d2s.c

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/*---------------------------------------------------------------------------
 *
 * Ryu floating-point output for double precision.
 *
 * Portions Copyright (c) 2018-2026, PostgreSQL Global Development Group
 *
 * IDENTIFICATION
 *    src/common/d2s.c
 *
 * This is a modification of code taken from github.com/ulfjack/ryu under the
 * terms of the Boost license (not the Apache license). The original copyright
 * notice follows:
 *
 * Copyright 2018 Ulf Adams
 *
 * The contents of this file may be used under the terms of the Apache
 * License, Version 2.0.
 *
 *     (See accompanying file LICENSE-Apache or copy at
 *      http://www.apache.org/licenses/LICENSE-2.0)
 *
 * Alternatively, the contents of this file may be used under the terms of the
 * Boost Software License, Version 1.0.
 *
 *     (See accompanying file LICENSE-Boost or copy at
 *      https://www.boost.org/LICENSE_1_0.txt)
 *
 * Unless required by applicable law or agreed to in writing, this software is
 * distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
 * KIND, either express or implied.
 *
 *---------------------------------------------------------------------------
 */
 
/*
 *  Runtime compiler options:
 *
 *  -DRYU_ONLY_64_BIT_OPS Avoid using uint128 or 64-bit intrinsics. Slower,
 *      depending on your compiler.
 */
 
#ifndef FRONTEND
#include "postgres.h"
#else
#include "postgres_fe.h"
#endif
 
#include "common/shortest_dec.h"
 
/*
 * For consistency, we use 128-bit types if and only if the rest of PG also
 * does, even though we could use them here without worrying about the
 * alignment concerns that apply elsewhere.
 */
#if !defined(HAVE_INT128) && defined(_MSC_VER) \
    && !defined(RYU_ONLY_64_BIT_OPS) && defined(_M_X64)
#define HAS_64_BIT_INTRINSICS
#endif
 
#include "ryu_common.h"
#include "digit_table.h"
#include "d2s_full_table.h"
#include "d2s_intrinsics.h"
 
#define DOUBLE_MANTISSA_BITS 52
#define DOUBLE_EXPONENT_BITS 11
#define DOUBLE_BIAS 1023
 
#define DOUBLE_POW5_INV_BITCOUNT 122
#define DOUBLE_POW5_BITCOUNT 121
 
 
static inline uint32
pow5Factor(uint64 value)
{
    uint32      count = 0;
 
    for (;;)
    {
        Assert(value != 0);
        const uint64 q = div5(value);
        const uint32 r = (uint32) (value - 5 * q);
 
        if (r != 0)
            break;
 
        value = q;
        ++count;
    }
    return count;
}
 
/*  Returns true if value is divisible by 5^p. */
static inline bool
multipleOfPowerOf5(const uint64 value, const uint32 p)
{
    /*
     * I tried a case distinction on p, but there was no performance
     * difference.
     */
    return pow5Factor(value) >= p;
}
 
/*  Returns true if value is divisible by 2^p. */
static inline bool
multipleOfPowerOf2(const uint64 value, const uint32 p)
{
    /* return __builtin_ctzll(value) >= p; */
    return (value & ((UINT64CONST(1) << p) - 1)) == 0;
}
 
