selfuncs.c

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/*-------------------------------------------------------------------------
 *
 * selfuncs.c
 *    Selectivity functions and index cost estimation functions for
 *    standard operators and index access methods.
 *
 *    Selectivity routines are registered in the pg_operator catalog
 *    in the "oprrest" and "oprjoin" attributes.
 *
 *    Index cost functions are located via the index AM's API struct,
 *    which is obtained from the handler function registered in pg_am.
 *
 * Portions Copyright (c) 1996-2020, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *    src/backend/utils/adt/selfuncs.c
 *
 *-------------------------------------------------------------------------
 */

/*----------
 * Operator selectivity estimation functions are called to estimate the
 * selectivity of WHERE clauses whose top-level operator is their operator.
 * We divide the problem into two cases:
 *      Restriction clause estimation: the clause involves vars of just
 *          one relation.
 *      Join clause estimation: the clause involves vars of multiple rels.
 * Join selectivity estimation is far more difficult and usually less accurate
 * than restriction estimation.
 *
 * When dealing with the inner scan of a nestloop join, we consider the
 * join's joinclauses as restriction clauses for the inner relation, and
 * treat vars of the outer relation as parameters (a/k/a constants of unknown
 * values).  So, restriction estimators need to be able to accept an argument
 * telling which relation is to be treated as the variable.
 *
 * The call convention for a restriction estimator (oprrest function) is
 *
 *      Selectivity oprrest (PlannerInfo *root,
 *                           Oid operator,
 *                           List *args,
 *                           int varRelid);
 *
 * root: general information about the query (rtable and RelOptInfo lists
 * are particularly important for the estimator).
 * operator: OID of the specific operator in question.
 * args: argument list from the operator clause.
 * varRelid: if not zero, the relid (rtable index) of the relation to
 * be treated as the variable relation.  May be zero if the args list
 * is known to contain vars of only one relation.
 *
 * This is represented at the SQL level (in pg_proc) as
 *
 *      float8 oprrest (internal, oid, internal, int4);
 *
 * The result is a selectivity, that is, a fraction (0 to 1) of the rows
 * of the relation that are expected to produce a TRUE result for the
 * given operator.
 *
 * The call convention for a join estimator (oprjoin function) is similar
 * except that varRelid is not needed, and instead join information is
 * supplied:
 *
 *      Selectivity oprjoin (PlannerInfo *root,
 *                           Oid operator,
 *                           List *args,
 *                           JoinType jointype,
 *                           SpecialJoinInfo *sjinfo);
 *
 *      float8 oprjoin (internal, oid, internal, int2, internal);
 *
 * (Before Postgres 8.4, join estimators had only the first four of these
 * parameters.  That signature is still allowed, but deprecated.)  The
 * relationship between jointype and sjinfo is explained in the comments for
 * clause_selectivity() --- the short version is that jointype is usually
 * best ignored in favor of examining sjinfo.
 *
 * Join selectivity for regular inner and outer joins is defined as the
 * fraction (0 to 1) of the cross product of the relations that is expected
 * to produce a TRUE result for the given operator.  For both semi and anti
 * joins, however, the selectivity is defined as the fraction of the left-hand
 * side relation's rows that are expected to have a match (ie, at least one
 * row with a TRUE result) in the right-hand side.
 *
 * For both oprrest and oprjoin functions, the operator's input collation OID
 * (if any) is passed using the standard fmgr mechanism, so that the estimator
 * function can fetch it with PG_GET_COLLATION().  Note, however, that all
 * statistics in pg_statistic are currently built using the relevant column's
 * collation.  Thus, in most cases where we are looking at statistics, we
 * should ignore the operator collation and use the stats entry's collation.
 * We expect that the error induced by doing this is usually not large enough
 * to justify complicating matters.  In any case, doing otherwise would yield
 * entirely garbage results for ordered stats data such as histograms.
 *----------
 */

