABS(var2)
                     * result = -(ABS(var2) - ABS(var1))
                     * ----------
                     */
                    sub_abs(var2, var1, result);
                    result->sign = NUMERIC_NEG;
                    break;
            }
        }
    }
    else
    {
        if (var2->sign == NUMERIC_NEG)
        {
            /* ----------
             * Both are negative
             * Must compare absolute values
             * ----------
             */
            switch (cmp_abs(var1, var2))
            {
                case 0:
                    /* ----------
                     * ABS(var1) == ABS(var2)
                     * result = ZERO
                     * ----------
                     */
                    zero_var(result);
                    result->dscale = Max(var1->dscale, var2->dscale);
                    break;
 
                case 1:
                    /* ----------
                     * ABS(var1) > ABS(var2)
                     * result = -(ABS(var1) - ABS(var2))
                     * ----------
                     */
                    sub_abs(var1, var2, result);
                    result->sign = NUMERIC_NEG;
                    break;
 
                case -1:
                    /* ----------
                     * ABS(var1) < ABS(var2)
                     * result = +(ABS(var2) - ABS(var1))
                     * ----------
                     */
                    sub_abs(var2, var1, result);
                    result->sign = NUMERIC_POS;
                    break;
            }
        }
        else
        {
            /* ----------
             * var1 is negative, var2 is positive
             * result = -(ABS(var1) + ABS(var2))
             * ----------
             */
            add_abs(var1, var2, result);
            result->sign = NUMERIC_NEG;
        }
    }
}
 
 
/*
 * mul_var() -
 *
 *  Multiplication on variable level. Product of var1 * var2 is stored
 *  in result.  Result is rounded to no more than rscale fractional digits.
 */
static void
mul_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result,
        int rscale)
{
    int         res_ndigits;
    int         res_ndigitpairs;
    int         res_sign;
    int         res_weight;
    int         pair_offset;
    int         maxdigits;
    int         maxdigitpairs;
    uint64     *dig,
               *dig_i1_off;
    uint64      maxdig;
    uint64      carry;
    uint64      newdig;
    int         var1ndigits;
    int         var2ndigits;
    int         var1ndigitpairs;
    int         var2ndigitpairs;
    NumericDigit *var1digits;
    NumericDigit *var2digits;
    uint32      var1digitpair;
    uint32     *var2digitpairs;
    NumericDigit *res_digits;
    int         i,
                i1,
                i2,
                i2limit;
 
    /*
     * Arrange for var1 to be the shorter of the two numbers.  This improves
     * performance because the inner multiplication loop is much simpler than
     * the outer loop, so it's better to have a smaller number of iterations
     * of the outer loop.  This also reduces the number of times that the
     * accumulator array needs to be normalized.
     */
    if (var1->ndigits > var2->ndigits)
    {
        const NumericVar *tmp = var1;
 
        var1 = var2;
        var2 = tmp;
    }
 
    /* copy these values into local vars for speed in inner loop */
    var1ndigits = var1->ndigits;
    var2ndigits = var2->ndigits;
    var1digits = var1->digits;
    var2digits = var2->digits;
 
    if (var1ndigits == 0)
    {
        /* one or both inputs is zero; so is result */
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * If var1 has 1-6 digits and the exact result was requested, delegate to
     * mul_var_short() which uses a faster direct multiplication algorithm.
     */
    if (var1ndigits <= 6 && rscale == var1->dscale + var2->dscale)
    {
        mul_var_short(var1, var2, result);
        return;
    }
 
    /* Determine result sign */
    if (var1->sign == var2->sign)
        res_sign = NUMERIC_POS;
    else
        res_sign = NUMERIC_NEG;
 
    /*
     * Determine the number of result digits to compute and the (maximum
     * possible) result weight.  If the exact result would have more than
     * rscale fractional digits, truncate the computation with
     * MUL_GUARD_DIGITS guard digits, i.e., ignore input digits that would
     * only contribute to the right of that.  (This will give the exact
     * rounded-to-rscale answer unless carries out of the ignored positions
     * would have propagated through more than MUL_GUARD_DIGITS digits.)
     *
     * Note: an exact computation could not produce more than var1ndigits +
     * var2ndigits digits, but we allocate at least one extra output digit in
     * case rscale-driven rounding produces a carry out of the highest exact
     * digit.
     *
     * The computation itself is done using base-NBASE^2 arithmetic, so we
     * actually process the input digits in pairs, producing a base-NBASE^2
     * intermediate result.  This significantly improves performance, since
     * schoolbook multiplication is O(N^2) in the number of input digits, and
     * working in base NBASE^2 effectively halves "N".
     *
     * Note: in a truncated computation, we must compute at least one extra
     * output digit to ensure that all the guard digits are fully computed.
     */
    /* digit pairs in each input */
    var1ndigitpairs = (var1ndigits + 1) / 2;
    var2ndigitpairs = (var2ndigits + 1) / 2;
 
    /* digits in exact result */
    res_ndigits = var1ndigits + var2ndigits;
 
    /* digit pairs in exact result with at least one extra output digit */
    res_ndigitpairs = res_ndigits / 2 + 1;
 
    /* pair offset to align result to end of dig[] */
    pair_offset = res_ndigitpairs - var1ndigitpairs - var2ndigitpairs + 1;
 
    /* maximum possible result weight (odd-length inputs shifted up below) */
    res_weight = var1->weight + var2->weight + 1 + 2 * res_ndigitpairs -
        res_ndigits - (var1ndigits & 1) - (var2ndigits & 1);
 
    /* rscale-based truncation with at least one extra output digit */
    maxdigits = res_weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS +
        MUL_GUARD_DIGITS;
    maxdigitpairs = maxdigits / 2 + 1;
 
    res_ndigitpairs = Min(res_ndigitpairs, maxdigitpairs);
    res_ndigits = 2 * res_ndigitpairs;
 
    /*
     * In the computation below, digit pair i1 of var1 and digit pair i2 of
     * var2 are multiplied and added to digit i1+i2+pair_offset of dig[]. Thus
     * input digit pairs with index >= res_ndigitpairs - pair_offset don't
     * contribute to the result, and can be ignored.
     */
    if (res_ndigitpairs <= pair_offset)
    {
        /* All input digits will be ignored; so result is zero */
        zero_var(result);
        result->dscale = rscale;
        return;
    }
    var1ndigitpairs = Min(var1ndigitpairs, res_ndigitpairs - pair_offset);
    var2ndigitpairs = Min(var2ndigitpairs, res_ndigitpairs - pair_offset);
 
    /*
     * We do the arithmetic in an array "dig[]" of unsigned 64-bit integers.
     * Since PG_UINT64_MAX is much larger than NBASE^4, this gives us a lot of
     * headroom to avoid normalizing carries immediately.
     *
     * maxdig tracks the maximum possible value of any dig[] entry; when this
     * threatens to exceed PG_UINT64_MAX, we take the time to propagate
     * carries.  Furthermore, we need to ensure that overflow doesn't occur
     * during the carry propagation passes either.  The carry values could be
     * as much as PG_UINT64_MAX / NBASE^2, so really we must normalize when
     * digits threaten to exceed PG_UINT64_MAX - PG_UINT64_MAX / NBASE^2.
     *
     * To avoid overflow in maxdig itself, it actually represents the maximum
     * possible value divided by NBASE^2-1, i.e., at the top of the loop it is
     * known that no dig[] entry exceeds maxdig * (NBASE^2-1).
     *
     * The conversion of var1 to base NBASE^2 is done on the fly, as each new
     * digit is required.  The digits of var2 are converted upfront, and
     * stored at the end of dig[].  To avoid loss of precision, the input
     * digits are aligned with the start of digit pair array, effectively
     * shifting them up (multiplying by NBASE) if the inputs have an odd
     * number of NBASE digits.
     */
    dig = (uint64 *) palloc(res_ndigitpairs * sizeof(uint64) +
                            var2ndigitpairs * sizeof(uint32));
 
    /* convert var2 to base NBASE^2, shifting up if its length is odd */
    var2digitpairs = (uint32 *) (dig + res_ndigitpairs);
 
    for (i2 = 0; i2 < var2ndigitpairs - 1; i2++)
        var2digitpairs[i2] = var2digits[2 * i2] * NBASE + var2digits[2 * i2 + 1];
 
    if (2 * i2 + 1 < var2ndigits)
        var2digitpairs[i2] = var2digits[2 * i2] * NBASE + var2digits[2 * i2 + 1];
    else
        var2digitpairs[i2] = var2digits[2 * i2] * NBASE;
 
    /*
     * Start by multiplying var2 by the least significant contributing digit
     * pair from var1, storing the results at the end of dig[], and filling
     * the leading digits with zeros.
     *
     * The loop here is the same as the inner loop below, except that we set
     * the results in dig[], rather than adding to them.  This is the
     * performance bottleneck for multiplication, so we want to keep it simple
     * enough so that it can be auto-vectorized.  Accordingly, process the
     * digits left-to-right even though schoolbook multiplication would
     * suggest right-to-left.  Since we aren't propagating carries in this
     * loop, the order does not matter.
     */
    i1 = var1ndigitpairs - 1;
    if (2 * i1 + 1 < var1ndigits)
        var1digitpair = var1digits[2 * i1] * NBASE + var1digits[2 * i1 + 1];
    else
        var1digitpair = var1digits[2 * i1] * NBASE;
    maxdig = var1digitpair;
 
    i2limit = Min(var2ndigitpairs, res_ndigitpairs - i1 - pair_offset);
    dig_i1_off = &dig[i1 + pair_offset];
 
    memset(dig, 0, (i1 + pair_offset) * sizeof(uint64));
    for (i2 = 0; i2 < i2limit; i2++)
        dig_i1_off[i2] = (uint64) var1digitpair * var2digitpairs[i2];
 
    /*
     * Next, multiply var2 by the remaining digit pairs from var1, adding the
     * results to dig[] at the appropriate offsets, and normalizing whenever
     * there is a risk of any dig[] entry overflowing.
     */
    for (i1 = i1 - 1; i1 >= 0; i1--)
    {
        var1digitpair = var1digits[2 * i1] * NBASE + var1digits[2 * i1 + 1];
        if (var1digitpair == 0)
            continue;
 
        /* Time to normalize? */
        maxdig += var1digitpair;
        if (maxdig > (PG_UINT64_MAX - PG_UINT64_MAX / NBASE_SQR) / (NBASE_SQR - 1))
        {
            /* Yes, do it (to base NBASE^2) */
            carry = 0;
            for (i = res_ndigitpairs - 1; i >= 0; i--)
            {
                newdig = dig[i] + carry;
                if (newdig >= NBASE_SQR)
                {
                    carry = newdig / NBASE_SQR;
                    newdig -= carry * NBASE_SQR;
                }
                else
                    carry = 0;
                dig[i] = newdig;
            }
            Assert(carry == 0);
            /* Reset maxdig to indicate new worst-case */
            maxdig = 1 + var1digitpair;
        }
 
        /* Multiply and add */
        i2limit = Min(var2ndigitpairs, res_ndigitpairs - i1 - pair_offset);
        dig_i1_off = &dig[i1 + pair_offset];
 
        for (i2 = 0; i2 < i2limit; i2++)
            dig_i1_off[i2] += (uint64) var1digitpair * var2digitpairs[i2];
    }
 
    /*
     * Now we do a final carry propagation pass to normalize back to base
     * NBASE^2, and construct the base-NBASE result digits.  Note that this is
     * still done at full precision w/guard digits.
     */
    alloc_var(result, res_ndigits);
    res_digits = result->digits;
    carry = 0;
    for (i = res_ndigitpairs - 1; i >= 0; i--)
    {
        newdig = dig[i] + carry;
        if (newdig >= NBASE_SQR)
        {
            carry = newdig / NBASE_SQR;
            newdig -= carry * NBASE_SQR;
        }
        else
            carry = 0;
        res_digits[2 * i + 1] = (NumericDigit) ((uint32) newdig % NBASE);
        res_digits[2 * i] = (NumericDigit) ((uint32) newdig / NBASE);
    }
    Assert(carry == 0);
 
    pfree(dig);
 
    /*
     * Finally, round the result to the requested precision.
     */
    result->weight = res_weight;
    result->sign = res_sign;
 
    /* Round to target rscale (and set result->dscale) */
    round_var(result, rscale);
 
    /* Strip leading and trailing zeroes */
    strip_var(result);
}
 
 
/*
 * mul_var_short() -
 *
 *  Special-case multiplication function used when var1 has 1-6 digits, var2
 *  has at least as many digits as var1, and the exact product var1 * var2 is
 *  requested.
 */
static void
mul_var_short(const NumericVar *var1, const NumericVar *var2,
              NumericVar *result)
{
    int         var1ndigits = var1->ndigits;
    int         var2ndigits = var2->ndigits;
    NumericDigit *var1digits = var1->digits;
    NumericDigit *var2digits = var2->digits;
    int         res_sign;
    int         res_weight;
    int         res_ndigits;
    NumericDigit *res_buf;
    NumericDigit *res_digits;
    uint32      carry = 0;
    uint32      term;
 
    /* Check preconditions */
    Assert(var1ndigits >= 1);
    Assert(var1ndigits <= 6);
    Assert(var2ndigits >= var1ndigits);
 
    /*
     * Determine the result sign, weight, and number of digits to calculate.
     * The weight figured here is correct if the product has no leading zero
     * digits; otherwise strip_var() will fix things up.  Note that, unlike
     * mul_var(), we do not need to allocate an extra output digit, because we
     * are not rounding here.
     */
    if (var1->sign == var2->sign)
        res_sign = NUMERIC_POS;
    else
        res_sign = NUMERIC_NEG;
    res_weight = var1->weight + var2->weight + 1;
    res_ndigits = var1ndigits + var2ndigits;
 
    /* Allocate result digit array */
    res_buf = digitbuf_alloc(res_ndigits + 1);
    res_buf[0] = 0;             /* spare digit for later rounding */
    res_digits = res_buf + 1;
 