/*
 * We need a 64x128-bit multiplication and a subsequent 128-bit shift.
 *
 * Multiplication:
 *
 *    The 64-bit factor is variable and passed in, the 128-bit factor comes
 *    from a lookup table. We know that the 64-bit factor only has 55
 *    significant bits (i.e., the 9 topmost bits are zeros). The 128-bit
 *    factor only has 124 significant bits (i.e., the 4 topmost bits are
 *    zeros).
 *
 * Shift:
 *
 *    In principle, the multiplication result requires 55 + 124 = 179 bits to
 *    represent. However, we then shift this value to the right by j, which is
 *    at least j >= 115, so the result is guaranteed to fit into 179 - 115 =
 *    64 bits. This means that we only need the topmost 64 significant bits of
 *    the 64x128-bit multiplication.
 *
 * There are several ways to do this:
 *
 *  1. Best case: the compiler exposes a 128-bit type.
 *     We perform two 64x64-bit multiplications, add the higher 64 bits of the
 *     lower result to the higher result, and shift by j - 64 bits.
 *
 *     We explicitly cast from 64-bit to 128-bit, so the compiler can tell
 *     that these are only 64-bit inputs, and can map these to the best
 *     possible sequence of assembly instructions. x86-64 machines happen to
 *     have matching assembly instructions for 64x64-bit multiplications and
 *     128-bit shifts.
 *
 *  2. Second best case: the compiler exposes intrinsics for the x86-64
 *     assembly instructions mentioned in 1.
 *
 *  3. We only have 64x64 bit instructions that return the lower 64 bits of
 *     the result, i.e., we have to use plain C.
 *
 *     Our inputs are less than the full width, so we have three options:
 *     a. Ignore this fact and just implement the intrinsics manually.
 *     b. Split both into 31-bit pieces, which guarantees no internal
 *        overflow, but requires extra work upfront (unless we change the
 *        lookup table).
 *     c. Split only the first factor into 31-bit pieces, which also
 *        guarantees no internal overflow, but requires extra work since the
 *        intermediate results are not perfectly aligned.
 */
#if defined(HAVE_INT128)
 
/*  Best case: use 128-bit type. */
static inline uint64
mulShift(const uint64 m, const uint64 *const mul, const int32 j)
{
    const uint128 b0 = ((uint128) m) * mul[0];
    const uint128 b2 = ((uint128) m) * mul[1];
 
    return (uint64) (((b0 >> 64) + b2) >> (j - 64));
}
 
static inline uint64
mulShiftAll(const uint64 m, const uint64 *const mul, const int32 j,
            uint64 *const vp, uint64 *const vm, const uint32 mmShift)
{
    *vp = mulShift(4 * m + 2, mul, j);
    *vm = mulShift(4 * m - 1 - mmShift, mul, j);
    return mulShift(4 * m, mul, j);
}
 
#elif defined(HAS_64_BIT_INTRINSICS)
 
static inline uint64
mulShift(const uint64 m, const uint64 *const mul, const int32 j)
{
    /* m is maximum 55 bits */
    uint64      high1;
 
    /* 128 */
    const uint64 low1 = umul128(m, mul[1], &high1);
 
    /* 64 */
    uint64      high0;
    uint64      sum;
 
    /* 64 */
    umul128(m, mul[0], &high0);
    /* 0 */
    sum = high0 + low1;
 
    if (sum < high0)
    {
        ++high1;
        /* overflow into high1 */
    }
    return shiftright128(sum, high1, j - 64);
}
 
static inline uint64
mulShiftAll(const uint64 m, const uint64 *const mul, const int32 j,
            uint64 *const vp, uint64 *const vm, const uint32 mmShift)
{
    *vp = mulShift(4 * m + 2, mul, j);
    *vm = mulShift(4 * m - 1 - mmShift, mul, j);
    return mulShift(4 * m, mul, j);
}
 
#else                           /* // !defined(HAVE_INT128) &&
                                 * !defined(HAS_64_BIT_INTRINSICS) */
 
static inline uint64
mulShiftAll(uint64 m, const uint64 *const mul, const int32 j,
            uint64 *const vp, uint64 *const vm, const uint32 mmShift)
{
    m <<= 1;                    /* m is maximum 55 bits */
 
    uint64      tmp;
    const uint64 lo = umul128(m, mul[0], &tmp);
    uint64      hi;
    const uint64 mid = tmp + umul128(m, mul[1], &hi);
 
    hi += mid < tmp;            /* overflow into hi */
 
    const uint64 lo2 = lo + mul[0];
    const uint64 mid2 = mid + mul[1] + (lo2 < lo);
    const uint64 hi2 = hi + (mid2 < mid);
 
    *vp = shiftright128(mid2, hi2, j - 64 - 1);
 
    if (mmShift == 1)
    {
        const uint64 lo3 = lo - mul[0];
        const uint64 mid3 = mid - mul[1] - (lo3 > lo);
        const uint64 hi3 = hi - (mid3 > mid);
 
        *vm = shiftright128(mid3, hi3, j - 64 - 1);
    }
    else
    {
        const uint64 lo3 = lo + lo;
        const uint64 mid3 = mid + mid + (lo3 < lo);
        const uint64 hi3 = hi + hi + (mid3 < mid);
        const uint64 lo4 = lo3 - mul[0];
        const uint64 mid4 = mid3 - mul[1] - (lo4 > lo3);
        const uint64 hi4 = hi3 - (mid4 > mid3);
 