#include "postgres.h"

#include <ctype.h>
#include <math.h>

#include "access/brin.h"
#include "access/brin_page.h"
#include "access/gin.h"
#include "access/table.h"
#include "access/tableam.h"
#include "access/visibilitymap.h"
#include "catalog/pg_am.h"
#include "catalog/pg_collation.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_statistic.h"
#include "catalog/pg_statistic_ext.h"
#include "executor/nodeAgg.h"
#include "miscadmin.h"
#include "nodes/makefuncs.h"
#include "nodes/nodeFuncs.h"
#include "optimizer/clauses.h"
#include "optimizer/cost.h"
#include "optimizer/optimizer.h"
#include "optimizer/pathnode.h"
#include "optimizer/paths.h"
#include "optimizer/plancat.h"
#include "parser/parse_clause.h"
#include "parser/parsetree.h"
#include "statistics/statistics.h"
#include "storage/bufmgr.h"
#include "utils/builtins.h"
#include "utils/date.h"
#include "utils/datum.h"
#include "utils/fmgroids.h"
#include "utils/index_selfuncs.h"
#include "utils/lsyscache.h"
#include "utils/memutils.h"
#include "utils/pg_locale.h"
#include "utils/rel.h"
#include "utils/selfuncs.h"
#include "utils/snapmgr.h"
#include "utils/spccache.h"
#include "utils/syscache.h"
#include "utils/timestamp.h"
#include "utils/typcache.h"


/* Hooks for plugins to get control when we ask for stats */
get_relation_stats_hook_type get_relation_stats_hook = NULL;
get_index_stats_hook_type get_index_stats_hook = NULL;

static double eqsel_internal(PG_FUNCTION_ARGS, bool negate);
static double eqjoinsel_inner(Oid opfuncoid,
                              VariableStatData *vardata1, VariableStatData *vardata2,
                              double nd1, double nd2,
                              bool isdefault1, bool isdefault2,
                              AttStatsSlot *sslot1, AttStatsSlot *sslot2,
                              Form_pg_statistic stats1, Form_pg_statistic stats2,
                              bool have_mcvs1, bool have_mcvs2);
static double eqjoinsel_semi(Oid opfuncoid,
                             VariableStatData *vardata1, VariableStatData *vardata2,
                             double nd1, double nd2,
                             bool isdefault1, bool isdefault2,
                             AttStatsSlot *sslot1, AttStatsSlot *sslot2,
                             Form_pg_statistic stats1, Form_pg_statistic stats2,
                             bool have_mcvs1, bool have_mcvs2,
                             RelOptInfo *inner_rel);
static bool estimate_multivariate_ndistinct(PlannerInfo *root,
                                            RelOptInfo *rel, List **varinfos, double *ndistinct);
static bool convert_to_scalar(Datum value, Oid valuetypid, Oid collid,
                              double *scaledvalue,
                              Datum lobound, Datum hibound, Oid boundstypid,
                              double *scaledlobound, double *scaledhibound);
static double convert_numeric_to_scalar(Datum value, Oid typid, bool *failure);
static void convert_string_to_scalar(char *value,
                                     double *scaledvalue,
                                     char *lobound,
                                     double *scaledlobound,
                                     char *hibound,
                                     double *scaledhibound);
static void convert_bytea_to_scalar(Datum value,
                                    double *scaledvalue,
                                    Datum lobound,
                                    double *scaledlobound,
                                    Datum hibound,
                                    double *scaledhibound);
static double convert_one_string_to_scalar(char *value,
                                           int rangelo, int rangehi);
static double convert_one_bytea_to_scalar(unsigned char *value, int valuelen,
                                          int rangelo, int rangehi);
static char *convert_string_datum(Datum value, Oid typid, Oid collid,
                                  bool *failure);
static double convert_timevalue_to_scalar(Datum value, Oid typid,
                                          bool *failure);
static void examine_simple_variable(PlannerInfo *root, Var *var,
                                    VariableStatData *vardata);
static bool get_variable_range(PlannerInfo *root, VariableStatData *vardata,
                               Oid sortop, Datum *min, Datum *max);
static bool get_actual_variable_range(PlannerInfo *root,
                                      VariableStatData *vardata,
                                      Oid sortop,
                                      Datum *min, Datum *max);
static bool get_actual_variable_endpoint(Relation heapRel,
                                         Relation indexRel,
                                         ScanDirection indexscandir,
                                         ScanKey scankeys,
                                         int16 typLen,
                                         bool typByVal,
                                         TupleTableSlot *tableslot,
                                         MemoryContext outercontext,
                                         Datum *endpointDatum);
static RelOptInfo *find_join_input_rel(PlannerInfo *root, Relids relids);