    /*
     * Compute the result digits in reverse, in one pass, propagating the
     * carry up as we go.  The i'th result digit consists of the sum of the
     * products var1digits[i1] * var2digits[i2] for which i = i1 + i2 + 1.
     */
#define PRODSUM1(v1,i1,v2,i2) ((v1)[(i1)] * (v2)[(i2)])
#define PRODSUM2(v1,i1,v2,i2) (PRODSUM1(v1,i1,v2,i2) + (v1)[(i1)+1] * (v2)[(i2)-1])
#define PRODSUM3(v1,i1,v2,i2) (PRODSUM2(v1,i1,v2,i2) + (v1)[(i1)+2] * (v2)[(i2)-2])
#define PRODSUM4(v1,i1,v2,i2) (PRODSUM3(v1,i1,v2,i2) + (v1)[(i1)+3] * (v2)[(i2)-3])
#define PRODSUM5(v1,i1,v2,i2) (PRODSUM4(v1,i1,v2,i2) + (v1)[(i1)+4] * (v2)[(i2)-4])
#define PRODSUM6(v1,i1,v2,i2) (PRODSUM5(v1,i1,v2,i2) + (v1)[(i1)+5] * (v2)[(i2)-5])
 
    switch (var1ndigits)
    {
        case 1:
            /* ---------
             * 1-digit case:
             *      var1ndigits = 1
             *      var2ndigits >= 1
             *      res_ndigits = var2ndigits + 1
             * ----------
             */
            for (int i = var2ndigits - 1; i >= 0; i--)
            {
                term = PRODSUM1(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            res_digits[0] = (NumericDigit) carry;
            break;
 
        case 2:
            /* ---------
             * 2-digit case:
             *      var1ndigits = 2
             *      var2ndigits >= 2
             *      res_ndigits = var2ndigits + 2
             * ----------
             */
            /* last result digit and carry */
            term = PRODSUM1(var1digits, 1, var2digits, var2ndigits - 1);
            res_digits[res_ndigits - 1] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            /* remaining digits, except for the first two */
            for (int i = var2ndigits - 1; i >= 1; i--)
            {
                term = PRODSUM2(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            break;
 
        case 3:
            /* ---------
             * 3-digit case:
             *      var1ndigits = 3
             *      var2ndigits >= 3
             *      res_ndigits = var2ndigits + 3
             * ----------
             */
            /* last two result digits */
            term = PRODSUM1(var1digits, 2, var2digits, var2ndigits - 1);
            res_digits[res_ndigits - 1] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM2(var1digits, 1, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 2] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            /* remaining digits, except for the first three */
            for (int i = var2ndigits - 1; i >= 2; i--)
            {
                term = PRODSUM3(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            break;
 
        case 4:
            /* ---------
             * 4-digit case:
             *      var1ndigits = 4
             *      var2ndigits >= 4
             *      res_ndigits = var2ndigits + 4
             * ----------
             */
            /* last three result digits */
            term = PRODSUM1(var1digits, 3, var2digits, var2ndigits - 1);
            res_digits[res_ndigits - 1] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM2(var1digits, 2, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 2] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM3(var1digits, 1, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 3] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            /* remaining digits, except for the first four */
            for (int i = var2ndigits - 1; i >= 3; i--)
            {
                term = PRODSUM4(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            break;
 
        case 5:
            /* ---------
             * 5-digit case:
             *      var1ndigits = 5
             *      var2ndigits >= 5
             *      res_ndigits = var2ndigits + 5
             * ----------
             */
            /* last four result digits */
            term = PRODSUM1(var1digits, 4, var2digits, var2ndigits - 1);
            res_digits[res_ndigits - 1] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM2(var1digits, 3, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 2] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM3(var1digits, 2, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 3] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM4(var1digits, 1, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 4] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            /* remaining digits, except for the first five */
            for (int i = var2ndigits - 1; i >= 4; i--)
            {
                term = PRODSUM5(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            break;
 
        case 6:
            /* ---------
             * 6-digit case:
             *      var1ndigits = 6
             *      var2ndigits >= 6
             *      res_ndigits = var2ndigits + 6
             * ----------
             */
            /* last five result digits */
            term = PRODSUM1(var1digits, 5, var2digits, var2ndigits - 1);
            res_digits[res_ndigits - 1] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM2(var1digits, 4, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 2] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM3(var1digits, 3, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 3] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM4(var1digits, 2, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 4] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            term = PRODSUM5(var1digits, 1, var2digits, var2ndigits - 1) + carry;
            res_digits[res_ndigits - 5] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
 
            /* remaining digits, except for the first six */
            for (int i = var2ndigits - 1; i >= 5; i--)
            {
                term = PRODSUM6(var1digits, 0, var2digits, i) + carry;
                res_digits[i + 1] = (NumericDigit) (term % NBASE);
                carry = term / NBASE;
            }
            break;
    }
 
    /*
     * Finally, for var1ndigits > 1, compute the remaining var1ndigits most
     * significant result digits.
     */
    switch (var1ndigits)
    {
        case 6:
            term = PRODSUM5(var1digits, 0, var2digits, 4) + carry;
            res_digits[5] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
            /* FALLTHROUGH */
        case 5:
            term = PRODSUM4(var1digits, 0, var2digits, 3) + carry;
            res_digits[4] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
            /* FALLTHROUGH */
        case 4:
            term = PRODSUM3(var1digits, 0, var2digits, 2) + carry;
            res_digits[3] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
            /* FALLTHROUGH */
        case 3:
            term = PRODSUM2(var1digits, 0, var2digits, 1) + carry;
            res_digits[2] = (NumericDigit) (term % NBASE);
            carry = term / NBASE;
            /* FALLTHROUGH */
        case 2:
            term = PRODSUM1(var1digits, 0, var2digits, 0) + carry;
            res_digits[1] = (NumericDigit) (term % NBASE);
            res_digits[0] = (NumericDigit) (term / NBASE);
            break;
    }
 
    /* Store the product in result */
    digitbuf_free(result->buf);
    result->ndigits = res_ndigits;
    result->buf = res_buf;
    result->digits = res_digits;
    result->weight = res_weight;
    result->sign = res_sign;
    result->dscale = var1->dscale + var2->dscale;
 
    /* Strip leading and trailing zeroes */
    strip_var(result);
}
 
 
/*
 * div_var() -
 *
 *  Compute the quotient var1 / var2 to rscale fractional digits.
 *
 *  If "round" is true, the result is rounded at the rscale'th digit; if
 *  false, it is truncated (towards zero) at that digit.
 *
 *  If "exact" is true, the exact result is computed to the specified rscale;
 *  if false, successive quotient digits are approximated up to rscale plus
 *  DIV_GUARD_DIGITS extra digits, ignoring all contributions from digits to
 *  the right of that, before rounding or truncating to the specified rscale.
 *  This can be significantly faster, and usually gives the same result as the
 *  exact computation, but it may occasionally be off by one in the final
 *  digit, if contributions from the ignored digits would have propagated
 *  through the guard digits.  This is good enough for the transcendental
 *  functions, where small errors are acceptable.
 */
static void
div_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result,
        int rscale, bool round, bool exact)
{
    int         var1ndigits = var1->ndigits;
    int         var2ndigits = var2->ndigits;
    int         res_sign;
    int         res_weight;
    int         res_ndigits;
    int         var1ndigitpairs;
    int         var2ndigitpairs;
    int         res_ndigitpairs;
    int         div_ndigitpairs;
    int64      *dividend;
    int32      *divisor;
    double      fdivisor,
                fdivisorinverse,
                fdividend,
                fquotient;
    int64       maxdiv;
    int         qi;
    int32       qdigit;
    int64       carry;
    int64       newdig;
    int64      *remainder;
    NumericDigit *res_digits;
    int         i;
 
    /*
     * First of all division by zero check; we must not be handed an
     * unnormalized divisor.
     */
    if (var2ndigits == 0 || var2->digits[0] == 0)
        ereport(ERROR,
                (errcode(ERRCODE_DIVISION_BY_ZERO),
                 errmsg("division by zero")));
 
    /*
     * If the divisor has just one or two digits, delegate to div_var_int(),
     * which uses fast short division.
     *
     * Similarly, on platforms with 128-bit integer support, delegate to
     * div_var_int64() for divisors with three or four digits.
     */
    if (var2ndigits <= 2)
    {
        int         idivisor;
        int         idivisor_weight;
 
        idivisor = var2->digits[0];
        idivisor_weight = var2->weight;
        if (var2ndigits == 2)
        {
            idivisor = idivisor * NBASE + var2->digits[1];
            idivisor_weight--;
        }
        if (var2->sign == NUMERIC_NEG)
            idivisor = -idivisor;
 
        div_var_int(var1, idivisor, idivisor_weight, result, rscale, round);
        return;
    }
#ifdef HAVE_INT128
    if (var2ndigits <= 4)
    {
        int64       idivisor;
        int         idivisor_weight;
 
        idivisor = var2->digits[0];
        idivisor_weight = var2->weight;
        for (i = 1; i < var2ndigits; i++)
        {
            idivisor = idivisor * NBASE + var2->digits[i];
            idivisor_weight--;
        }
        if (var2->sign == NUMERIC_NEG)
            idivisor = -idivisor;
 
        div_var_int64(var1, idivisor, idivisor_weight, result, rscale, round);
        return;
    }
#endif
 
    /*
     * Otherwise, perform full long division.
     */
 
    /* Result zero check */
    if (var1ndigits == 0)
    {
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * The approximate computation can be significantly faster than the exact
     * one, since the working dividend is var2ndigitpairs base-NBASE^2 digits
     * shorter below.  However, that comes with the tradeoff of computing
     * DIV_GUARD_DIGITS extra base-NBASE result digits.  Ignoring all other
     * overheads, that suggests that, in theory, the approximate computation
     * will only be faster than the exact one when var2ndigits is greater than
     * 2 * (DIV_GUARD_DIGITS + 1), independent of the size of var1.
     *
     * Thus, we're better off doing an exact computation when var2 is shorter
     * than this.  Empirically, it has been found that the exact threshold is
     * a little higher, due to other overheads in the outer division loop.
     */
    if (var2ndigits <= 2 * (DIV_GUARD_DIGITS + 2))
        exact = true;
 
    /*
     * Determine the result sign, weight and number of digits to calculate.
     * The weight figured here is correct if the emitted quotient has no
     * leading zero digits; otherwise strip_var() will fix things up.
     */
    if (var1->sign == var2->sign)
        res_sign = NUMERIC_POS;
    else
        res_sign = NUMERIC_NEG;
    res_weight = var1->weight - var2->weight + 1;
    /* The number of accurate result digits we need to produce: */
    res_ndigits = res_weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS;
    /* ... but always at least 1 */
    res_ndigits = Max(res_ndigits, 1);
    /* If rounding needed, figure one more digit to ensure correct result */
    if (round)
        res_ndigits++;
    /* Add guard digits for roundoff error when producing approx result */
    if (!exact)
        res_ndigits += DIV_GUARD_DIGITS;
 
    /*
     * The computation itself is done using base-NBASE^2 arithmetic, so we
     * actually process the input digits in pairs, producing a base-NBASE^2
     * intermediate result.  This significantly improves performance, since
     * the computation is O(N^2) in the number of input digits, and working in
     * base NBASE^2 effectively halves "N".
     */
    var1ndigitpairs = (var1ndigits + 1) / 2;
    var2ndigitpairs = (var2ndigits + 1) / 2;
    res_ndigitpairs = (res_ndigits + 1) / 2;
    res_ndigits = 2 * res_ndigitpairs;
 
    /*
     * We do the arithmetic in an array "dividend[]" of signed 64-bit
     * integers.  Since PG_INT64_MAX is much larger than NBASE^4, this gives
     * us a lot of headroom to avoid normalizing carries immediately.
     *
     * When performing an exact computation, the working dividend requires
     * res_ndigitpairs + var2ndigitpairs digits.  If var1 is larger than that,
     * the extra digits do not contribute to the result, and are ignored.
     *
     * When performing an approximate computation, the working dividend only
     * requires res_ndigitpairs digits (which includes the extra guard
     * digits).  All input digits beyond that are ignored.
     */
    if (exact)
    {
        div_ndigitpairs = res_ndigitpairs + var2ndigitpairs;
        var1ndigitpairs = Min(var1ndigitpairs, div_ndigitpairs);
    }
    else
    {
        div_ndigitpairs = res_ndigitpairs;
        var1ndigitpairs = Min(var1ndigitpairs, div_ndigitpairs);
        var2ndigitpairs = Min(var2ndigitpairs, div_ndigitpairs);
    }
 
    /*
     * Allocate room for the working dividend (div_ndigitpairs 64-bit digits)
     * plus the divisor (var2ndigitpairs 32-bit base-NBASE^2 digits).
     *
     * For convenience, we allocate one extra dividend digit, which is set to
     * zero and not counted in div_ndigitpairs, so that the main loop below
     * can safely read and write the (qi+1)'th digit in the approximate case.
     */
    dividend = (int64 *) palloc((div_ndigitpairs + 1) * sizeof(int64) +
                                var2ndigitpairs * sizeof(int32));
    divisor = (int32 *) (dividend + div_ndigitpairs + 1);
 
    /* load var1 into dividend[0 .. var1ndigitpairs-1], zeroing the rest */
    for (i = 0; i < var1ndigitpairs - 1; i++)
        dividend[i] = var1->digits[2 * i] * NBASE + var1->digits[2 * i + 1];
 
    if (2 * i + 1 < var1ndigits)
        dividend[i] = var1->digits[2 * i] * NBASE + var1->digits[2 * i + 1];
    else
        dividend[i] = var1->digits[2 * i] * NBASE;
 
    memset(dividend + i + 1, 0, (div_ndigitpairs - i) * sizeof(int64));
 
    /* load var2 into divisor[0 .. var2ndigitpairs-1] */
    for (i = 0; i < var2ndigitpairs - 1; i++)
        divisor[i] = var2->digits[2 * i] * NBASE + var2->digits[2 * i + 1];
 
    if (2 * i + 1 < var2ndigits)
        divisor[i] = var2->digits[2 * i] * NBASE + var2->digits[2 * i + 1];
    else
        divisor[i] = var2->digits[2 * i] * NBASE;
 
    /*
     * We estimate each quotient digit using floating-point arithmetic, taking
     * the first 2 base-NBASE^2 digits of the (current) dividend and divisor.
     * This must be float to avoid overflow.
     *
     * Since the floating-point dividend and divisor use 4 base-NBASE input
     * digits, they include roughly 40-53 bits of information from their
     * respective inputs (assuming NBASE is 10000), which fits well in IEEE
     * double-precision variables.  The relative error in the floating-point
     * quotient digit will then be less than around 2/NBASE^3, so the
     * estimated base-NBASE^2 quotient digit will typically be correct, and
     * should not be off by more than one from the correct value.
     */
    fdivisor = (double) divisor[0] * NBASE_SQR;
    if (var2ndigitpairs > 1)
        fdivisor += (double) divisor[1];
    fdivisorinverse = 1.0 / fdivisor;
 
    /*
     * maxdiv tracks the maximum possible absolute value of any dividend[]
     * entry; when this threatens to exceed PG_INT64_MAX, we take the time to
     * propagate carries.  Furthermore, we need to ensure that overflow
     * doesn't occur during the carry propagation passes either.  The carry
     * values may have an absolute value as high as PG_INT64_MAX/NBASE^2 + 1,
     * so really we must normalize when digits threaten to exceed PG_INT64_MAX
     * - PG_INT64_MAX/NBASE^2 - 1.
     *
     * To avoid overflow in maxdiv itself, it represents the max absolute
     * value divided by NBASE^2-1, i.e., at the top of the loop it is known
     * that no dividend[] entry has an absolute value exceeding maxdiv *
     * (NBASE^2-1).
     *
     * Actually, though, that holds good only for dividend[] entries after
     * dividend[qi]; the adjustment done at the bottom of the loop may cause
     * dividend[qi + 1] to exceed the maxdiv limit, so that dividend[qi] in
     * the next iteration is beyond the limit.  This does not cause problems,
     * as explained below.
     */
    maxdiv = 1;
 