        *vm = shiftright128(mid4, hi4, j - 64);
    }
 
    return shiftright128(mid, hi, j - 64 - 1);
}
 
#endif                          /* // HAS_64_BIT_INTRINSICS */
 
static inline uint32
decimalLength(const uint64 v)
{
    /* This is slightly faster than a loop. */
    /* The average output length is 16.38 digits, so we check high-to-low. */
    /* Function precondition: v is not an 18, 19, or 20-digit number. */
    /* (17 digits are sufficient for round-tripping.) */
    Assert(v < 100000000000000000L);
    if (v >= 10000000000000000L)
    {
        return 17;
    }
    if (v >= 1000000000000000L)
    {
        return 16;
    }
    if (v >= 100000000000000L)
    {
        return 15;
    }
    if (v >= 10000000000000L)
    {
        return 14;
    }
    if (v >= 1000000000000L)
    {
        return 13;
    }
    if (v >= 100000000000L)
    {
        return 12;
    }
    if (v >= 10000000000L)
    {
        return 11;
    }
    if (v >= 1000000000L)
    {
        return 10;
    }
    if (v >= 100000000L)
    {
        return 9;
    }
    if (v >= 10000000L)
    {
        return 8;
    }
    if (v >= 1000000L)
    {
        return 7;
    }
    if (v >= 100000L)
    {
        return 6;
    }
    if (v >= 10000L)
    {
        return 5;
    }
    if (v >= 1000L)
    {
        return 4;
    }
    if (v >= 100L)
    {
        return 3;
    }
    if (v >= 10L)
    {
        return 2;
    }
    return 1;
}
 
/*  A floating decimal representing m * 10^e. */
typedef struct floating_decimal_64
{
    uint64      mantissa;
    int32       exponent;
} floating_decimal_64;
 
static inline floating_decimal_64
d2d(const uint64 ieeeMantissa, const uint32 ieeeExponent)
{
    int32       e2;
    uint64      m2;
 
    if (ieeeExponent == 0)
    {
        /* We subtract 2 so that the bounds computation has 2 additional bits. */
        e2 = 1 - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS - 2;
        m2 = ieeeMantissa;
    }
    else
    {
        e2 = ieeeExponent - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS - 2;
        m2 = (UINT64CONST(1) << DOUBLE_MANTISSA_BITS) | ieeeMantissa;
    }
 
#if STRICTLY_SHORTEST
    const bool  even = (m2 & 1) == 0;
    const bool  acceptBounds = even;
#else
    const bool  acceptBounds = false;
#endif
 
    /* Step 2: Determine the interval of legal decimal representations. */
    const uint64 mv = 4 * m2;
 
    /* Implicit bool -> int conversion. True is 1, false is 0. */
    const uint32 mmShift = ieeeMantissa != 0 || ieeeExponent <= 1;
 
    /* We would compute mp and mm like this: */
    /* uint64 mp = 4 * m2 + 2; */
    /* uint64 mm = mv - 1 - mmShift; */
 
    /* Step 3: Convert to a decimal power base using 128-bit arithmetic. */
    uint64      vr,
                vp,
                vm;
    int32       e10;
    bool        vmIsTrailingZeros = false;
    bool        vrIsTrailingZeros = false;
 
    if (e2 >= 0)
    {
        /*
         * I tried special-casing q == 0, but there was no effect on
         * performance.
         *
         * This expr is slightly faster than max(0, log10Pow2(e2) - 1).
         */
        const uint32 q = log10Pow2(e2) - (e2 > 3);
        const int32 k = DOUBLE_POW5_INV_BITCOUNT + pow5bits(q) - 1;
        const int32 i = -e2 + q + k;
 
        e10 = q;
 
        vr = mulShiftAll(m2, DOUBLE_POW5_INV_SPLIT[q], i, &vp, &vm, mmShift);
 
        if (q <= 21)
        {
            /*
             * This should use q <= 22, but I think 21 is also safe. Smaller
             * values may still be safe, but it's more difficult to reason
             * about them.
             *
             * Only one of mp, mv, and mm can be a multiple of 5, if any.
             */
            const uint32 mvMod5 = (uint32) (mv - 5 * div5(mv));
 