/*
 *      eqsel           - Selectivity of "=" for any data types.
 *
 * Note: this routine is also used to estimate selectivity for some
 * operators that are not "=" but have comparable selectivity behavior,
 * such as "~=" (geometric approximate-match).  Even for "=", we must
 * keep in mind that the left and right datatypes may differ.
 */
Datum
eqsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8((float8) eqsel_internal(fcinfo, false));
}

/*
 * Common code for eqsel() and neqsel()
 */
static double
eqsel_internal(PG_FUNCTION_ARGS, bool negate)
{
    PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
    Oid         operator = PG_GETARG_OID(1);
    List       *args = (List *) PG_GETARG_POINTER(2);
    int         varRelid = PG_GETARG_INT32(3);
    VariableStatData vardata;
    Node       *other;
    bool        varonleft;
    double      selec;

    /*
     * When asked about <>, we do the estimation using the corresponding =
     * operator, then convert to <> via "1.0 - eq_selectivity - nullfrac".
     */
    if (negate)
    {
        operator = get_negator(operator);
        if (!OidIsValid(operator))
        {
            /* Use default selectivity (should we raise an error instead?) */
            return 1.0 - DEFAULT_EQ_SEL;
        }
    }

    /*
     * If expression is not variable = something or something = variable, then
     * punt and return a default estimate.
     */
    if (!get_restriction_variable(root, args, varRelid,
                                  &vardata, &other, &varonleft))
        return negate ? (1.0 - DEFAULT_EQ_SEL) : DEFAULT_EQ_SEL;

    /*
     * We can do a lot better if the something is a constant.  (Note: the
     * Const might result from estimation rather than being a simple constant
     * in the query.)
     */
    if (IsA(other, Const))
        selec = var_eq_const(&vardata, operator,
                             ((Const *) other)->constvalue,
                             ((Const *) other)->constisnull,
                             varonleft, negate);
    else
        selec = var_eq_non_const(&vardata, operator, other,
                                 varonleft, negate);

    ReleaseVariableStats(vardata);

    return selec;
}

/*
 * var_eq_const --- eqsel for var = const case
 *
 * This is exported so that some other estimation functions can use it.
 */
double
var_eq_const(VariableStatData *vardata, Oid operator,
             Datum constval, bool constisnull,
             bool varonleft, bool negate)
{
    double      selec;
    double      nullfrac = 0.0;
    bool        isdefault;
    Oid         opfuncoid;

    /*
     * If the constant is NULL, assume operator is strict and return zero, ie,
     * operator will never return TRUE.  (It's zero even for a negator op.)
     */
    if (constisnull)
        return 0.0;

    /*
     * Grab the nullfrac for use below.  Note we allow use of nullfrac
     * regardless of security check.
     */
    if (HeapTupleIsValid(vardata->statsTuple))
    {
        Form_pg_statistic stats;

        stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
        nullfrac = stats->stanullfrac;
    }