    /*
     * Outer loop computes next quotient digit, which goes in dividend[qi].
     */
    for (qi = 0; qi < res_ndigitpairs; qi++)
    {
        /* Approximate the current dividend value */
        fdividend = (double) dividend[qi] * NBASE_SQR;
        fdividend += (double) dividend[qi + 1];
 
        /* Compute the (approximate) quotient digit */
        fquotient = fdividend * fdivisorinverse;
        qdigit = (fquotient >= 0.0) ? ((int32) fquotient) :
            (((int32) fquotient) - 1);  /* truncate towards -infinity */
 
        if (qdigit != 0)
        {
            /* Do we need to normalize now? */
            maxdiv += i64abs(qdigit);
            if (maxdiv > (PG_INT64_MAX - PG_INT64_MAX / NBASE_SQR - 1) / (NBASE_SQR - 1))
            {
                /*
                 * Yes, do it.  Note that if var2ndigitpairs is much smaller
                 * than div_ndigitpairs, we can save a significant amount of
                 * effort here by noting that we only need to normalise those
                 * dividend[] entries touched where prior iterations
                 * subtracted multiples of the divisor.
                 */
                carry = 0;
                for (i = Min(qi + var2ndigitpairs - 2, div_ndigitpairs - 1); i > qi; i--)
                {
                    newdig = dividend[i] + carry;
                    if (newdig < 0)
                    {
                        carry = -((-newdig - 1) / NBASE_SQR) - 1;
                        newdig -= carry * NBASE_SQR;
                    }
                    else if (newdig >= NBASE_SQR)
                    {
                        carry = newdig / NBASE_SQR;
                        newdig -= carry * NBASE_SQR;
                    }
                    else
                        carry = 0;
                    dividend[i] = newdig;
                }
                dividend[qi] += carry;
 
                /*
                 * All the dividend[] digits except possibly dividend[qi] are
                 * now in the range 0..NBASE^2-1.  We do not need to consider
                 * dividend[qi] in the maxdiv value anymore, so we can reset
                 * maxdiv to 1.
                 */
                maxdiv = 1;
 
                /*
                 * Recompute the quotient digit since new info may have
                 * propagated into the top two dividend digits.
                 */
                fdividend = (double) dividend[qi] * NBASE_SQR;
                fdividend += (double) dividend[qi + 1];
                fquotient = fdividend * fdivisorinverse;
                qdigit = (fquotient >= 0.0) ? ((int32) fquotient) :
                    (((int32) fquotient) - 1);  /* truncate towards -infinity */
 
                maxdiv += i64abs(qdigit);
            }
 
            /*
             * Subtract off the appropriate multiple of the divisor.
             *
             * The digits beyond dividend[qi] cannot overflow, because we know
             * they will fall within the maxdiv limit.  As for dividend[qi]
             * itself, note that qdigit is approximately trunc(dividend[qi] /
             * divisor[0]), which would make the new value simply dividend[qi]
             * mod divisor[0].  The lower-order terms in qdigit can change
             * this result by not more than about twice PG_INT64_MAX/NBASE^2,
             * so overflow is impossible.
             *
             * This inner loop is the performance bottleneck for division, so
             * code it in the same way as the inner loop of mul_var() so that
             * it can be auto-vectorized.
             */
            if (qdigit != 0)
            {
                int         istop = Min(var2ndigitpairs, div_ndigitpairs - qi);
                int64      *dividend_qi = &dividend[qi];
 
                for (i = 0; i < istop; i++)
                    dividend_qi[i] -= (int64) qdigit * divisor[i];
            }
        }
 
        /*
         * The dividend digit we are about to replace might still be nonzero.
         * Fold it into the next digit position.
         *
         * There is no risk of overflow here, although proving that requires
         * some care.  Much as with the argument for dividend[qi] not
         * overflowing, if we consider the first two terms in the numerator
         * and denominator of qdigit, we can see that the final value of
         * dividend[qi + 1] will be approximately a remainder mod
         * (divisor[0]*NBASE^2 + divisor[1]).  Accounting for the lower-order
         * terms is a bit complicated but ends up adding not much more than
         * PG_INT64_MAX/NBASE^2 to the possible range.  Thus, dividend[qi + 1]
         * cannot overflow here, and in its role as dividend[qi] in the next
         * loop iteration, it can't be large enough to cause overflow in the
         * carry propagation step (if any), either.
         *
         * But having said that: dividend[qi] can be more than
         * PG_INT64_MAX/NBASE^2, as noted above, which means that the product
         * dividend[qi] * NBASE^2 *can* overflow.  When that happens, adding
         * it to dividend[qi + 1] will always cause a canceling overflow so
         * that the end result is correct.  We could avoid the intermediate
         * overflow by doing the multiplication and addition using unsigned
         * int64 arithmetic, which is modulo 2^64, but so far there appears no
         * need.
         */
        dividend[qi + 1] += dividend[qi] * NBASE_SQR;
 
        dividend[qi] = qdigit;
    }
 
    /*
     * If an exact result was requested, use the remainder to correct the
     * approximate quotient.  The remainder is in dividend[], immediately
     * after the quotient digits.  Note, however, that although the remainder
     * starts at dividend[qi = res_ndigitpairs], the first digit is the result
     * of folding two remainder digits into one above, and the remainder
     * currently only occupies var2ndigitpairs - 1 digits (the last digit of
     * the working dividend was untouched by the computation above).  Thus we
     * expand the remainder down by one base-NBASE^2 digit when we normalize
     * it, so that it completely fills the last var2ndigitpairs digits of the
     * dividend array.
     */
    if (exact)
    {
        /* Normalize the remainder, expanding it down by one digit */
        remainder = &dividend[qi];
        carry = 0;
        for (i = var2ndigitpairs - 2; i >= 0; i--)
        {
            newdig = remainder[i] + carry;
            if (newdig < 0)
            {
                carry = -((-newdig - 1) / NBASE_SQR) - 1;
                newdig -= carry * NBASE_SQR;
            }
            else if (newdig >= NBASE_SQR)
            {
                carry = newdig / NBASE_SQR;
                newdig -= carry * NBASE_SQR;
            }
            else
                carry = 0;
            remainder[i + 1] = newdig;
        }
        remainder[0] = carry;
 
        if (remainder[0] < 0)
        {
            /*
             * The remainder is negative, so the approximate quotient is too
             * large.  Correct by reducing the quotient by one and adding the
             * divisor to the remainder until the remainder is positive.  We
             * expect the quotient to be off by at most one, which has been
             * borne out in all testing, but not conclusively proven, so we
             * allow for larger corrections, just in case.
             */
            do
            {
                /* Add the divisor to the remainder */
                carry = 0;
                for (i = var2ndigitpairs - 1; i > 0; i--)
                {
                    newdig = remainder[i] + divisor[i] + carry;
                    if (newdig >= NBASE_SQR)
                    {
                        remainder[i] = newdig - NBASE_SQR;
                        carry = 1;
                    }
                    else
                    {
                        remainder[i] = newdig;
                        carry = 0;
                    }
                }
                remainder[0] += divisor[0] + carry;
 
                /* Subtract 1 from the quotient (propagating carries later) */
                dividend[qi - 1]--;
 
            } while (remainder[0] < 0);
        }
        else
        {
            /*
             * The remainder is nonnegative.  If it's greater than or equal to
             * the divisor, then the approximate quotient is too small and
             * must be corrected.  As above, we don't expect to have to apply
             * more than one correction, but allow for it just in case.
             */
            while (true)
            {
                bool        less = false;
 
                /* Is remainder < divisor? */
                for (i = 0; i < var2ndigitpairs; i++)
                {
                    if (remainder[i] < divisor[i])
                    {
                        less = true;
                        break;
                    }
                    if (remainder[i] > divisor[i])
                        break;  /* remainder > divisor */
                }
                if (less)
                    break;      /* quotient is correct */
 
                /* Subtract the divisor from the remainder */
                carry = 0;
                for (i = var2ndigitpairs - 1; i > 0; i--)
                {
                    newdig = remainder[i] - divisor[i] + carry;
                    if (newdig < 0)
                    {
                        remainder[i] = newdig + NBASE_SQR;
                        carry = -1;
                    }
                    else
                    {
                        remainder[i] = newdig;
                        carry = 0;
                    }
                }
                remainder[0] = remainder[0] - divisor[0] + carry;
 
                /* Add 1 to the quotient (propagating carries later) */
                dividend[qi - 1]++;
            }
        }
    }
 
    /*
     * Because the quotient digits were estimates that might have been off by
     * one (and we didn't bother propagating carries when adjusting the
     * quotient above), some quotient digits might be out of range, so do a
     * final carry propagation pass to normalize back to base NBASE^2, and
     * construct the base-NBASE result digits.  Note that this is still done
     * at full precision w/guard digits.
     */
    alloc_var(result, res_ndigits);
    res_digits = result->digits;
    carry = 0;
    for (i = res_ndigitpairs - 1; i >= 0; i--)
    {
        newdig = dividend[i] + carry;
        if (newdig < 0)
        {
            carry = -((-newdig - 1) / NBASE_SQR) - 1;
            newdig -= carry * NBASE_SQR;
        }
        else if (newdig >= NBASE_SQR)
        {
            carry = newdig / NBASE_SQR;
            newdig -= carry * NBASE_SQR;
        }
        else
            carry = 0;
        res_digits[2 * i + 1] = (NumericDigit) ((uint32) newdig % NBASE);
        res_digits[2 * i] = (NumericDigit) ((uint32) newdig / NBASE);
    }
    Assert(carry == 0);
 
    pfree(dividend);
 
    /*
     * Finally, round or truncate the result to the requested precision.
     */
    result->weight = res_weight;
    result->sign = res_sign;
 
    /* Round or truncate to target rscale (and set result->dscale) */
    if (round)
        round_var(result, rscale);
    else
        trunc_var(result, rscale);
 
    /* Strip leading and trailing zeroes */
    strip_var(result);
}
 
 
/*
 * div_var_int() -
 *
 *  Divide a numeric variable by a 32-bit integer with the specified weight.
 *  The quotient var / (ival * NBASE^ival_weight) is stored in result.
 */
static void
div_var_int(const NumericVar *var, int ival, int ival_weight,
            NumericVar *result, int rscale, bool round)
{
    NumericDigit *var_digits = var->digits;
    int         var_ndigits = var->ndigits;
    int         res_sign;
    int         res_weight;
    int         res_ndigits;
    NumericDigit *res_buf;
    NumericDigit *res_digits;
    uint32      divisor;
    int         i;
 
    /* Guard against division by zero */
    if (ival == 0)
        ereport(ERROR,
                errcode(ERRCODE_DIVISION_BY_ZERO),
                errmsg("division by zero"));
 
    /* Result zero check */
    if (var_ndigits == 0)
    {
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * Determine the result sign, weight and number of digits to calculate.
     * The weight figured here is correct if the emitted quotient has no
     * leading zero digits; otherwise strip_var() will fix things up.
     */
    if (var->sign == NUMERIC_POS)
        res_sign = ival > 0 ? NUMERIC_POS : NUMERIC_NEG;
    else
        res_sign = ival > 0 ? NUMERIC_NEG : NUMERIC_POS;
    res_weight = var->weight - ival_weight;
    /* The number of accurate result digits we need to produce: */
    res_ndigits = res_weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS;
    /* ... but always at least 1 */
    res_ndigits = Max(res_ndigits, 1);
    /* If rounding needed, figure one more digit to ensure correct result */
    if (round)
        res_ndigits++;
 
    res_buf = digitbuf_alloc(res_ndigits + 1);
    res_buf[0] = 0;             /* spare digit for later rounding */
    res_digits = res_buf + 1;
 
    /*
     * Now compute the quotient digits.  This is the short division algorithm
     * described in Knuth volume 2, section 4.3.1 exercise 16, except that we
     * allow the divisor to exceed the internal base.
     *
     * In this algorithm, the carry from one digit to the next is at most
     * divisor - 1.  Therefore, while processing the next digit, carry may
     * become as large as divisor * NBASE - 1, and so it requires a 64-bit
     * integer if this exceeds UINT_MAX.
     */
    divisor = abs(ival);
 
    if (divisor <= UINT_MAX / NBASE)
    {
        /* carry cannot overflow 32 bits */
        uint32      carry = 0;
 
        for (i = 0; i < res_ndigits; i++)
        {
            carry = carry * NBASE + (i < var_ndigits ? var_digits[i] : 0);
            res_digits[i] = (NumericDigit) (carry / divisor);
            carry = carry % divisor;
        }
    }
    else
    {
        /* carry may exceed 32 bits */
        uint64      carry = 0;
 
        for (i = 0; i < res_ndigits; i++)
        {
            carry = carry * NBASE + (i < var_ndigits ? var_digits[i] : 0);
            res_digits[i] = (NumericDigit) (carry / divisor);
            carry = carry % divisor;
        }
    }
 
    /* Store the quotient in result */
    digitbuf_free(result->buf);
    result->ndigits = res_ndigits;
    result->buf = res_buf;
    result->digits = res_digits;
    result->weight = res_weight;
    result->sign = res_sign;
 
    /* Round or truncate to target rscale (and set result->dscale) */
    if (round)
        round_var(result, rscale);
    else
        trunc_var(result, rscale);
 
    /* Strip leading/trailing zeroes */
    strip_var(result);
}
 
 
#ifdef HAVE_INT128
/*
 * div_var_int64() -
 *
 *  Divide a numeric variable by a 64-bit integer with the specified weight.
 *  The quotient var / (ival * NBASE^ival_weight) is stored in result.
 *
 *  This duplicates the logic in div_var_int(), so any changes made there
 *  should be made here too.
 */
static void
div_var_int64(const NumericVar *var, int64 ival, int ival_weight,
              NumericVar *result, int rscale, bool round)
{
    NumericDigit *var_digits = var->digits;
    int         var_ndigits = var->ndigits;
    int         res_sign;
    int         res_weight;
    int         res_ndigits;
    NumericDigit *res_buf;
    NumericDigit *res_digits;
    uint64      divisor;
    int         i;
 
    /* Guard against division by zero */
    if (ival == 0)
        ereport(ERROR,
                errcode(ERRCODE_DIVISION_BY_ZERO),
                errmsg("division by zero"));
 
    /* Result zero check */
    if (var_ndigits == 0)
    {
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * Determine the result sign, weight and number of digits to calculate.
     * The weight figured here is correct if the emitted quotient has no
     * leading zero digits; otherwise strip_var() will fix things up.
     */
    if (var->sign == NUMERIC_POS)
        res_sign = ival > 0 ? NUMERIC_POS : NUMERIC_NEG;
    else
        res_sign = ival > 0 ? NUMERIC_NEG : NUMERIC_POS;
    res_weight = var->weight - ival_weight;
    /* The number of accurate result digits we need to produce: */
    res_ndigits = res_weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS;
    /* ... but always at least 1 */
    res_ndigits = Max(res_ndigits, 1);
    /* If rounding needed, figure one more digit to ensure correct result */
    if (round)
        res_ndigits++;
 
    res_buf = digitbuf_alloc(res_ndigits + 1);
    res_buf[0] = 0;             /* spare digit for later rounding */
    res_digits = res_buf + 1;
 