            if (mvMod5 == 0)
            {
                vrIsTrailingZeros = multipleOfPowerOf5(mv, q);
            }
            else if (acceptBounds)
            {
                /*----
                 * Same as min(e2 + (~mm & 1), pow5Factor(mm)) >= q
                 * <=> e2 + (~mm & 1) >= q && pow5Factor(mm) >= q
                 * <=> true && pow5Factor(mm) >= q, since e2 >= q.
                 *----
                 */
                vmIsTrailingZeros = multipleOfPowerOf5(mv - 1 - mmShift, q);
            }
            else
            {
                /* Same as min(e2 + 1, pow5Factor(mp)) >= q. */
                vp -= multipleOfPowerOf5(mv + 2, q);
            }
        }
    }
    else
    {
        /*
         * This expression is slightly faster than max(0, log10Pow5(-e2) - 1).
         */
        const uint32 q = log10Pow5(-e2) - (-e2 > 1);
        const int32 i = -e2 - q;
        const int32 k = pow5bits(i) - DOUBLE_POW5_BITCOUNT;
        const int32 j = q - k;
 
        e10 = q + e2;
 
        vr = mulShiftAll(m2, DOUBLE_POW5_SPLIT[i], j, &vp, &vm, mmShift);
 
        if (q <= 1)
        {
            /*
             * {vr,vp,vm} is trailing zeros if {mv,mp,mm} has at least q
             * trailing 0 bits.
             */
            /* mv = 4 * m2, so it always has at least two trailing 0 bits. */
            vrIsTrailingZeros = true;
            if (acceptBounds)
            {
                /*
                 * mm = mv - 1 - mmShift, so it has 1 trailing 0 bit iff
                 * mmShift == 1.
                 */
                vmIsTrailingZeros = mmShift == 1;
            }
            else
            {
                /*
                 * mp = mv + 2, so it always has at least one trailing 0 bit.
                 */
                --vp;
            }
        }
        else if (q < 63)
        {
            /* TODO(ulfjack):Use a tighter bound here. */
            /*
             * We need to compute min(ntz(mv), pow5Factor(mv) - e2) >= q - 1
             */
            /* <=> ntz(mv) >= q - 1 && pow5Factor(mv) - e2 >= q - 1 */
            /* <=> ntz(mv) >= q - 1 (e2 is negative and -e2 >= q) */
            /* <=> (mv & ((1 << (q - 1)) - 1)) == 0 */
 
            /*
             * We also need to make sure that the left shift does not
             * overflow.
             */
            vrIsTrailingZeros = multipleOfPowerOf2(mv, q - 1);
        }
    }
 
    /*
     * Step 4: Find the shortest decimal representation in the interval of
     * legal representations.
     */
    uint32      removed = 0;
    uint8       lastRemovedDigit = 0;
    uint64      output;
 
    /* On average, we remove ~2 digits. */
    if (vmIsTrailingZeros || vrIsTrailingZeros)
    {
        /* General case, which happens rarely (~0.7%). */
        for (;;)
        {
            const uint64 vpDiv10 = div10(vp);
            const uint64 vmDiv10 = div10(vm);
 
            if (vpDiv10 <= vmDiv10)
                break;
 
            const uint32 vmMod10 = (uint32) (vm - 10 * vmDiv10);
            const uint64 vrDiv10 = div10(vr);
            const uint32 vrMod10 = (uint32) (vr - 10 * vrDiv10);
 
            vmIsTrailingZeros &= vmMod10 == 0;
            vrIsTrailingZeros &= lastRemovedDigit == 0;
            lastRemovedDigit = (uint8) vrMod10;
            vr = vrDiv10;
            vp = vpDiv10;
            vm = vmDiv10;
            ++removed;
        }
 
        if (vmIsTrailingZeros)
        {
            for (;;)
            {
                const uint64 vmDiv10 = div10(vm);
                const uint32 vmMod10 = (uint32) (vm - 10 * vmDiv10);
 
                if (vmMod10 != 0)
                    break;
 
                const uint64 vpDiv10 = div10(vp);
                const uint64 vrDiv10 = div10(vr);
                const uint32 vrMod10 = (uint32) (vr - 10 * vrDiv10);
 
                vrIsTrailingZeros &= lastRemovedDigit == 0;
                lastRemovedDigit = (uint8) vrMod10;
                vr = vrDiv10;
                vp = vpDiv10;
                vm = vmDiv10;
                ++removed;
            }
        }
 
        if (vrIsTrailingZeros && lastRemovedDigit == 5 && vr % 2 == 0)
        {
            /* Round even if the exact number is .....50..0. */
            lastRemovedDigit = 4;
        }
 