    /*
     * If we matched the var to a unique index or DISTINCT clause, assume
     * there is exactly one match regardless of anything else.  (This is
     * slightly bogus, since the index or clause's equality operator might be
     * different from ours, but it's much more likely to be right than
     * ignoring the information.)
     */
    if (vardata->isunique && vardata->rel && vardata->rel->tuples >= 1.0)
    {
        selec = 1.0 / vardata->rel->tuples;
    }
    else if (HeapTupleIsValid(vardata->statsTuple) &&
             statistic_proc_security_check(vardata,
                                           (opfuncoid = get_opcode(operator))))
    {
        AttStatsSlot sslot;
        bool        match = false;
        int         i;

        /*
         * Is the constant "=" to any of the column's most common values?
         * (Although the given operator may not really be "=", we will assume
         * that seeing whether it returns TRUE is an appropriate test.  If you
         * don't like this, maybe you shouldn't be using eqsel for your
         * operator...)
         */
        if (get_attstatsslot(&sslot, vardata->statsTuple,
                             STATISTIC_KIND_MCV, InvalidOid,
                             ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
        {
            FmgrInfo    eqproc;

            fmgr_info(opfuncoid, &eqproc);

            for (i = 0; i < sslot.nvalues; i++)
            {
                /* be careful to apply operator right way 'round */
                if (varonleft)
                    match = DatumGetBool(FunctionCall2Coll(&eqproc,
                                                           sslot.stacoll,
                                                           sslot.values[i],
                                                           constval));
                else
                    match = DatumGetBool(FunctionCall2Coll(&eqproc,
                                                           sslot.stacoll,
                                                           constval,
                                                           sslot.values[i]));
                if (match)
                    break;
            }
        }
        else
        {
            /* no most-common-value info available */
            i = 0;              /* keep compiler quiet */
        }

        if (match)
        {
            /*
             * Constant is "=" to this common value.  We know selectivity
             * exactly (or as exactly as ANALYZE could calculate it, anyway).
             */
            selec = sslot.numbers[i];
        }
        else
        {
            /*
             * Comparison is against a constant that is neither NULL nor any
             * of the common values.  Its selectivity cannot be more than
             * this:
             */
            double      sumcommon = 0.0;
            double      otherdistinct;

            for (i = 0; i < sslot.nnumbers; i++)
                sumcommon += sslot.numbers[i];
            selec = 1.0 - sumcommon - nullfrac;
            CLAMP_PROBABILITY(selec);

            /*
             * and in fact it's probably a good deal less. We approximate that
             * all the not-common values share this remaining fraction
             * equally, so we divide by the number of other distinct values.
             */
            otherdistinct = get_variable_numdistinct(vardata, &isdefault) -
                sslot.nnumbers;
            if (otherdistinct > 1)
                selec /= otherdistinct;

            /*
             * Another cross-check: selectivity shouldn't be estimated as more
             * than the least common "most common value".
             */
            if (sslot.nnumbers > 0 && selec > sslot.numbers[sslot.nnumbers - 1])
                selec = sslot.numbers[sslot.nnumbers - 1];
        }

        free_attstatsslot(&sslot);
    }
    else
    {
        /*
         * No ANALYZE stats available, so make a guess using estimated number
         * of distinct values and assuming they are equally common. (The guess
         * is unlikely to be very good, but we do know a few special cases.)
         */
        selec = 1.0 / get_variable_numdistinct(vardata, &isdefault);
    }

    /* now adjust if we wanted <> rather than = */
    if (negate)
        selec = 1.0 - selec - nullfrac;

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(selec);

    return selec;
}

/*
 * var_eq_non_const --- eqsel for var = something-other-than-const case
 *
 * This is exported so that some other estimation functions can use it.
 */
double
var_eq_non_const(VariableStatData *vardata, Oid operator,
                 Node *other,
                 bool varonleft, bool negate)
{
    double      selec;
    double      nullfrac = 0.0;
    bool        isdefault;