    /*
     * Now compute the quotient digits.  This is the short division algorithm
     * described in Knuth volume 2, section 4.3.1 exercise 16, except that we
     * allow the divisor to exceed the internal base.
     *
     * In this algorithm, the carry from one digit to the next is at most
     * divisor - 1.  Therefore, while processing the next digit, carry may
     * become as large as divisor * NBASE - 1, and so it requires a 128-bit
     * integer if this exceeds PG_UINT64_MAX.
     */
    divisor = i64abs(ival);
 
    if (divisor <= PG_UINT64_MAX / NBASE)
    {
        /* carry cannot overflow 64 bits */
        uint64      carry = 0;
 
        for (i = 0; i < res_ndigits; i++)
        {
            carry = carry * NBASE + (i < var_ndigits ? var_digits[i] : 0);
            res_digits[i] = (NumericDigit) (carry / divisor);
            carry = carry % divisor;
        }
    }
    else
    {
        /* carry may exceed 64 bits */
        uint128     carry = 0;
 
        for (i = 0; i < res_ndigits; i++)
        {
            carry = carry * NBASE + (i < var_ndigits ? var_digits[i] : 0);
            res_digits[i] = (NumericDigit) (carry / divisor);
            carry = carry % divisor;
        }
    }
 
    /* Store the quotient in result */
    digitbuf_free(result->buf);
    result->ndigits = res_ndigits;
    result->buf = res_buf;
    result->digits = res_digits;
    result->weight = res_weight;
    result->sign = res_sign;
 
    /* Round or truncate to target rscale (and set result->dscale) */
    if (round)
        round_var(result, rscale);
    else
        trunc_var(result, rscale);
 
    /* Strip leading/trailing zeroes */
    strip_var(result);
}
#endif
 
 
/*
 * Default scale selection for division
 *
 * Returns the appropriate result scale for the division result.
 */
static int
select_div_scale(const NumericVar *var1, const NumericVar *var2)
{
    int         weight1,
                weight2,
                qweight,
                i;
    NumericDigit firstdigit1,
                firstdigit2;
    int         rscale;
 
    /*
     * The result scale of a division isn't specified in any SQL standard. For
     * PostgreSQL we select a result scale that will give at least
     * NUMERIC_MIN_SIG_DIGITS significant digits, so that numeric gives a
     * result no less accurate than float8; but use a scale not less than
     * either input's display scale.
     */
 
    /* Get the actual (normalized) weight and first digit of each input */
 
    weight1 = 0;                /* values to use if var1 is zero */
    firstdigit1 = 0;
    for (i = 0; i < var1->ndigits; i++)
    {
        firstdigit1 = var1->digits[i];
        if (firstdigit1 != 0)
        {
            weight1 = var1->weight - i;
            break;
        }
    }
 
    weight2 = 0;                /* values to use if var2 is zero */
    firstdigit2 = 0;
    for (i = 0; i < var2->ndigits; i++)
    {
        firstdigit2 = var2->digits[i];
        if (firstdigit2 != 0)
        {
            weight2 = var2->weight - i;
            break;
        }
    }
 
    /*
     * Estimate weight of quotient.  If the two first digits are equal, we
     * can't be sure, but assume that var1 is less than var2.
     */
    qweight = weight1 - weight2;
    if (firstdigit1 <= firstdigit2)
        qweight--;
 
    /* Select result scale */
    rscale = NUMERIC_MIN_SIG_DIGITS - qweight * DEC_DIGITS;
    rscale = Max(rscale, var1->dscale);
    rscale = Max(rscale, var2->dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    return rscale;
}
 
 
/*
 * mod_var() -
 *
 *  Calculate the modulo of two numerics at variable level
 */
static void
mod_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    NumericVar  tmp;
 
    init_var(&tmp);
 
    /* ---------
     * We do this using the equation
     *      mod(x,y) = x - trunc(x/y)*y
     * div_var can be persuaded to give us trunc(x/y) directly.
     * ----------
     */
    div_var(var1, var2, &tmp, 0, false, true);
 
    mul_var(var2, &tmp, &tmp, var2->dscale);
 
    sub_var(var1, &tmp, result);
 
    free_var(&tmp);
}
 
 
/*
 * div_mod_var() -
 *
 *  Calculate the truncated integer quotient and numeric remainder of two
 *  numeric variables.  The remainder is precise to var2's dscale.
 */
static void
div_mod_var(const NumericVar *var1, const NumericVar *var2,
            NumericVar *quot, NumericVar *rem)
{
    NumericVar  q;
    NumericVar  r;
 
    init_var(&q);
    init_var(&r);
 
    /*
     * Use div_var() with exact = false to get an initial estimate for the
     * integer quotient (truncated towards zero).  This might be slightly
     * inaccurate, but we correct it below.
     */
    div_var(var1, var2, &q, 0, false, false);
 
    /* Compute initial estimate of remainder using the quotient estimate. */
    mul_var(var2, &q, &r, var2->dscale);
    sub_var(var1, &r, &r);
 
    /*
     * Adjust the results if necessary --- the remainder should have the same
     * sign as var1, and its absolute value should be less than the absolute
     * value of var2.
     */
    while (r.ndigits != 0 && r.sign != var1->sign)
    {
        /* The absolute value of the quotient is too large */
        if (var1->sign == var2->sign)
        {
            sub_var(&q, &const_one, &q);
            add_var(&r, var2, &r);
        }
        else
        {
            add_var(&q, &const_one, &q);
            sub_var(&r, var2, &r);
        }
    }
 
    while (cmp_abs(&r, var2) >= 0)
    {
        /* The absolute value of the quotient is too small */
        if (var1->sign == var2->sign)
        {
            add_var(&q, &const_one, &q);
            sub_var(&r, var2, &r);
        }
        else
        {
            sub_var(&q, &const_one, &q);
            add_var(&r, var2, &r);
        }
    }
 
    set_var_from_var(&q, quot);
    set_var_from_var(&r, rem);
 
    free_var(&q);
    free_var(&r);
}
 
 
/*
 * ceil_var() -
 *
 *  Return the smallest integer greater than or equal to the argument
 *  on variable level
 */
static void
ceil_var(const NumericVar *var, NumericVar *result)
{
    NumericVar  tmp;
 
    init_var(&tmp);
    set_var_from_var(var, &tmp);
 
    trunc_var(&tmp, 0);
 
    if (var->sign == NUMERIC_POS && cmp_var(var, &tmp) != 0)
        add_var(&tmp, &const_one, &tmp);
 
    set_var_from_var(&tmp, result);
    free_var(&tmp);
}
 
 
/*
 * floor_var() -
 *
 *  Return the largest integer equal to or less than the argument
 *  on variable level
 */
static void
floor_var(const NumericVar *var, NumericVar *result)
{
    NumericVar  tmp;
 
    init_var(&tmp);
    set_var_from_var(var, &tmp);
 
    trunc_var(&tmp, 0);
 
    if (var->sign == NUMERIC_NEG && cmp_var(var, &tmp) != 0)
        sub_var(&tmp, &const_one, &tmp);
 
    set_var_from_var(&tmp, result);
    free_var(&tmp);
}
 
 
/*
 * gcd_var() -
 *
 *  Calculate the greatest common divisor of two numerics at variable level
 */
static void
gcd_var(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    int         res_dscale;
    int         cmp;
    NumericVar  tmp_arg;
    NumericVar  mod;
 
    res_dscale = Max(var1->dscale, var2->dscale);
 
    /*
     * Arrange for var1 to be the number with the greater absolute value.
     *
     * This would happen automatically in the loop below, but avoids an
     * expensive modulo operation.
     */
    cmp = cmp_abs(var1, var2);
    if (cmp < 0)
    {
        const NumericVar *tmp = var1;
 
        var1 = var2;
        var2 = tmp;
    }
 
    /*
     * Also avoid the taking the modulo if the inputs have the same absolute
     * value, or if the smaller input is zero.
     */
    if (cmp == 0 || var2->ndigits == 0)
    {
        set_var_from_var(var1, result);
        result->sign = NUMERIC_POS;
        result->dscale = res_dscale;
        return;
    }
 
    init_var(&tmp_arg);
    init_var(&mod);
 
    /* Use the Euclidean algorithm to find the GCD */
    set_var_from_var(var1, &tmp_arg);
    set_var_from_var(var2, result);
 
    for (;;)
    {
        /* this loop can take a while, so allow it to be interrupted */
        CHECK_FOR_INTERRUPTS();
 
        mod_var(&tmp_arg, result, &mod);
        if (mod.ndigits == 0)
            break;
        set_var_from_var(result, &tmp_arg);
        set_var_from_var(&mod, result);
    }
    result->sign = NUMERIC_POS;
    result->dscale = res_dscale;
 
    free_var(&tmp_arg);
    free_var(&mod);
}
 
 
/*
 * sqrt_var() -
 *
 *  Compute the square root of x using the Karatsuba Square Root algorithm.
 *  NOTE: we allow rscale < 0 here, implying rounding before the decimal
 *  point.
 */
static void
sqrt_var(const NumericVar *arg, NumericVar *result, int rscale)
{
    int         stat;
    int         res_weight;
    int         res_ndigits;
    int         src_ndigits;
    int         step;
    int         ndigits[32];
    int         blen;
    int64       arg_int64;
    int         src_idx;
    int64       s_int64;
    int64       r_int64;
    NumericVar  s_var;
    NumericVar  r_var;
    NumericVar  a0_var;
    NumericVar  a1_var;
    NumericVar  q_var;
    NumericVar  u_var;
 
    stat = cmp_var(arg, &const_zero);
    if (stat == 0)
    {
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * SQL2003 defines sqrt() in terms of power, so we need to emit the right
     * SQLSTATE error code if the operand is negative.
     */
    if (stat < 0)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                 errmsg("cannot take square root of a negative number")));
 
    init_var(&s_var);
    init_var(&r_var);
    init_var(&a0_var);
    init_var(&a1_var);
    init_var(&q_var);
    init_var(&u_var);
 
    /*
     * The result weight is half the input weight, rounded towards minus
     * infinity --- res_weight = floor(arg->weight / 2).
     */
    if (arg->weight >= 0)
        res_weight = arg->weight / 2;
    else
        res_weight = -((-arg->weight - 1) / 2 + 1);
 
    /*
     * Number of NBASE digits to compute.  To ensure correct rounding, compute
     * at least 1 extra decimal digit.  We explicitly allow rscale to be
     * negative here, but must always compute at least 1 NBASE digit.  Thus
     * res_ndigits = res_weight + 1 + ceil((rscale + 1) / DEC_DIGITS) or 1.
     */
    if (rscale + 1 >= 0)
        res_ndigits = res_weight + 1 + (rscale + DEC_DIGITS) / DEC_DIGITS;
    else
        res_ndigits = res_weight + 1 - (-rscale - 1) / DEC_DIGITS;
    res_ndigits = Max(res_ndigits, 1);
 
    /*
     * Number of source NBASE digits logically required to produce a result
     * with this precision --- every digit before the decimal point, plus 2
     * for each result digit after the decimal point (or minus 2 for each
     * result digit we round before the decimal point).
     */
    src_ndigits = arg->weight + 1 + (res_ndigits - res_weight - 1) * 2;
    src_ndigits = Max(src_ndigits, 1);
 
    /* ----------
     * From this point on, we treat the input and the result as integers and
     * compute the integer square root and remainder using the Karatsuba
     * Square Root algorithm, which may be written recursively as follows:
     *
     *  SqrtRem(n = a3*b^3 + a2*b^2 + a1*b + a0):
     *      [ for some base b, and coefficients a0,a1,a2,a3 chosen so that
     *        0 <= a0,a1,a2 < b and a3 >= b/4 ]
     *      Let (s,r) = SqrtRem(a3*b + a2)
     *      Let (q,u) = DivRem(r*b + a1, 2*s)
     *      Let s = s*b + q
     *      Let r = u*b + a0 - q^2
     *      If r < 0 Then
     *          Let r = r + s
     *          Let s = s - 1
     *          Let r = r + s
     *      Return (s,r)
     *
     * See "Karatsuba Square Root", Paul Zimmermann, INRIA Research Report
     * RR-3805, November 1999.  At the time of writing this was available
     * on the net at <https://hal.inria.fr/inria-00072854>.
     *
     * The way to read the assumption "n = a3*b^3 + a2*b^2 + a1*b + a0" is
     * "choose a base b such that n requires at least four base-b digits to
     * express; then those digits are a3,a2,a1,a0, with a3 possibly larger
     * than b".  For optimal performance, b should have approximately a
     * quarter the number of digits in the input, so that the outer square
     * root computes roughly twice as many digits as the inner one.  For
     * simplicity, we choose b = NBASE^blen, an integer power of NBASE.
     *
     * We implement the algorithm iteratively rather than recursively, to
     * allow the working variables to be reused.  With this approach, each
     * digit of the input is read precisely once --- src_idx tracks the number
     * of input digits used so far.
     *
     * The array ndigits[] holds the number of NBASE digits of the input that
     * will have been used at the end of each iteration, which roughly doubles
     * each time.  Note that the array elements are stored in reverse order,
     * so if the final iteration requires src_ndigits = 37 input digits, the
     * array will contain [37,19,11,7,5,3], and we would start by computing
     * the square root of the 3 most significant NBASE digits.
     *
     * In each iteration, we choose blen to be the largest integer for which
     * the input number has a3 >= b/4, when written in the form above.  In
     * general, this means blen = src_ndigits / 4 (truncated), but if
     * src_ndigits is a multiple of 4, that might lead to the coefficient a3
     * being less than b/4 (if the first input digit is less than NBASE/4), in
     * which case we choose blen = src_ndigits / 4 - 1.  The number of digits
     * in the inner square root is then src_ndigits - 2*blen.  So, for
     * example, if we have src_ndigits = 26 initially, the array ndigits[]
     * will be either [26,14,8,4] or [26,14,8,6,4], depending on the size of
     * the first input digit.
     *
     * Additionally, we can put an upper bound on the number of steps required
     * as follows --- suppose that the number of source digits is an n-bit
     * number in the range [2^(n-1), 2^n-1], then blen will be in the range
     * [2^(n-3)-1, 2^(n-2)-1] and the number of digits in the inner square
     * root will be in the range [2^(n-2), 2^(n-1)+1].  In the next step, blen
     * will be in the range [2^(n-4)-1, 2^(n-3)] and the number of digits in
     * the next inner square root will be in the range [2^(n-3), 2^(n-2)+1].
     * This pattern repeats, and in the worst case the array ndigits[] will
     * contain [2^n-1, 2^(n-1)+1, 2^(n-2)+1, ... 9, 5, 3], and the computation
     * will require n steps.  Therefore, since all digit array sizes are
     * signed 32-bit integers, the number of steps required is guaranteed to
     * be less than 32.
     * ----------
     */
    step = 0;
    while ((ndigits[step] = src_ndigits) > 4)
    {
        /* Choose b so that a3 >= b/4, as described above */
        blen = src_ndigits / 4;
        if (blen * 4 == src_ndigits && arg->digits[0] < NBASE / 4)
            blen--;
 
        /* Number of digits in the next step (inner square root) */
        src_ndigits -= 2 * blen;
        step++;
    }
 