        /*
         * We need to take vr + 1 if vr is outside bounds or we need to round
         * up.
         */
        output = vr + ((vr == vm && (!acceptBounds || !vmIsTrailingZeros)) || lastRemovedDigit >= 5);
    }
    else
    {
        /*
         * Specialized for the common case (~99.3%). Percentages below are
         * relative to this.
         */
        bool        roundUp = false;
        const uint64 vpDiv100 = div100(vp);
        const uint64 vmDiv100 = div100(vm);
 
        if (vpDiv100 > vmDiv100)
        {
            /* Optimization:remove two digits at a time(~86.2 %). */
            const uint64 vrDiv100 = div100(vr);
            const uint32 vrMod100 = (uint32) (vr - 100 * vrDiv100);
 
            roundUp = vrMod100 >= 50;
            vr = vrDiv100;
            vp = vpDiv100;
            vm = vmDiv100;
            removed += 2;
        }
 
        /*----
         * Loop iterations below (approximately), without optimization
         * above:
         *
         * 0: 0.03%, 1: 13.8%, 2: 70.6%, 3: 14.0%, 4: 1.40%, 5: 0.14%,
         * 6+: 0.02%
         *
         * Loop iterations below (approximately), with optimization
         * above:
         *
         * 0: 70.6%, 1: 27.8%, 2: 1.40%, 3: 0.14%, 4+: 0.02%
         *----
         */
        for (;;)
        {
            const uint64 vpDiv10 = div10(vp);
            const uint64 vmDiv10 = div10(vm);
 
            if (vpDiv10 <= vmDiv10)
                break;
 
            const uint64 vrDiv10 = div10(vr);
            const uint32 vrMod10 = (uint32) (vr - 10 * vrDiv10);
 
            roundUp = vrMod10 >= 5;
            vr = vrDiv10;
            vp = vpDiv10;
            vm = vmDiv10;
            ++removed;
        }
 
        /*
         * We need to take vr + 1 if vr is outside bounds or we need to round
         * up.
         */
        output = vr + (vr == vm || roundUp);
    }
 
    const int32 exp = e10 + removed;
 
    floating_decimal_64 fd;
 
    fd.exponent = exp;
    fd.mantissa = output;
    return fd;
}
 
static inline int
to_chars_df(const floating_decimal_64 v, const uint32 olength, char *const result)
{
    /* Step 5: Print the decimal representation. */
    int         index = 0;
 
    uint64      output = v.mantissa;
    int32       exp = v.exponent;
 
    /*----
     * On entry, mantissa * 10^exp is the result to be output.
     * Caller has already done the - sign if needed.
     *
     * We want to insert the point somewhere depending on the output length
     * and exponent, which might mean adding zeros:
     *
     *            exp  | format
     *            1+   |  ddddddddd000000
     *            0    |  ddddddddd
     *  -1 .. -len+1   |  dddddddd.d to d.ddddddddd
     *  -len ...       |  0.ddddddddd to 0.000dddddd
     */
    uint32      i = 0;
    int32       nexp = exp + olength;
 
    if (nexp <= 0)
    {
        /* -nexp is number of 0s to add after '.' */
        Assert(nexp >= -3);
        /* 0.000ddddd */
        index = 2 - nexp;
        /* won't need more than this many 0s */
        memcpy(result, "0.000000", 8);
    }
    else if (exp < 0)
    {
        /*
         * dddd.dddd; leave space at the start and move the '.' in after
         */
        index = 1;
    }
    else
    {
        /*
         * We can save some code later by pre-filling with zeros. We know that
         * there can be no more than 16 output digits in this form, otherwise
         * we would not choose fixed-point output.
         */
        Assert(exp < 16 && exp + olength <= 16);
        memset(result, '0', 16);
    }
 