    /*
     * Grab the nullfrac for use below.
     */
    if (HeapTupleIsValid(vardata->statsTuple))
    {
        Form_pg_statistic stats;

        stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
        nullfrac = stats->stanullfrac;
    }

    /*
     * If we matched the var to a unique index or DISTINCT clause, assume
     * there is exactly one match regardless of anything else.  (This is
     * slightly bogus, since the index or clause's equality operator might be
     * different from ours, but it's much more likely to be right than
     * ignoring the information.)
     */
    if (vardata->isunique && vardata->rel && vardata->rel->tuples >= 1.0)
    {
        selec = 1.0 / vardata->rel->tuples;
    }
    else if (HeapTupleIsValid(vardata->statsTuple))
    {
        double      ndistinct;
        AttStatsSlot sslot;

        /*
         * Search is for a value that we do not know a priori, but we will
         * assume it is not NULL.  Estimate the selectivity as non-null
         * fraction divided by number of distinct values, so that we get a
         * result averaged over all possible values whether common or
         * uncommon.  (Essentially, we are assuming that the not-yet-known
         * comparison value is equally likely to be any of the possible
         * values, regardless of their frequency in the table.  Is that a good
         * idea?)
         */
        selec = 1.0 - nullfrac;
        ndistinct = get_variable_numdistinct(vardata, &isdefault);
        if (ndistinct > 1)
            selec /= ndistinct;

        /*
         * Cross-check: selectivity should never be estimated as more than the
         * most common value's.
         */
        if (get_attstatsslot(&sslot, vardata->statsTuple,
                             STATISTIC_KIND_MCV, InvalidOid,
                             ATTSTATSSLOT_NUMBERS))
        {
            if (sslot.nnumbers > 0 && selec > sslot.numbers[0])
                selec = sslot.numbers[0];
            free_attstatsslot(&sslot);
        }
    }
    else
    {
        /*
         * No ANALYZE stats available, so make a guess using estimated number
         * of distinct values and assuming they are equally common. (The guess
         * is unlikely to be very good, but we do know a few special cases.)
         */
        selec = 1.0 / get_variable_numdistinct(vardata, &isdefault);
    }

    /* now adjust if we wanted <> rather than = */
    if (negate)
        selec = 1.0 - selec - nullfrac;

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(selec);

    return selec;
}

/*
 *      neqsel          - Selectivity of "!=" for any data types.
 *
 * This routine is also used for some operators that are not "!="
 * but have comparable selectivity behavior.  See above comments
 * for eqsel().
 */
Datum
neqsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8((float8) eqsel_internal(fcinfo, true));
}

/*
 *  scalarineqsel       - Selectivity of "<", "<=", ">", ">=" for scalars.
 *
 * This is the guts of scalarltsel/scalarlesel/scalargtsel/scalargesel.
 * The isgt and iseq flags distinguish which of the four cases apply.
 *
 * The caller has commuted the clause, if necessary, so that we can treat
 * the variable as being on the left.  The caller must also make sure that
 * the other side of the clause is a non-null Const, and dissect that into
 * a value and datatype.  (This definition simplifies some callers that
 * want to estimate against a computed value instead of a Const node.)
 *
 * This routine works for any datatype (or pair of datatypes) known to
 * convert_to_scalar().  If it is applied to some other datatype,
 * it will return an approximate estimate based on assuming that the constant
 * value falls in the middle of the bin identified by binary search.
 */
static double
scalarineqsel(PlannerInfo *root, Oid operator, bool isgt, bool iseq,
              VariableStatData *vardata, Datum constval, Oid consttype)
{
    Form_pg_statistic stats;
    FmgrInfo    opproc;
    double      mcv_selec,
                hist_selec,
                sumcommon;
    double      selec;

    if (!HeapTupleIsValid(vardata->statsTuple))
    {
        /*
         * No stats are available.  Typically this means we have to fall back
         * on the default estimate; but if the variable is CTID then we can
         * make an estimate based on comparing the constant to the table size.
         */
        if (vardata->var && IsA(vardata->var, Var) &&
            ((Var *) vardata->var)->varattno == SelfItemPointerAttributeNumber)
        {
            ItemPointer itemptr;
            double      block;
            double      density;