    /*
     * First iteration (innermost square root and remainder):
     *
     * Here src_ndigits <= 4, and the input fits in an int64.  Its square root
     * has at most 9 decimal digits, so estimate it using double precision
     * arithmetic, which will in fact almost certainly return the correct
     * result with no further correction required.
     */
    arg_int64 = arg->digits[0];
    for (src_idx = 1; src_idx < src_ndigits; src_idx++)
    {
        arg_int64 *= NBASE;
        if (src_idx < arg->ndigits)
            arg_int64 += arg->digits[src_idx];
    }
 
    s_int64 = (int64) sqrt((double) arg_int64);
    r_int64 = arg_int64 - s_int64 * s_int64;
 
    /*
     * Use Newton's method to correct the result, if necessary.
     *
     * This uses integer division with truncation to compute the truncated
     * integer square root by iterating using the formula x -> (x + n/x) / 2.
     * This is known to converge to isqrt(n), unless n+1 is a perfect square.
     * If n+1 is a perfect square, the sequence will oscillate between the two
     * values isqrt(n) and isqrt(n)+1, so we can be assured of convergence by
     * checking the remainder.
     */
    while (r_int64 < 0 || r_int64 > 2 * s_int64)
    {
        s_int64 = (s_int64 + arg_int64 / s_int64) / 2;
        r_int64 = arg_int64 - s_int64 * s_int64;
    }
 
    /*
     * Iterations with src_ndigits <= 8:
     *
     * The next 1 or 2 iterations compute larger (outer) square roots with
     * src_ndigits <= 8, so the result still fits in an int64 (even though the
     * input no longer does) and we can continue to compute using int64
     * variables to avoid more expensive numeric computations.
     *
     * It is fairly easy to see that there is no risk of the intermediate
     * values below overflowing 64-bit integers.  In the worst case, the
     * previous iteration will have computed a 3-digit square root (of a
     * 6-digit input less than NBASE^6 / 4), so at the start of this
     * iteration, s will be less than NBASE^3 / 2 = 10^12 / 2, and r will be
     * less than 10^12.  In this case, blen will be 1, so numer will be less
     * than 10^17, and denom will be less than 10^12 (and hence u will also be
     * less than 10^12).  Finally, since q^2 = u*b + a0 - r, we can also be
     * sure that q^2 < 10^17.  Therefore all these quantities fit comfortably
     * in 64-bit integers.
     */
    step--;
    while (step >= 0 && (src_ndigits = ndigits[step]) <= 8)
    {
        int         b;
        int         a0;
        int         a1;
        int         i;
        int64       numer;
        int64       denom;
        int64       q;
        int64       u;
 
        blen = (src_ndigits - src_idx) / 2;
 
        /* Extract a1 and a0, and compute b */
        a0 = 0;
        a1 = 0;
        b = 1;
 
        for (i = 0; i < blen; i++, src_idx++)
        {
            b *= NBASE;
            a1 *= NBASE;
            if (src_idx < arg->ndigits)
                a1 += arg->digits[src_idx];
        }
 
        for (i = 0; i < blen; i++, src_idx++)
        {
            a0 *= NBASE;
            if (src_idx < arg->ndigits)
                a0 += arg->digits[src_idx];
        }
 
        /* Compute (q,u) = DivRem(r*b + a1, 2*s) */
        numer = r_int64 * b + a1;
        denom = 2 * s_int64;
        q = numer / denom;
        u = numer - q * denom;
 
        /* Compute s = s*b + q and r = u*b + a0 - q^2 */
        s_int64 = s_int64 * b + q;
        r_int64 = u * b + a0 - q * q;
 
        if (r_int64 < 0)
        {
            /* s is too large by 1; set r += s, s--, r += s */
            r_int64 += s_int64;
            s_int64--;
            r_int64 += s_int64;
        }
 
        Assert(src_idx == src_ndigits); /* All input digits consumed */
        step--;
    }
 
    /*
     * On platforms with 128-bit integer support, we can further delay the
     * need to use numeric variables.
     */
#ifdef HAVE_INT128
    if (step >= 0)
    {
        int128      s_int128;
        int128      r_int128;
 
        s_int128 = s_int64;
        r_int128 = r_int64;
 
        /*
         * Iterations with src_ndigits <= 16:
         *
         * The result fits in an int128 (even though the input doesn't) so we
         * use int128 variables to avoid more expensive numeric computations.
         */
        while (step >= 0 && (src_ndigits = ndigits[step]) <= 16)
        {
            int64       b;
            int64       a0;
            int64       a1;
            int64       i;
            int128      numer;
            int128      denom;
            int128      q;
            int128      u;
 
            blen = (src_ndigits - src_idx) / 2;
 
            /* Extract a1 and a0, and compute b */
            a0 = 0;
            a1 = 0;
            b = 1;
 
            for (i = 0; i < blen; i++, src_idx++)
            {
                b *= NBASE;
                a1 *= NBASE;
                if (src_idx < arg->ndigits)
                    a1 += arg->digits[src_idx];
            }
 
            for (i = 0; i < blen; i++, src_idx++)
            {
                a0 *= NBASE;
                if (src_idx < arg->ndigits)
                    a0 += arg->digits[src_idx];
            }
 
            /* Compute (q,u) = DivRem(r*b + a1, 2*s) */
            numer = r_int128 * b + a1;
            denom = 2 * s_int128;
            q = numer / denom;
            u = numer - q * denom;
 
            /* Compute s = s*b + q and r = u*b + a0 - q^2 */
            s_int128 = s_int128 * b + q;
            r_int128 = u * b + a0 - q * q;
 
            if (r_int128 < 0)
            {
                /* s is too large by 1; set r += s, s--, r += s */
                r_int128 += s_int128;
                s_int128--;
                r_int128 += s_int128;
            }
 
            Assert(src_idx == src_ndigits); /* All input digits consumed */
            step--;
        }
 
        /*
         * All remaining iterations require numeric variables.  Convert the
         * integer values to NumericVar and continue.  Note that in the final
         * iteration we don't need the remainder, so we can save a few cycles
         * there by not fully computing it.
         */
        int128_to_numericvar(s_int128, &s_var);
        if (step >= 0)
            int128_to_numericvar(r_int128, &r_var);
    }
    else
    {
        int64_to_numericvar(s_int64, &s_var);
        /* step < 0, so we certainly don't need r */
    }
#else                           /* !HAVE_INT128 */
    int64_to_numericvar(s_int64, &s_var);
    if (step >= 0)
        int64_to_numericvar(r_int64, &r_var);
#endif                          /* HAVE_INT128 */
 
    /*
     * The remaining iterations with src_ndigits > 8 (or 16, if have int128)
     * use numeric variables.
     */
    while (step >= 0)
    {
        int         tmp_len;
 
        src_ndigits = ndigits[step];
        blen = (src_ndigits - src_idx) / 2;
 
        /* Extract a1 and a0 */
        if (src_idx < arg->ndigits)
        {
            tmp_len = Min(blen, arg->ndigits - src_idx);
            alloc_var(&a1_var, tmp_len);
            memcpy(a1_var.digits, arg->digits + src_idx,
                   tmp_len * sizeof(NumericDigit));
            a1_var.weight = blen - 1;
            a1_var.sign = NUMERIC_POS;
            a1_var.dscale = 0;
            strip_var(&a1_var);
        }
        else
        {
            zero_var(&a1_var);
            a1_var.dscale = 0;
        }
        src_idx += blen;
 
        if (src_idx < arg->ndigits)
        {
            tmp_len = Min(blen, arg->ndigits - src_idx);
            alloc_var(&a0_var, tmp_len);
            memcpy(a0_var.digits, arg->digits + src_idx,
                   tmp_len * sizeof(NumericDigit));
            a0_var.weight = blen - 1;
            a0_var.sign = NUMERIC_POS;
            a0_var.dscale = 0;
            strip_var(&a0_var);
        }
        else
        {
            zero_var(&a0_var);
            a0_var.dscale = 0;
        }
        src_idx += blen;
 
        /* Compute (q,u) = DivRem(r*b + a1, 2*s) */
        set_var_from_var(&r_var, &q_var);
        q_var.weight += blen;
        add_var(&q_var, &a1_var, &q_var);
        add_var(&s_var, &s_var, &u_var);
        div_mod_var(&q_var, &u_var, &q_var, &u_var);
 
        /* Compute s = s*b + q */
        s_var.weight += blen;
        add_var(&s_var, &q_var, &s_var);
 
        /*
         * Compute r = u*b + a0 - q^2.
         *
         * In the final iteration, we don't actually need r; we just need to
         * know whether it is negative, so that we know whether to adjust s.
         * So instead of the final subtraction we can just compare.
         */
        u_var.weight += blen;
        add_var(&u_var, &a0_var, &u_var);
        mul_var(&q_var, &q_var, &q_var, 0);
 
        if (step > 0)
        {
            /* Need r for later iterations */
            sub_var(&u_var, &q_var, &r_var);
            if (r_var.sign == NUMERIC_NEG)
            {
                /* s is too large by 1; set r += s, s--, r += s */
                add_var(&r_var, &s_var, &r_var);
                sub_var(&s_var, &const_one, &s_var);
                add_var(&r_var, &s_var, &r_var);
            }
        }
        else
        {
            /* Don't need r anymore, except to test if s is too large by 1 */
            if (cmp_var(&u_var, &q_var) < 0)
                sub_var(&s_var, &const_one, &s_var);
        }
 
        Assert(src_idx == src_ndigits); /* All input digits consumed */
        step--;
    }
 
    /*
     * Construct the final result, rounding it to the requested precision.
     */
    set_var_from_var(&s_var, result);
    result->weight = res_weight;
    result->sign = NUMERIC_POS;
 
    /* Round to target rscale (and set result->dscale) */
    round_var(result, rscale);
 
    /* Strip leading and trailing zeroes */
    strip_var(result);
 
    free_var(&s_var);
    free_var(&r_var);
    free_var(&a0_var);
    free_var(&a1_var);
    free_var(&q_var);
    free_var(&u_var);
}
 
 
/*
 * exp_var() -
 *
 *  Raise e to the power of x, computed to rscale fractional digits
 */
static void
exp_var(const NumericVar *arg, NumericVar *result, int rscale)
{
    NumericVar  x;
    NumericVar  elem;
    int         ni;
    double      val;
    int         dweight;
    int         ndiv2;
    int         sig_digits;
    int         local_rscale;
 
    init_var(&x);
    init_var(&elem);
 
    set_var_from_var(arg, &x);
 
    /*
     * Estimate the dweight of the result using floating point arithmetic, so
     * that we can choose an appropriate local rscale for the calculation.
     */
    val = numericvar_to_double_no_overflow(&x);
 
    /* Guard against overflow/underflow */
    /* If you change this limit, see also power_var()'s limit */
    if (fabs(val) >= NUMERIC_MAX_RESULT_SCALE * 3)
    {
        if (val > 0)
            ereport(ERROR,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("value overflows numeric format")));
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /* decimal weight = log10(e^x) = x * log10(e) */
    dweight = (int) (val * 0.434294481903252);
 
    /*
     * Reduce x to the range -0.01 <= x <= 0.01 (approximately) by dividing by
     * 2^ndiv2, to improve the convergence rate of the Taylor series.
     *
     * Note that the overflow check above ensures that fabs(x) < 6000, which
     * means that ndiv2 <= 20 here.
     */
    if (fabs(val) > 0.01)
    {
        ndiv2 = 1;
        val /= 2;
 
        while (fabs(val) > 0.01)
        {
            ndiv2++;
            val /= 2;
        }
 
        local_rscale = x.dscale + ndiv2;
        div_var_int(&x, 1 << ndiv2, 0, &x, local_rscale, true);
    }
    else
        ndiv2 = 0;
 
    /*
     * Set the scale for the Taylor series expansion.  The final result has
     * (dweight + rscale + 1) significant digits.  In addition, we have to
     * raise the Taylor series result to the power 2^ndiv2, which introduces
     * an error of up to around log10(2^ndiv2) digits, so work with this many
     * extra digits of precision (plus a few more for good measure).
     */
    sig_digits = 1 + dweight + rscale + (int) (ndiv2 * 0.301029995663981);
    sig_digits = Max(sig_digits, 0) + 8;
 
    local_rscale = sig_digits - 1;
 
    /*
     * Use the Taylor series
     *
     * exp(x) = 1 + x + x^2/2! + x^3/3! + ...
     *
     * Given the limited range of x, this should converge reasonably quickly.
     * We run the series until the terms fall below the local_rscale limit.
     */
    add_var(&const_one, &x, result);
 
    mul_var(&x, &x, &elem, local_rscale);
    ni = 2;
    div_var_int(&elem, ni, 0, &elem, local_rscale, true);
 
    while (elem.ndigits != 0)
    {
        add_var(result, &elem, result);
 
        mul_var(&elem, &x, &elem, local_rscale);
        ni++;
        div_var_int(&elem, ni, 0, &elem, local_rscale, true);
    }
 
    /*
     * Compensate for the argument range reduction.  Since the weight of the
     * result doubles with each multiplication, we can reduce the local rscale
     * as we proceed.
     */
    while (ndiv2-- > 0)
    {
        local_rscale = sig_digits - result->weight * 2 * DEC_DIGITS;
        local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
        mul_var(result, result, result, local_rscale);
    }
 
    /* Round to requested rscale */
    round_var(result, rscale);
 
    free_var(&x);
    free_var(&elem);
}
 
 
/*
 * Estimate the dweight of the most significant decimal digit of the natural
 * logarithm of a number.
 *
 * Essentially, we're approximating log10(abs(ln(var))).  This is used to
 * determine the appropriate rscale when computing natural logarithms.
 *
 * Note: many callers call this before range-checking the input.  Therefore,
 * we must be robust against values that are invalid to apply ln() to.
 * We don't wish to throw an error here, so just return zero in such cases.
 */
static int
estimate_ln_dweight(const NumericVar *var)
{
    int         ln_dweight;
 
    /* Caller should fail on ln(negative), but for the moment return zero */
    if (var->sign != NUMERIC_POS)
        return 0;
 
    if (cmp_var(var, &const_zero_point_nine) >= 0 &&
        cmp_var(var, &const_one_point_one) <= 0)
    {
        /*
         * 0.9 <= var <= 1.1
         *
         * ln(var) has a negative weight (possibly very large).  To get a
         * reasonably accurate result, estimate it using ln(1+x) ~= x.
         */
        NumericVar  x;
 
        init_var(&x);
        sub_var(var, &const_one, &x);
 
        if (x.ndigits > 0)
        {
            /* Use weight of most significant decimal digit of x */
            ln_dweight = x.weight * DEC_DIGITS + (int) log10(x.digits[0]);
        }
        else
        {
            /* x = 0.  Since ln(1) = 0 exactly, we don't need extra digits */
            ln_dweight = 0;
        }
 
        free_var(&x);
    }
    else
    {
        /*
         * Estimate the logarithm using the first couple of digits from the
         * input number.  This will give an accurate result whenever the input
         * is not too close to 1.
         */
        if (var->ndigits > 0)
        {
            int         digits;
            int         dweight;
            double      ln_var;
 
            digits = var->digits[0];
            dweight = var->weight * DEC_DIGITS;
 
            if (var->ndigits > 1)
            {
                digits = digits * NBASE + var->digits[1];
                dweight -= DEC_DIGITS;
            }
 