    /*
     * We prefer 32-bit operations, even on 64-bit platforms. We have at most
     * 17 digits, and uint32 can store 9 digits. If output doesn't fit into
     * uint32, we cut off 8 digits, so the rest will fit into uint32.
     */
    if ((output >> 32) != 0)
    {
        /* Expensive 64-bit division. */
        const uint64 q = div1e8(output);
        uint32      output2 = (uint32) (output - 100000000 * q);
        const uint32 c = output2 % 10000;
 
        output = q;
        output2 /= 10000;
 
        const uint32 d = output2 % 10000;
        const uint32 c0 = (c % 100) << 1;
        const uint32 c1 = (c / 100) << 1;
        const uint32 d0 = (d % 100) << 1;
        const uint32 d1 = (d / 100) << 1;
 
        memcpy(result + index + olength - i - 2, DIGIT_TABLE + c0, 2);
        memcpy(result + index + olength - i - 4, DIGIT_TABLE + c1, 2);
        memcpy(result + index + olength - i - 6, DIGIT_TABLE + d0, 2);
        memcpy(result + index + olength - i - 8, DIGIT_TABLE + d1, 2);
        i += 8;
    }
 
    uint32      output2 = (uint32) output;
 
    while (output2 >= 10000)
    {
        const uint32 c = output2 - 10000 * (output2 / 10000);
        const uint32 c0 = (c % 100) << 1;
        const uint32 c1 = (c / 100) << 1;
 
        output2 /= 10000;
        memcpy(result + index + olength - i - 2, DIGIT_TABLE + c0, 2);
        memcpy(result + index + olength - i - 4, DIGIT_TABLE + c1, 2);
        i += 4;
    }
    if (output2 >= 100)
    {
        const uint32 c = (output2 % 100) << 1;
 
        output2 /= 100;
        memcpy(result + index + olength - i - 2, DIGIT_TABLE + c, 2);
        i += 2;
    }
    if (output2 >= 10)
    {
        const uint32 c = output2 << 1;
 
        memcpy(result + index + olength - i - 2, DIGIT_TABLE + c, 2);
    }
    else
    {
        result[index] = (char) ('0' + output2);
    }
 
    if (index == 1)
    {
        /*
         * nexp is 1..15 here, representing the number of digits before the
         * point. A value of 16 is not possible because we switch to
         * scientific notation when the display exponent reaches 15.
         */
        Assert(nexp < 16);
        /* gcc only seems to want to optimize memmove for small 2^n */
        if (nexp & 8)
        {
            memmove(result + index - 1, result + index, 8);
            index += 8;
        }
        if (nexp & 4)
        {
            memmove(result + index - 1, result + index, 4);
            index += 4;
        }
        if (nexp & 2)
        {
            memmove(result + index - 1, result + index, 2);
            index += 2;
        }
        if (nexp & 1)
        {
            result[index - 1] = result[index];
        }
        result[nexp] = '.';
        index = olength + 1;
    }
    else if (exp >= 0)
    {
        /* we supplied the trailing zeros earlier, now just set the length. */
        index = olength + exp;
    }
    else
    {
        index = olength + (2 - nexp);
    }
 
    return index;
}
 
static inline int
to_chars(floating_decimal_64 v, const bool sign, char *const result)
{
    /* Step 5: Print the decimal representation. */
    int         index = 0;
 
    uint64      output = v.mantissa;
    uint32      olength = decimalLength(output);
    int32       exp = v.exponent + olength - 1;
 
    if (sign)
    {
        result[index++] = '-';
    }
 
    /*
     * The thresholds for fixed-point output are chosen to match printf
     * defaults. Beware that both the code of to_chars_df and the value of
     * DOUBLE_SHORTEST_DECIMAL_LEN are sensitive to these thresholds.
     */
    if (exp >= -4 && exp < 15)
        return to_chars_df(v, olength, result + index) + sign;
 