            /*
             * If the relation's empty, we're going to include all of it.
             * (This is mostly to avoid divide-by-zero below.)
             */
            if (vardata->rel->pages == 0)
                return 1.0;

            itemptr = (ItemPointer) DatumGetPointer(constval);
            block = ItemPointerGetBlockNumberNoCheck(itemptr);

            /*
             * Determine the average number of tuples per page (density).
             *
             * Since the last page will, on average, be only half full, we can
             * estimate it to have half as many tuples as earlier pages.  So
             * give it half the weight of a regular page.
             */
            density = vardata->rel->tuples / (vardata->rel->pages - 0.5);

            /* If target is the last page, use half the density. */
            if (block >= vardata->rel->pages - 1)
                density *= 0.5;

            /*
             * Using the average tuples per page, calculate how far into the
             * page the itemptr is likely to be and adjust block accordingly,
             * by adding that fraction of a whole block (but never more than a
             * whole block, no matter how high the itemptr's offset is).  Here
             * we are ignoring the possibility of dead-tuple line pointers,
             * which is fairly bogus, but we lack the info to do better.
             */
            if (density > 0.0)
            {
                OffsetNumber offset = ItemPointerGetOffsetNumberNoCheck(itemptr);

                block += Min(offset / density, 1.0);
            }

            /*
             * Convert relative block number to selectivity.  Again, the last
             * page has only half weight.
             */
            selec = block / (vardata->rel->pages - 0.5);

            /*
             * The calculation so far gave us a selectivity for the "<=" case.
             * We'll have one less tuple for "<" and one additional tuple for
             * ">=", the latter of which we'll reverse the selectivity for
             * below, so we can simply subtract one tuple for both cases.  The
             * cases that need this adjustment can be identified by iseq being
             * equal to isgt.
             */
            if (iseq == isgt && vardata->rel->tuples >= 1.0)
                selec -= (1.0 / vardata->rel->tuples);

            /* Finally, reverse the selectivity for the ">", ">=" cases. */
            if (isgt)
                selec = 1.0 - selec;

            CLAMP_PROBABILITY(selec);
            return selec;
        }

        /* no stats available, so default result */
        return DEFAULT_INEQ_SEL;
    }
    stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);

    fmgr_info(get_opcode(operator), &opproc);

    /*
     * If we have most-common-values info, add up the fractions of the MCV
     * entries that satisfy MCV OP CONST.  These fractions contribute directly
     * to the result selectivity.  Also add up the total fraction represented
     * by MCV entries.
     */
    mcv_selec = mcv_selectivity(vardata, &opproc, constval, true,
                                &sumcommon);

    /*
     * If there is a histogram, determine which bin the constant falls in, and
     * compute the resulting contribution to selectivity.
     */
    hist_selec = ineq_histogram_selectivity(root, vardata,
                                            &opproc, isgt, iseq,
                                            constval, consttype);

    /*
     * Now merge the results from the MCV and histogram calculations,
     * realizing that the histogram covers only the non-null values that are
     * not listed in MCV.
     */
    selec = 1.0 - stats->stanullfrac - sumcommon;

    if (hist_selec >= 0.0)
        selec *= hist_selec;
    else
    {
        /*
         * If no histogram but there are values not accounted for by MCV,
         * arbitrarily assume half of them will match.
         */
        selec *= 0.5;
    }

    selec += mcv_selec;