            /*----------
             * We have var ~= digits * 10^dweight
             * so ln(var) ~= ln(digits) + dweight * ln(10)
             *----------
             */
            ln_var = log((double) digits) + dweight * 2.302585092994046;
            ln_dweight = (int) log10(fabs(ln_var));
        }
        else
        {
            /* Caller should fail on ln(0), but for the moment return zero */
            ln_dweight = 0;
        }
    }
 
    return ln_dweight;
}
 
 
/*
 * ln_var() -
 *
 *  Compute the natural log of x
 */
static void
ln_var(const NumericVar *arg, NumericVar *result, int rscale)
{
    NumericVar  x;
    NumericVar  xx;
    int         ni;
    NumericVar  elem;
    NumericVar  fact;
    int         nsqrt;
    int         local_rscale;
    int         cmp;
 
    cmp = cmp_var(arg, &const_zero);
    if (cmp == 0)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_ARGUMENT_FOR_LOG),
                 errmsg("cannot take logarithm of zero")));
    else if (cmp < 0)
        ereport(ERROR,
                (errcode(ERRCODE_INVALID_ARGUMENT_FOR_LOG),
                 errmsg("cannot take logarithm of a negative number")));
 
    init_var(&x);
    init_var(&xx);
    init_var(&elem);
    init_var(&fact);
 
    set_var_from_var(arg, &x);
    set_var_from_var(&const_two, &fact);
 
    /*
     * Reduce input into range 0.9 < x < 1.1 with repeated sqrt() operations.
     *
     * The final logarithm will have up to around rscale+6 significant digits.
     * Each sqrt() will roughly halve the weight of x, so adjust the local
     * rscale as we work so that we keep this many significant digits at each
     * step (plus a few more for good measure).
     *
     * Note that we allow local_rscale < 0 during this input reduction
     * process, which implies rounding before the decimal point.  sqrt_var()
     * explicitly supports this, and it significantly reduces the work
     * required to reduce very large inputs to the required range.  Once the
     * input reduction is complete, x.weight will be 0 and its display scale
     * will be non-negative again.
     */
    nsqrt = 0;
    while (cmp_var(&x, &const_zero_point_nine) <= 0)
    {
        local_rscale = rscale - x.weight * DEC_DIGITS / 2 + 8;
        sqrt_var(&x, &x, local_rscale);
        mul_var(&fact, &const_two, &fact, 0);
        nsqrt++;
    }
    while (cmp_var(&x, &const_one_point_one) >= 0)
    {
        local_rscale = rscale - x.weight * DEC_DIGITS / 2 + 8;
        sqrt_var(&x, &x, local_rscale);
        mul_var(&fact, &const_two, &fact, 0);
        nsqrt++;
    }
 
    /*
     * We use the Taylor series for 0.5 * ln((1+z)/(1-z)),
     *
     * z + z^3/3 + z^5/5 + ...
     *
     * where z = (x-1)/(x+1) is in the range (approximately) -0.053 .. 0.048
     * due to the above range-reduction of x.
     *
     * The convergence of this is not as fast as one would like, but is
     * tolerable given that z is small.
     *
     * The Taylor series result will be multiplied by 2^(nsqrt+1), which has a
     * decimal weight of (nsqrt+1) * log10(2), so work with this many extra
     * digits of precision (plus a few more for good measure).
     */
    local_rscale = rscale + (int) ((nsqrt + 1) * 0.301029995663981) + 8;
 
    sub_var(&x, &const_one, result);
    add_var(&x, &const_one, &elem);
    div_var(result, &elem, result, local_rscale, true, false);
    set_var_from_var(result, &xx);
    mul_var(result, result, &x, local_rscale);
 
    ni = 1;
 
    for (;;)
    {
        ni += 2;
        mul_var(&xx, &x, &xx, local_rscale);
        div_var_int(&xx, ni, 0, &elem, local_rscale, true);
 
        if (elem.ndigits == 0)
            break;
 
        add_var(result, &elem, result);
 
        if (elem.weight < (result->weight - local_rscale * 2 / DEC_DIGITS))
            break;
    }
 
    /* Compensate for argument range reduction, round to requested rscale */
    mul_var(result, &fact, result, rscale);
 
    free_var(&x);
    free_var(&xx);
    free_var(&elem);
    free_var(&fact);
}
 
 
/*
 * log_var() -
 *
 *  Compute the logarithm of num in a given base.
 *
 *  Note: this routine chooses dscale of the result.
 */
static void
log_var(const NumericVar *base, const NumericVar *num, NumericVar *result)
{
    NumericVar  ln_base;
    NumericVar  ln_num;
    int         ln_base_dweight;
    int         ln_num_dweight;
    int         result_dweight;
    int         rscale;
    int         ln_base_rscale;
    int         ln_num_rscale;
 
    init_var(&ln_base);
    init_var(&ln_num);
 
    /* Estimated dweights of ln(base), ln(num) and the final result */
    ln_base_dweight = estimate_ln_dweight(base);
    ln_num_dweight = estimate_ln_dweight(num);
    result_dweight = ln_num_dweight - ln_base_dweight;
 
    /*
     * Select the scale of the result so that it will have at least
     * NUMERIC_MIN_SIG_DIGITS significant digits and is not less than either
     * input's display scale.
     */
    rscale = NUMERIC_MIN_SIG_DIGITS - result_dweight;
    rscale = Max(rscale, base->dscale);
    rscale = Max(rscale, num->dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    /*
     * Set the scales for ln(base) and ln(num) so that they each have more
     * significant digits than the final result.
     */
    ln_base_rscale = rscale + result_dweight - ln_base_dweight + 8;
    ln_base_rscale = Max(ln_base_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
    ln_num_rscale = rscale + result_dweight - ln_num_dweight + 8;
    ln_num_rscale = Max(ln_num_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
    /* Form natural logarithms */
    ln_var(base, &ln_base, ln_base_rscale);
    ln_var(num, &ln_num, ln_num_rscale);
 
    /* Divide and round to the required scale */
    div_var(&ln_num, &ln_base, result, rscale, true, false);
 
    free_var(&ln_num);
    free_var(&ln_base);
}
 
 
/*
 * power_var() -
 *
 *  Raise base to the power of exp
 *
 *  Note: this routine chooses dscale of the result.
 */
static void
power_var(const NumericVar *base, const NumericVar *exp, NumericVar *result)
{
    int         res_sign;
    NumericVar  abs_base;
    NumericVar  ln_base;
    NumericVar  ln_num;
    int         ln_dweight;
    int         rscale;
    int         sig_digits;
    int         local_rscale;
    double      val;
 
    /* If exp can be represented as an integer, use power_var_int */
    if (exp->ndigits == 0 || exp->ndigits <= exp->weight + 1)
    {
        /* exact integer, but does it fit in int? */
        int64       expval64;
 
        if (numericvar_to_int64(exp, &expval64))
        {
            if (expval64 >= PG_INT32_MIN && expval64 <= PG_INT32_MAX)
            {
                /* Okay, use power_var_int */
                power_var_int(base, (int) expval64, exp->dscale, result);
                return;
            }
        }
    }
 
    /*
     * This avoids log(0) for cases of 0 raised to a non-integer.  0 ^ 0 is
     * handled by power_var_int().
     */
    if (cmp_var(base, &const_zero) == 0)
    {
        set_var_from_var(&const_zero, result);
        result->dscale = NUMERIC_MIN_SIG_DIGITS;    /* no need to round */
        return;
    }
 
    init_var(&abs_base);
    init_var(&ln_base);
    init_var(&ln_num);
 
    /*
     * If base is negative, insist that exp be an integer.  The result is then
     * positive if exp is even and negative if exp is odd.
     */
    if (base->sign == NUMERIC_NEG)
    {
        /*
         * Check that exp is an integer.  This error code is defined by the
         * SQL standard, and matches other errors in numeric_power().
         */
        if (exp->ndigits > 0 && exp->ndigits > exp->weight + 1)
            ereport(ERROR,
                    (errcode(ERRCODE_INVALID_ARGUMENT_FOR_POWER_FUNCTION),
                     errmsg("a negative number raised to a non-integer power yields a complex result")));
 
        /* Test if exp is odd or even */
        if (exp->ndigits > 0 && exp->ndigits == exp->weight + 1 &&
            (exp->digits[exp->ndigits - 1] & 1))
            res_sign = NUMERIC_NEG;
        else
            res_sign = NUMERIC_POS;
 
        /* Then work with abs(base) below */
        set_var_from_var(base, &abs_base);
        abs_base.sign = NUMERIC_POS;
        base = &abs_base;
    }
    else
        res_sign = NUMERIC_POS;
 
    /*----------
     * Decide on the scale for the ln() calculation.  For this we need an
     * estimate of the weight of the result, which we obtain by doing an
     * initial low-precision calculation of exp * ln(base).
     *
     * We want result = e ^ (exp * ln(base))
     * so result dweight = log10(result) = exp * ln(base) * log10(e)
     *
     * We also perform a crude overflow test here so that we can exit early if
     * the full-precision result is sure to overflow, and to guard against
     * integer overflow when determining the scale for the real calculation.
     * exp_var() supports inputs up to NUMERIC_MAX_RESULT_SCALE * 3, so the
     * result will overflow if exp * ln(base) >= NUMERIC_MAX_RESULT_SCALE * 3.
     * Since the values here are only approximations, we apply a small fuzz
     * factor to this overflow test and let exp_var() determine the exact
     * overflow threshold so that it is consistent for all inputs.
     *----------
     */
    ln_dweight = estimate_ln_dweight(base);
 
    /*
     * Set the scale for the low-precision calculation, computing ln(base) to
     * around 8 significant digits.  Note that ln_dweight may be as small as
     * -NUMERIC_DSCALE_MAX, so the scale may exceed NUMERIC_MAX_DISPLAY_SCALE
     * here.
     */
    local_rscale = 8 - ln_dweight;
    local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
    ln_var(base, &ln_base, local_rscale);
 
    mul_var(&ln_base, exp, &ln_num, local_rscale);
 
    val = numericvar_to_double_no_overflow(&ln_num);
 
    /* initial overflow/underflow test with fuzz factor */
    if (fabs(val) > NUMERIC_MAX_RESULT_SCALE * 3.01)
    {
        if (val > 0)
            ereport(ERROR,
                    (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                     errmsg("value overflows numeric format")));
        zero_var(result);
        result->dscale = NUMERIC_MAX_DISPLAY_SCALE;
        return;
    }
 
    val *= 0.434294481903252;   /* approximate decimal result weight */
 
    /* choose the result scale */
    rscale = NUMERIC_MIN_SIG_DIGITS - (int) val;
    rscale = Max(rscale, base->dscale);
    rscale = Max(rscale, exp->dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    /* significant digits required in the result */
    sig_digits = rscale + (int) val;
    sig_digits = Max(sig_digits, 0);
 
    /* set the scale for the real exp * ln(base) calculation */
    local_rscale = sig_digits - ln_dweight + 8;
    local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
    /* and do the real calculation */
 
    ln_var(base, &ln_base, local_rscale);
 
    mul_var(&ln_base, exp, &ln_num, local_rscale);
 
    exp_var(&ln_num, result, rscale);
 
    if (res_sign == NUMERIC_NEG && result->ndigits > 0)
        result->sign = NUMERIC_NEG;
 
    free_var(&ln_num);
    free_var(&ln_base);
    free_var(&abs_base);
}
 
/*
 * power_var_int() -
 *
 *  Raise base to the power of exp, where exp is an integer.
 *
 *  Note: this routine chooses dscale of the result.
 */
static void
power_var_int(const NumericVar *base, int exp, int exp_dscale,
              NumericVar *result)
{
    double      f;
    int         p;
    int         i;
    int         rscale;
    int         sig_digits;
    unsigned int mask;
    bool        neg;
    NumericVar  base_prod;
    int         local_rscale;
 
    /*
     * Choose the result scale.  For this we need an estimate of the decimal
     * weight of the result, which we obtain by approximating using double
     * precision arithmetic.
     *
     * We also perform crude overflow/underflow tests here so that we can exit
     * early if the result is sure to overflow/underflow, and to guard against
     * integer overflow when choosing the result scale.
     */
    if (base->ndigits != 0)
    {
        /*----------
         * Choose f (double) and p (int) such that base ~= f * 10^p.
         * Then log10(result) = log10(base^exp) ~= exp * (log10(f) + p).
         *----------
         */
        f = base->digits[0];
        p = base->weight * DEC_DIGITS;
 
        for (i = 1; i < base->ndigits && i * DEC_DIGITS < 16; i++)
        {
            f = f * NBASE + base->digits[i];
            p -= DEC_DIGITS;
        }
 
        f = exp * (log10(f) + p);   /* approximate decimal result weight */
    }
    else
        f = 0;                  /* result is 0 or 1 (weight 0), or error */
 
    /* overflow/underflow tests with fuzz factors */
    if (f > (NUMERIC_WEIGHT_MAX + 1) * DEC_DIGITS)
        ereport(ERROR,
                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                 errmsg("value overflows numeric format")));
    if (f + 1 < -NUMERIC_MAX_DISPLAY_SCALE)
    {
        zero_var(result);
        result->dscale = NUMERIC_MAX_DISPLAY_SCALE;
        return;
    }
 
    /*
     * Choose the result scale in the same way as power_var(), so it has at
     * least NUMERIC_MIN_SIG_DIGITS significant digits and is not less than
     * either input's display scale.
     */
    rscale = NUMERIC_MIN_SIG_DIGITS - (int) f;
    rscale = Max(rscale, base->dscale);
    rscale = Max(rscale, exp_dscale);
    rscale = Max(rscale, NUMERIC_MIN_DISPLAY_SCALE);
    rscale = Min(rscale, NUMERIC_MAX_DISPLAY_SCALE);
 
    /* Handle some common special cases, as well as corner cases */
    switch (exp)
    {
        case 0:
 
            /*
             * While 0 ^ 0 can be either 1 or indeterminate (error), we treat
             * it as 1 because most programming languages do this. SQL:2003
             * also requires a return value of 1.
             * https://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_power
             */
            set_var_from_var(&const_one, result);
            result->dscale = rscale;    /* no need to round */
            return;
        case 1:
            set_var_from_var(base, result);
            round_var(result, rscale);
            return;
        case -1:
            div_var(&const_one, base, result, rscale, true, true);
            return;
        case 2:
            mul_var(base, base, result, rscale);
            return;
        default:
            break;
    }
 
    /* Handle the special case where the base is zero */
    if (base->ndigits == 0)
    {
        if (exp < 0)
            ereport(ERROR,
                    (errcode(ERRCODE_DIVISION_BY_ZERO),
                     errmsg("division by zero")));
        zero_var(result);
        result->dscale = rscale;
        return;
    }
 
    /*
     * The general case repeatedly multiplies base according to the bit
     * pattern of exp.
     *
     * The local rscale used for each multiplication is varied to keep a fixed
     * number of significant digits, sufficient to give the required result
     * scale.
     */
 