    /*
     * If v.exponent is exactly 0, we might have reached here via the small
     * integer fast path, in which case v.mantissa might contain trailing
     * (decimal) zeros. For scientific notation we need to move these zeros
     * into the exponent. (For fixed point this doesn't matter, which is why
     * we do this here rather than above.)
     *
     * Since we already calculated the display exponent (exp) above based on
     * the old decimal length, that value does not change here. Instead, we
     * just reduce the display length for each digit removed.
     *
     * If we didn't get here via the fast path, the raw exponent will not
     * usually be 0, and there will be no trailing zeros, so we pay no more
     * than one div10/multiply extra cost. We claw back half of that by
     * checking for divisibility by 2 before dividing by 10.
     */
    if (v.exponent == 0)
    {
        while ((output & 1) == 0)
        {
            const uint64 q = div10(output);
            const uint32 r = (uint32) (output - 10 * q);
 
            if (r != 0)
                break;
            output = q;
            --olength;
        }
    }
 
    /*----
     * Print the decimal digits.
     *
     * The following code is equivalent to:
     *
     * for (uint32 i = 0; i < olength - 1; ++i) {
     *   const uint32 c = output % 10; output /= 10;
     *   result[index + olength - i] = (char) ('0' + c);
     * }
     * result[index] = '0' + output % 10;
     *----
     */
 
    uint32      i = 0;
 
    /*
     * We prefer 32-bit operations, even on 64-bit platforms. We have at most
     * 17 digits, and uint32 can store 9 digits. If output doesn't fit into
     * uint32, we cut off 8 digits, so the rest will fit into uint32.
     */
    if ((output >> 32) != 0)
    {
        /* Expensive 64-bit division. */
        const uint64 q = div1e8(output);
        uint32      output2 = (uint32) (output - 100000000 * q);
 
        output = q;
 
        const uint32 c = output2 % 10000;
 
        output2 /= 10000;
 
        const uint32 d = output2 % 10000;
        const uint32 c0 = (c % 100) << 1;
        const uint32 c1 = (c / 100) << 1;
        const uint32 d0 = (d % 100) << 1;
        const uint32 d1 = (d / 100) << 1;
 
        memcpy(result + index + olength - i - 1, DIGIT_TABLE + c0, 2);
        memcpy(result + index + olength - i - 3, DIGIT_TABLE + c1, 2);
        memcpy(result + index + olength - i - 5, DIGIT_TABLE + d0, 2);
        memcpy(result + index + olength - i - 7, DIGIT_TABLE + d1, 2);
        i += 8;
    }
 
    uint32      output2 = (uint32) output;
 
    while (output2 >= 10000)
    {
        const uint32 c = output2 - 10000 * (output2 / 10000);
 
        output2 /= 10000;
 
        const uint32 c0 = (c % 100) << 1;
        const uint32 c1 = (c / 100) << 1;
 
        memcpy(result + index + olength - i - 1, DIGIT_TABLE + c0, 2);
        memcpy(result + index + olength - i - 3, DIGIT_TABLE + c1, 2);
        i += 4;
    }
    if (output2 >= 100)
    {
        const uint32 c = (output2 % 100) << 1;
 
        output2 /= 100;
        memcpy(result + index + olength - i - 1, DIGIT_TABLE + c, 2);
        i += 2;
    }
    if (output2 >= 10)
    {
        const uint32 c = output2 << 1;
 
        /*
         * We can't use memcpy here: the decimal dot goes between these two
         * digits.
         */
        result[index + olength - i] = DIGIT_TABLE[c + 1];
        result[index] = DIGIT_TABLE[c];
    }
    else
    {
        result[index] = (char) ('0' + output2);
    }
 
    /* Print decimal point if needed. */
    if (olength > 1)
    {
        result[index + 1] = '.';
        index += olength + 1;
    }
    else
    {
        ++index;
    }
 
    /* Print the exponent. */
    result[index++] = 'e';
    if (exp < 0)
    {
        result[index++] = '-';
        exp = -exp;
    }
    else
        result[index++] = '+';
 
    if (exp >= 100)
    {
        const int32 c = exp % 10;
 
        memcpy(result + index, DIGIT_TABLE + 2 * (exp / 10), 2);
        result[index + 2] = (char) ('0' + c);
        index += 3;
    }
    else
    {
        memcpy(result + index, DIGIT_TABLE + 2 * exp, 2);
        index += 2;
    }
 
    return index;
}
 
static inline bool
d2d_small_int(const uint64 ieeeMantissa,
              const uint32 ieeeExponent,
              floating_decimal_64 *v)
{
    const int32 e2 = (int32) ieeeExponent - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS;
 