    /*
     * Approximate number of significant digits in the result.  Note that the
     * underflow test above, together with the choice of rscale, ensures that
     * this approximation is necessarily > 0.
     */
    sig_digits = 1 + rscale + (int) f;
 
    /*
     * The multiplications to produce the result may introduce an error of up
     * to around log10(abs(exp)) digits, so work with this many extra digits
     * of precision (plus a few more for good measure).
     */
    sig_digits += (int) log(fabs((double) exp)) + 8;
 
    /*
     * Now we can proceed with the multiplications.
     */
    neg = (exp < 0);
    mask = pg_abs_s32(exp);
 
    init_var(&base_prod);
    set_var_from_var(base, &base_prod);
 
    if (mask & 1)
        set_var_from_var(base, result);
    else
        set_var_from_var(&const_one, result);
 
    while ((mask >>= 1) > 0)
    {
        /*
         * Do the multiplications using rscales large enough to hold the
         * results to the required number of significant digits, but don't
         * waste time by exceeding the scales of the numbers themselves.
         */
        local_rscale = sig_digits - 2 * base_prod.weight * DEC_DIGITS;
        local_rscale = Min(local_rscale, 2 * base_prod.dscale);
        local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
        mul_var(&base_prod, &base_prod, &base_prod, local_rscale);
 
        if (mask & 1)
        {
            local_rscale = sig_digits -
                (base_prod.weight + result->weight) * DEC_DIGITS;
            local_rscale = Min(local_rscale,
                               base_prod.dscale + result->dscale);
            local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 
            mul_var(&base_prod, result, result, local_rscale);
        }
 
        /*
         * When abs(base) > 1, the number of digits to the left of the decimal
         * point in base_prod doubles at each iteration, so if exp is large we
         * could easily spend large amounts of time and memory space doing the
         * multiplications.  But once the weight exceeds what will fit in
         * int16, the final result is guaranteed to overflow (or underflow, if
         * exp < 0), so we can give up before wasting too many cycles.
         */
        if (base_prod.weight > NUMERIC_WEIGHT_MAX ||
            result->weight > NUMERIC_WEIGHT_MAX)
        {
            /* overflow, unless neg, in which case result should be 0 */
            if (!neg)
                ereport(ERROR,
                        (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                         errmsg("value overflows numeric format")));
            zero_var(result);
            neg = false;
            break;
        }
    }
 
    free_var(&base_prod);
 
    /* Compensate for input sign, and round to requested rscale */
    if (neg)
        div_var(&const_one, result, result, rscale, true, false);
    else
        round_var(result, rscale);
}
 
/*
 * power_ten_int() -
 *
 *  Raise ten to the power of exp, where exp is an integer.  Note that unlike
 *  power_var_int(), this does no overflow/underflow checking or rounding.
 */
static void
power_ten_int(int exp, NumericVar *result)
{
    /* Construct the result directly, starting from 10^0 = 1 */
    set_var_from_var(&const_one, result);
 
    /* Scale needed to represent the result exactly */
    result->dscale = exp < 0 ? -exp : 0;
 
    /* Base-NBASE weight of result and remaining exponent */
    if (exp >= 0)
        result->weight = exp / DEC_DIGITS;
    else
        result->weight = (exp + 1) / DEC_DIGITS - 1;
 
    exp -= result->weight * DEC_DIGITS;
 
    /* Final adjustment of the result's single NBASE digit */
    while (exp-- > 0)
        result->digits[0] *= 10;
}
 
/*
 * random_var() - return a random value in the range [rmin, rmax].
 */
static void
random_var(pg_prng_state *state, const NumericVar *rmin,
           const NumericVar *rmax, NumericVar *result)
{
    int         rscale;
    NumericVar  rlen;
    int         res_ndigits;
    int         n;
    int         pow10;
    int         i;
    uint64      rlen64;
    int         rlen64_ndigits;
 
    rscale = Max(rmin->dscale, rmax->dscale);
 
    /* Compute rlen = rmax - rmin and check the range bounds */
    init_var(&rlen);
    sub_var(rmax, rmin, &rlen);
 
    if (rlen.sign == NUMERIC_NEG)
        ereport(ERROR,
                errcode(ERRCODE_INVALID_PARAMETER_VALUE),
                errmsg("lower bound must be less than or equal to upper bound"));
 
    /* Special case for an empty range */
    if (rlen.ndigits == 0)
    {
        set_var_from_var(rmin, result);
        result->dscale = rscale;
        free_var(&rlen);
        return;
    }
 
    /*
     * Otherwise, select a random value in the range [0, rlen = rmax - rmin],
     * and shift it to the required range by adding rmin.
     */
 
    /* Required result digits */
    res_ndigits = rlen.weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS;
 
    /*
     * To get the required rscale, the final result digit must be a multiple
     * of pow10 = 10^n, where n = (-rscale) mod DEC_DIGITS.
     */
    n = ((rscale + DEC_DIGITS - 1) / DEC_DIGITS) * DEC_DIGITS - rscale;
    pow10 = 1;
    for (i = 0; i < n; i++)
        pow10 *= 10;
 
    /*
     * To choose a random value uniformly from the range [0, rlen], we choose
     * from the slightly larger range [0, rlen2], where rlen2 is formed from
     * rlen by copying the first 4 NBASE digits, and setting all remaining
     * decimal digits to "9".
     *
     * Without loss of generality, we can ignore the weight of rlen2 and treat
     * it as a pure integer for the purposes of this discussion.  The process
     * above gives rlen2 + 1 = rlen64 * 10^N, for some integer N, where rlen64
     * is a 64-bit integer formed from the first 4 NBASE digits copied from
     * rlen.  Since this trivially factors into smaller pieces that fit in
     * 64-bit integers, the task of choosing a random value uniformly from the
     * rlen2 + 1 possible values in [0, rlen2] is much simpler.
     *
     * If the random value selected is too large, it is rejected, and we try
     * again until we get a result <= rlen, ensuring that the overall result
     * is uniform (no particular value is any more likely than any other).
     *
     * Since rlen64 holds 4 NBASE digits from rlen, it contains at least
     * DEC_DIGITS * 3 + 1 decimal digits (i.e., at least 13 decimal digits,
     * when DEC_DIGITS is 4). Therefore the probability of needing to reject
     * the value chosen and retry is less than 1e-13.
     */
    rlen64 = (uint64) rlen.digits[0];
    rlen64_ndigits = 1;
    while (rlen64_ndigits < res_ndigits && rlen64_ndigits < 4)
    {
        rlen64 *= NBASE;
        if (rlen64_ndigits < rlen.ndigits)
            rlen64 += rlen.digits[rlen64_ndigits];
        rlen64_ndigits++;
    }
 
    /* Loop until we get a result <= rlen */
    do
    {
        NumericDigit *res_digits;
        uint64      rand;
        int         whole_ndigits;
 
        alloc_var(result, res_ndigits);
        result->sign = NUMERIC_POS;
        result->weight = rlen.weight;
        result->dscale = rscale;
        res_digits = result->digits;
 
        /*
         * Set the first rlen64_ndigits using a random value in [0, rlen64].
         *
         * If this is the whole result, and rscale is not a multiple of
         * DEC_DIGITS (pow10 from above is not 1), then we need this to be a
         * multiple of pow10.
         */
        if (rlen64_ndigits == res_ndigits && pow10 != 1)
            rand = pg_prng_uint64_range(state, 0, rlen64 / pow10) * pow10;
        else
            rand = pg_prng_uint64_range(state, 0, rlen64);
 
        for (i = rlen64_ndigits - 1; i >= 0; i--)
        {
            res_digits[i] = (NumericDigit) (rand % NBASE);
            rand = rand / NBASE;
        }
 
        /*
         * Set the remaining digits to random values in range [0, NBASE),
         * noting that the last digit needs to be a multiple of pow10.
         */
        whole_ndigits = res_ndigits;
        if (pow10 != 1)
            whole_ndigits--;
 
        /* Set whole digits in groups of 4 for best performance */
        i = rlen64_ndigits;
        while (i < whole_ndigits - 3)
        {
            rand = pg_prng_uint64_range(state, 0,
                                        (uint64) NBASE * NBASE * NBASE * NBASE - 1);
            res_digits[i++] = (NumericDigit) (rand % NBASE);
            rand = rand / NBASE;
            res_digits[i++] = (NumericDigit) (rand % NBASE);
            rand = rand / NBASE;
            res_digits[i++] = (NumericDigit) (rand % NBASE);
            rand = rand / NBASE;
            res_digits[i++] = (NumericDigit) rand;
        }
 
        /* Remaining whole digits */
        while (i < whole_ndigits)
        {
            rand = pg_prng_uint64_range(state, 0, NBASE - 1);
            res_digits[i++] = (NumericDigit) rand;
        }
 
        /* Final partial digit (multiple of pow10) */
        if (i < res_ndigits)
        {
            rand = pg_prng_uint64_range(state, 0, NBASE / pow10 - 1) * pow10;
            res_digits[i] = (NumericDigit) rand;
        }
 
        /* Remove leading/trailing zeroes */
        strip_var(result);
 
        /* If result > rlen, try again */
 
    } while (cmp_var(result, &rlen) > 0);
 
    /* Offset the result to the required range */
    add_var(result, rmin, result);
 
    free_var(&rlen);
}
 
 
/* ----------------------------------------------------------------------
 *
 * Following are the lowest level functions that operate unsigned
 * on the variable level
 *
 * ----------------------------------------------------------------------
 */
 
 
/* ----------
 * cmp_abs() -
 *
 *  Compare the absolute values of var1 and var2
 *  Returns:    -1 for ABS(var1) < ABS(var2)
 *              0  for ABS(var1) == ABS(var2)
 *              1  for ABS(var1) > ABS(var2)
 * ----------
 */
static int
cmp_abs(const NumericVar *var1, const NumericVar *var2)
{
    return cmp_abs_common(var1->digits, var1->ndigits, var1->weight,
                          var2->digits, var2->ndigits, var2->weight);
}
 
/* ----------
 * cmp_abs_common() -
 *
 *  Main routine of cmp_abs(). This function can be used by both
 *  NumericVar and Numeric.
 * ----------
 */
static int
cmp_abs_common(const NumericDigit *var1digits, int var1ndigits, int var1weight,
               const NumericDigit *var2digits, int var2ndigits, int var2weight)
{
    int         i1 = 0;
    int         i2 = 0;
 
    /* Check any digits before the first common digit */
 
    while (var1weight > var2weight && i1 < var1ndigits)
    {
        if (var1digits[i1++] != 0)
            return 1;
        var1weight--;
    }
    while (var2weight > var1weight && i2 < var2ndigits)
    {
        if (var2digits[i2++] != 0)
            return -1;
        var2weight--;
    }
 
    /* At this point, either w1 == w2 or we've run out of digits */
 
    if (var1weight == var2weight)
    {
        while (i1 < var1ndigits && i2 < var2ndigits)
        {
            int         stat = var1digits[i1++] - var2digits[i2++];
 
            if (stat)
            {
                if (stat > 0)
                    return 1;
                return -1;
            }
        }
    }
 
    /*
     * At this point, we've run out of digits on one side or the other; so any
     * remaining nonzero digits imply that side is larger
     */
    while (i1 < var1ndigits)
    {
        if (var1digits[i1++] != 0)
            return 1;
    }
    while (i2 < var2ndigits)
    {
        if (var2digits[i2++] != 0)
            return -1;
    }
 
    return 0;
}
 
 
/*
 * add_abs() -
 *
 *  Add the absolute values of two variables into result.
 *  result might point to one of the operands without danger.
 */
static void
add_abs(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    NumericDigit *res_buf;
    NumericDigit *res_digits;
    int         res_ndigits;
    int         res_weight;
    int         res_rscale,
                rscale1,
                rscale2;
    int         res_dscale;
    int         i,
                i1,
                i2;
    int         carry = 0;
 
    /* copy these values into local vars for speed in inner loop */
    int         var1ndigits = var1->ndigits;
    int         var2ndigits = var2->ndigits;
    NumericDigit *var1digits = var1->digits;
    NumericDigit *var2digits = var2->digits;
 
    res_weight = Max(var1->weight, var2->weight) + 1;
 
    res_dscale = Max(var1->dscale, var2->dscale);
 
    /* Note: here we are figuring rscale in base-NBASE digits */
    rscale1 = var1->ndigits - var1->weight - 1;
    rscale2 = var2->ndigits - var2->weight - 1;
    res_rscale = Max(rscale1, rscale2);
 
    res_ndigits = res_rscale + res_weight + 1;
    if (res_ndigits <= 0)
        res_ndigits = 1;
 
    res_buf = digitbuf_alloc(res_ndigits + 1);
    res_buf[0] = 0;             /* spare digit for later rounding */
    res_digits = res_buf + 1;
 
    i1 = res_rscale + var1->weight + 1;
    i2 = res_rscale + var2->weight + 1;
    for (i = res_ndigits - 1; i >= 0; i--)
    {
        i1--;
        i2--;
        if (i1 >= 0 && i1 < var1ndigits)
            carry += var1digits[i1];
        if (i2 >= 0 && i2 < var2ndigits)
            carry += var2digits[i2];
 
        if (carry >= NBASE)
        {
            res_digits[i] = carry - NBASE;
            carry = 1;
        }
        else
        {
            res_digits[i] = carry;
            carry = 0;
        }
    }
 
    Assert(carry == 0);         /* else we failed to allow for carry out */
 
    digitbuf_free(result->buf);
    result->ndigits = res_ndigits;
    result->buf = res_buf;
    result->digits = res_digits;
    result->weight = res_weight;
    result->dscale = res_dscale;
 
    /* Remove leading/trailing zeroes */
    strip_var(result);
}
 
 
/*
 * sub_abs()
 *
 *  Subtract the absolute value of var2 from the absolute value of var1
 *  and store in result. result might point to one of the operands
 *  without danger.
 *
 *  ABS(var1) MUST BE GREATER OR EQUAL ABS(var2) !!!
 */
static void
sub_abs(const NumericVar *var1, const NumericVar *var2, NumericVar *result)
{
    NumericDigit *res_buf;
    NumericDigit *res_digits;
    int         res_ndigits;
    int         res_weight;
    int         res_rscale,
                rscale1,
                rscale2;
    int         res_dscale;
    int         i,
                i1,
                i2;
    int         borrow = 0;
 
    /* copy these values into local vars for speed in inner loop */
    int         var1ndigits = var1->ndigits;
    int         var2ndigits = var2->ndigits;
    NumericDigit *var1digits = var1->digits;
    NumericDigit *var2digits = var2->digits;
 
    res_weight = var1->weight;
 
    res_dscale = Max(var1->dscale, var2->dscale);
 
    /* Note: here we are figuring rscale in base-NBASE digits */
    rscale1 = var1->ndigits - var1->weight - 1;
    rscale2 = var2->ndigits - var2->weight - 1;
    res_rscale = Max(rscale1, rscale2);
 
    res_ndigits = res_rscale + res_weight + 1;
    if (res_ndigits <= 0)
        res_ndigits = 1;
 
    res_buf = digitbuf_alloc(res_ndigits + 1);
    res_buf[0] = 0;             /* spare digit for later rounding */
    res_digits = res_buf + 1;
 
    i1 = res_rscale + var1->weight + 1;
    i2 = res_rscale + var2->weight + 1;
    for (i = res_ndigits - 1; i >= 0; i--)
    {
        i1--;
        i2--;
        if (i1 >= 0 && i1 < var1ndigits)
            borrow += var1digits[i1];
        if (i2 >= 0 && i2 < var2ndigits)
            borrow -= var2digits[i2];
 
        if (borrow < 0)
        {
            res_digits[i] = borrow + NBASE;
            borrow = -1;
        }
        else
        {
            res_digits[i] = borrow;
            borrow = 0;
        }
    }
 
    Assert(borrow == 0);        /* else caller gave us var1 < var2 */
 
    digitbuf_free(result->buf);
    result->ndigits = res_ndigits;
    result->buf = res_buf;
    result->digits = res_digits;
    result->weight = res_weight;
    result->dscale = res_dscale;
 
    /* Remove leading/trailing zeroes */
    strip_var(result);
}
 
/*
 * round_var
 *
 * Round the value of a variable to no more than rscale decimal digits
 * after the decimal point.  NOTE: we allow rscale < 0 here, implying
 * rounding before the decimal point.
 */
static void
round_var(NumericVar *var, int rscale)
{
    NumericDigit *digits = var->digits;
    int         di;
    int         ndigits;
    int         carry;
 
    var->dscale = rscale;
 
    /* decimal digits wanted */
    di = (var->weight + 1) * DEC_DIGITS + rscale;
 
    /*
     * If di = 0, the value loses all digits, but could round up to 1 if its
     * first extra digit is >= 5.  If di < 0 the result must be 0.
     */
    if (di < 0)
    {
        var->ndigits = 0;
        var->weight = 0;
        var->sign = NUMERIC_POS;
    }
    else
    {
        /* NBASE digits wanted */
        ndigits = (di + DEC_DIGITS - 1) / DEC_DIGITS;
 
        /* 0, or number of decimal digits to keep in last NBASE digit */
        di %= DEC_DIGITS;
 
        if (ndigits < var->ndigits ||
            (ndigits == var->ndigits && di > 0))
        {
            var->ndigits = ndigits;
 
#if DEC_DIGITS == 1
            /* di must be zero */
            carry = (digits[ndigits] >= HALF_NBASE) ? 1 : 0;
#else
            if (di == 0)
                carry = (digits[ndigits] >= HALF_NBASE) ? 1 : 0;
            else
            {
                /* Must round within last NBASE digit */
                int         extra,
                            pow10;
 
#if DEC_DIGITS == 4
                pow10 = round_powers[di];
#elif DEC_DIGITS == 2
                pow10 = 10;
#else
#error unsupported NBASE
#endif
                extra = digits[--ndigits] % pow10;
                digits[ndigits] -= extra;
                carry = 0;
                if (extra >= pow10 / 2)
                {
                    pow10 += digits[ndigits];
                    if (pow10 >= NBASE)
                    {
                        pow10 -= NBASE;
                        carry = 1;
                    }
                    digits[ndigits] = pow10;
                }
            }
#endif
 
            /* Propagate carry if needed */
            while (carry)
            {
                carry += digits[--ndigits];
                if (carry >= NBASE)
                {
                    digits[ndigits] = carry - NBASE;
                    carry = 1;
                }
                else
                {
                    digits[ndigits] = carry;
                    carry = 0;
                }
            }
 
            if (ndigits < 0)
            {
                Assert(ndigits == -1);  /* better not have added > 1 digit */
                Assert(var->digits > var->buf);
                var->digits--;
                var->ndigits++;
                var->weight++;
            }
        }
    }
}
 
/*
 * trunc_var
 *
 * Truncate (towards zero) the value of a variable at rscale decimal digits
 * after the decimal point.  NOTE: we allow rscale < 0 here, implying
 * truncation before the decimal point.
 */
static void
trunc_var(NumericVar *var, int rscale)
{
    int         di;
    int         ndigits;
 
    var->dscale = rscale;
 
    /* decimal digits wanted */
    di = (var->weight + 1) * DEC_DIGITS + rscale;
 
    /*
     * If di <= 0, the value loses all digits.
     */
    if (di <= 0)
    {
        var->ndigits = 0;
        var->weight = 0;
        var->sign = NUMERIC_POS;
    }
    else
    {
        /* NBASE digits wanted */
        ndigits = (di + DEC_DIGITS - 1) / DEC_DIGITS;
 
        if (ndigits <= var->ndigits)
        {
            var->ndigits = ndigits;
 
#if DEC_DIGITS == 1
            /* no within-digit stuff to worry about */
#else
            /* 0, or number of decimal digits to keep in last NBASE digit */
            di %= DEC_DIGITS;
 
            if (di > 0)
            {
                /* Must truncate within last NBASE digit */
                NumericDigit *digits = var->digits;
                int         extra,
                            pow10;
 
#if DEC_DIGITS == 4
                pow10 = round_powers[di];
#elif DEC_DIGITS == 2
                pow10 = 10;
#else
#error unsupported NBASE
#endif
                extra = digits[--ndigits] % pow10;
                digits[ndigits] -= extra;
            }
#endif
        }
    }
}
 
/*
 * strip_var
 *
 * Strip any leading and trailing zeroes from a numeric variable
 */
static void
strip_var(NumericVar *var)
{
    NumericDigit *digits = var->digits;
    int         ndigits = var->ndigits;
 
    /* Strip leading zeroes */
    while (ndigits > 0 && *digits == 0)
    {
        digits++;
        var->weight--;
        ndigits--;
    }
 
    /* Strip trailing zeroes */
    while (ndigits > 0 && digits[ndigits - 1] == 0)
        ndigits--;
 
    /* If it's zero, normalize the sign and weight */
    if (ndigits == 0)
    {
        var->sign = NUMERIC_POS;
        var->weight = 0;
    }
 
    var->digits = digits;
    var->ndigits = ndigits;
}
 
 
/* ----------------------------------------------------------------------
 *
 * Fast sum accumulator functions
 *
 * ----------------------------------------------------------------------
 */
 
/*
 * Reset the accumulator's value to zero.  The buffers to hold the digits
 * are not free'd.
 */
static void
accum_sum_reset(NumericSumAccum *accum)
{
    int         i;
 
    accum->dscale = 0;
    for (i = 0; i < accum->ndigits; i++)
    {
        accum->pos_digits[i] = 0;
        accum->neg_digits[i] = 0;
    }
}
 
/*
 * Accumulate a new value.
 */
static void
accum_sum_add(NumericSumAccum *accum, const NumericVar *val)
{
    int32      *accum_digits;
    int         i,
                val_i;
    int         val_ndigits;
    NumericDigit *val_digits;
 
    /*
     * If we have accumulated too many values since the last carry
     * propagation, do it now, to avoid overflowing.  (We could allow more
     * than NBASE - 1, if we reserved two extra digits, rather than one, for
     * carry propagation.  But even with NBASE - 1, this needs to be done so
     * seldom, that the performance difference is negligible.)
     */
    if (accum->num_uncarried == NBASE - 1)
        accum_sum_carry(accum);
 
    /*
     * Adjust the weight or scale of the old value, so that it can accommodate
     * the new value.
     */
    accum_sum_rescale(accum, val);
 
    /* */
    if (val->sign == NUMERIC_POS)
        accum_digits = accum->pos_digits;
    else
        accum_digits = accum->neg_digits;
 
    /* copy these values into local vars for speed in loop */
    val_ndigits = val->ndigits;
    val_digits = val->digits;
 
    i = accum->weight - val->weight;
    for (val_i = 0; val_i < val_ndigits; val_i++)
    {
        accum_digits[i] += (int32) val_digits[val_i];
        i++;
    }
 
    accum->num_uncarried++;
}
 
/*
 * Propagate carries.
 */
static void
accum_sum_carry(NumericSumAccum *accum)
{
    int         i;
    int         ndigits;
    int32      *dig;
    int32       carry;
    int32       newdig = 0;
 
    /*
     * If no new values have been added since last carry propagation, nothing
     * to do.
     */
    if (accum->num_uncarried == 0)
        return;
 
    /*
     * We maintain that the weight of the accumulator is always one larger
     * than needed to hold the current value, before carrying, to make sure
     * there is enough space for the possible extra digit when carry is
     * propagated.  We cannot expand the buffer here, unless we require
     * callers of accum_sum_final() to switch to the right memory context.
     */
    Assert(accum->pos_digits[0] == 0 && accum->neg_digits[0] == 0);
 
    ndigits = accum->ndigits;
 
    /* Propagate carry in the positive sum */
    dig = accum->pos_digits;
    carry = 0;
    for (i = ndigits - 1; i >= 0; i--)
    {
        newdig = dig[i] + carry;
        if (newdig >= NBASE)
        {
            carry = newdig / NBASE;
            newdig -= carry * NBASE;
        }
        else
            carry = 0;
        dig[i] = newdig;
    }
    /* Did we use up the digit reserved for carry propagation? */
    if (newdig > 0)
        accum->have_carry_space = false;
 
    /* And the same for the negative sum */
    dig = accum->neg_digits;
    carry = 0;
    for (i = ndigits - 1; i >= 0; i--)
    {
        newdig = dig[i] + carry;
        if (newdig >= NBASE)
        {
            carry = newdig / NBASE;
            newdig -= carry * NBASE;
        }
        else
            carry = 0;
        dig[i] = newdig;
    }
    if (newdig > 0)
        accum->have_carry_space = false;
 
    accum->num_uncarried = 0;
}
 
/*
 * Re-scale accumulator to accommodate new value.
 *
 * If the new value has more digits than the current digit buffers in the
 * accumulator, enlarge the buffers.
 */
static void
accum_sum_rescale(NumericSumAccum *accum, const NumericVar *val)
{
    int         old_weight = accum->weight;
    int         old_ndigits = accum->ndigits;
    int         accum_ndigits;
    int         accum_weight;
    int         accum_rscale;
    int         val_rscale;
 
    accum_weight = old_weight;
    accum_ndigits = old_ndigits;
 
    /*
     * Does the new value have a larger weight? If so, enlarge the buffers,
     * and shift the existing value to the new weight, by adding leading
     * zeros.
     *
     * We enforce that the accumulator always has a weight one larger than
     * needed for the inputs, so that we have space for an extra digit at the
     * final carry-propagation phase, if necessary.
     */
    if (val->weight >= accum_weight)
    {
        accum_weight = val->weight + 1;
        accum_ndigits = accum_ndigits + (accum_weight - old_weight);
    }
 
    /*
     * Even though the new value is small, we might've used up the space
     * reserved for the carry digit in the last call to accum_sum_carry().  If
     * so, enlarge to make room for another one.
     */
    else if (!accum->have_carry_space)
    {
        accum_weight++;
        accum_ndigits++;
    }
 
    /* Is the new value wider on the right side? */
    accum_rscale = accum_ndigits - accum_weight - 1;
    val_rscale = val->ndigits - val->weight - 1;
    if (val_rscale > accum_rscale)
        accum_ndigits = accum_ndigits + (val_rscale - accum_rscale);
 
    if (accum_ndigits != old_ndigits ||
        accum_weight != old_weight)
    {
        int32      *new_pos_digits;
        int32      *new_neg_digits;
        int         weightdiff;
 
        weightdiff = accum_weight - old_weight;
 
        new_pos_digits = palloc0(accum_ndigits * sizeof(int32));
        new_neg_digits = palloc0(accum_ndigits * sizeof(int32));
 
        if (accum->pos_digits)
        {
            memcpy(&new_pos_digits[weightdiff], accum->pos_digits,
                   old_ndigits * sizeof(int32));
            pfree(accum->pos_digits);
 
            memcpy(&new_neg_digits[weightdiff], accum->neg_digits,
                   old_ndigits * sizeof(int32));
            pfree(accum->neg_digits);
        }
 
        accum->pos_digits = new_pos_digits;
        accum->neg_digits = new_neg_digits;
 
        accum->weight = accum_weight;
        accum->ndigits = accum_ndigits;
 
        Assert(accum->pos_digits[0] == 0 && accum->neg_digits[0] == 0);
        accum->have_carry_space = true;
    }
 
    if (val->dscale > accum->dscale)
        accum->dscale = val->dscale;
}
 
/*
 * Return the current value of the accumulator.  This perform final carry
 * propagation, and adds together the positive and negative sums.
 *
 * Unlike all the other routines, the caller is not required to switch to
 * the memory context that holds the accumulator.
 */
static void
accum_sum_final(NumericSumAccum *accum, NumericVar *result)
{
    int         i;
    NumericVar  pos_var;
    NumericVar  neg_var;
 
    if (accum->ndigits == 0)
    {
        set_var_from_var(&const_zero, result);
        return;
    }
 
    /* Perform final carry */
    accum_sum_carry(accum);
 
    /* Create NumericVars representing the positive and negative sums */
    init_var(&pos_var);
    init_var(&neg_var);
 
    pos_var.ndigits = neg_var.ndigits = accum->ndigits;
    pos_var.weight = neg_var.weight = accum->weight;
    pos_var.dscale = neg_var.dscale = accum->dscale;
    pos_var.sign = NUMERIC_POS;
    neg_var.sign = NUMERIC_NEG;
 
    pos_var.buf = pos_var.digits = digitbuf_alloc(accum->ndigits);
    neg_var.buf = neg_var.digits = digitbuf_alloc(accum->ndigits);
 
    for (i = 0; i < accum->ndigits; i++)
    {
        Assert(accum->pos_digits[i] < NBASE);
        pos_var.digits[i] = (int16) accum->pos_digits[i];
 
        Assert(accum->neg_digits[i] < NBASE);
        neg_var.digits[i] = (int16) accum->neg_digits[i];
    }
 
    /* And add them together */
    add_var(&pos_var, &neg_var, result);
 
    /* Remove leading/trailing zeroes */
    strip_var(result);
}
 
/*
 * Copy an accumulator's state.
 *
 * 'dst' is assumed to be uninitialized beforehand.  No attempt is made at
 * freeing old values.
 */
static void
accum_sum_copy(NumericSumAccum *dst, NumericSumAccum *src)
{
    dst->pos_digits = palloc(src->ndigits * sizeof(int32));
    dst->neg_digits = palloc(src->ndigits * sizeof(int32));
 
    memcpy(dst->pos_digits, src->pos_digits, src->ndigits * sizeof(int32));
    memcpy(dst->neg_digits, src->neg_digits, src->ndigits * sizeof(int32));
    dst->num_uncarried = src->num_uncarried;
    dst->ndigits = src->ndigits;
    dst->weight = src->weight;
    dst->dscale = src->dscale;
}
 
/*
 * Add the current value of 'accum2' into 'accum'.
 */
static void
accum_sum_combine(NumericSumAccum *accum, NumericSumAccum *accum2)
{
    NumericVar  tmp_var;
 
    init_var(&tmp_var);
 
    accum_sum_final(accum2, &tmp_var);
    accum_sum_add(accum, &tmp_var);
 
    free_var(&tmp_var);
}