    /*
     * Avoid using multiple "return false;" here since it tends to provoke the
     * compiler into inlining multiple copies of d2d, which is undesirable.
     */
 
    if (e2 >= -DOUBLE_MANTISSA_BITS && e2 <= 0)
    {
        /*----
         * Since 2^52 <= m2 < 2^53 and 0 <= -e2 <= 52:
         *   1 <= f = m2 / 2^-e2 < 2^53.
         *
         * Test if the lower -e2 bits of the significand are 0, i.e. whether
         * the fraction is 0. We can use ieeeMantissa here, since the implied
         * 1 bit can never be tested by this; the implied 1 can only be part
         * of a fraction if e2 < -DOUBLE_MANTISSA_BITS which we already
         * checked. (e.g. 0.5 gives ieeeMantissa == 0 and e2 == -53)
         */
        const uint64 mask = (UINT64CONST(1) << -e2) - 1;
        const uint64 fraction = ieeeMantissa & mask;
 
        if (fraction == 0)
        {
            /*----
             * f is an integer in the range [1, 2^53).
             * Note: mantissa might contain trailing (decimal) 0's.
             * Note: since 2^53 < 10^16, there is no need to adjust
             * decimalLength().
             */
            const uint64 m2 = (UINT64CONST(1) << DOUBLE_MANTISSA_BITS) | ieeeMantissa;
 
            v->mantissa = m2 >> -e2;
            v->exponent = 0;
            return true;
        }
    }
 
    return false;
}
 
/*
 * Store the shortest decimal representation of the given double as an
 * UNTERMINATED string in the caller's supplied buffer (which must be at least
 * DOUBLE_SHORTEST_DECIMAL_LEN-1 bytes long).
 *
 * Returns the number of bytes stored.
 */
int
double_to_shortest_decimal_bufn(double f, char *result)
{
    /*
     * Step 1: Decode the floating-point number, and unify normalized and
     * subnormal cases.
     */
    const uint64 bits = double_to_bits(f);
 
    /* Decode bits into sign, mantissa, and exponent. */
    const bool  ieeeSign = ((bits >> (DOUBLE_MANTISSA_BITS + DOUBLE_EXPONENT_BITS)) & 1) != 0;
    const uint64 ieeeMantissa = bits & ((UINT64CONST(1) << DOUBLE_MANTISSA_BITS) - 1);
    const uint32 ieeeExponent = (uint32) ((bits >> DOUBLE_MANTISSA_BITS) & ((1u << DOUBLE_EXPONENT_BITS) - 1));
 
    /* Case distinction; exit early for the easy cases. */
    if (ieeeExponent == ((1u << DOUBLE_EXPONENT_BITS) - 1u) || (ieeeExponent == 0 && ieeeMantissa == 0))
    {
        return copy_special_str(result, ieeeSign, (ieeeExponent != 0), (ieeeMantissa != 0));
    }
 
    floating_decimal_64 v;
    const bool  isSmallInt = d2d_small_int(ieeeMantissa, ieeeExponent, &v);
 
    if (!isSmallInt)
    {
        v = d2d(ieeeMantissa, ieeeExponent);
    }
 
    return to_chars(v, ieeeSign, result);
}
 
/*
 * Store the shortest decimal representation of the given double as a
 * null-terminated string in the caller's supplied buffer (which must be at
 * least DOUBLE_SHORTEST_DECIMAL_LEN bytes long).
 *
 * Returns the string length.
 */
int
double_to_shortest_decimal_buf(double f, char *result)
{
    const int   index = double_to_shortest_decimal_bufn(f, result);
 
    /* Terminate the string. */
    Assert(index < DOUBLE_SHORTEST_DECIMAL_LEN);
    result[index] = '\0';
    return index;
}
 
/*
 * Return the shortest decimal representation as a null-terminated palloc'd
 * string (outside the backend, uses malloc() instead).
 *
 * Caller is responsible for freeing the result.
 */
char *
double_to_shortest_decimal(double f)
{
    char       *const result = (char *) palloc(DOUBLE_SHORTEST_DECIMAL_LEN);
 
    double_to_shortest_decimal_buf(f, result);
    return result;
}