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(selec);

    return selec;
}

/*
 *  mcv_selectivity         - Examine the MCV list for selectivity estimates
 *
 * Determine the fraction of the variable's MCV population that satisfies
 * the predicate (VAR OP CONST), or (CONST OP VAR) if !varonleft.  Also
 * compute the fraction of the total column population represented by the MCV
 * list.  This code will work for any boolean-returning predicate operator.
 *
 * The function result is the MCV selectivity, and the fraction of the
 * total population is returned into *sumcommonp.  Zeroes are returned
 * if there is no MCV list.
 */
double
mcv_selectivity(VariableStatData *vardata, FmgrInfo *opproc,
                Datum constval, bool varonleft,
                double *sumcommonp)
{
    double      mcv_selec,
                sumcommon;
    AttStatsSlot sslot;
    int         i;

    mcv_selec = 0.0;
    sumcommon = 0.0;

    if (HeapTupleIsValid(vardata->statsTuple) &&
        statistic_proc_security_check(vardata, opproc->fn_oid) &&
        get_attstatsslot(&sslot, vardata->statsTuple,
                         STATISTIC_KIND_MCV, InvalidOid,
                         ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS))
    {
        for (i = 0; i < sslot.nvalues; i++)
        {
            if (varonleft ?
                DatumGetBool(FunctionCall2Coll(opproc,
                                               sslot.stacoll,
                                               sslot.values[i],
                                               constval)) :
                DatumGetBool(FunctionCall2Coll(opproc,
                                               sslot.stacoll,
                                               constval,
                                               sslot.values[i])))
                mcv_selec += sslot.numbers[i];
            sumcommon += sslot.numbers[i];
        }
        free_attstatsslot(&sslot);
    }

    *sumcommonp = sumcommon;
    return mcv_selec;
}

/*
 *  histogram_selectivity   - Examine the histogram for selectivity estimates
 *
 * Determine the fraction of the variable's histogram entries that satisfy
 * the predicate (VAR OP CONST), or (CONST OP VAR) if !varonleft.
 *
 * This code will work for any boolean-returning predicate operator, whether
 * or not it has anything to do with the histogram sort operator.  We are
 * essentially using the histogram just as a representative sample.  However,
 * small histograms are unlikely to be all that representative, so the caller
 * should be prepared to fall back on some other estimation approach when the
 * histogram is missing or very small.  It may also be prudent to combine this
 * approach with another one when the histogram is small.
 *
 * If the actual histogram size is not at least min_hist_size, we won't bother
 * to do the calculation at all.  Also, if the n_skip parameter is > 0, we
 * ignore the first and last n_skip histogram elements, on the grounds that
 * they are outliers and hence not very representative.  Typical values for
 * these parameters are 10 and 1.
 *
 * The function result is the selectivity, or -1 if there is no histogram
 * or it's smaller than min_hist_size.
 *
 * The output parameter *hist_size receives the actual histogram size,
 * or zero if no histogram.  Callers may use this number to decide how
 * much faith to put in the function result.
 *
 * Note that the result disregards both the most-common-values (if any) and
 * null entries.  The caller is expected to combine this result with
 * statistics for those portions of the column population.  It may also be
 * prudent to clamp the result range, ie, disbelieve exact 0 or 1 outputs.
 */
double
histogram_selectivity(VariableStatData *vardata, FmgrInfo *opproc,
                      Datum constval, bool varonleft,
                      int min_hist_size, int n_skip,
                      int *hist_size)
{
    double      result;
    AttStatsSlot sslot;

    /* check sanity of parameters */
    Assert(n_skip >= 0);
    Assert(min_hist_size > 2 * n_skip);

    if (HeapTupleIsValid(vardata->statsTuple) &&
        statistic_proc_security_check(vardata, opproc->fn_oid) &&
        get_attstatsslot(&sslot, vardata->statsTuple,
                         STATISTIC_KIND_HISTOGRAM, InvalidOid,
                         ATTSTATSSLOT_VALUES))
    {
        *hist_size = sslot.nvalues;
        if (sslot.nvalues >= min_hist_size)
        {
            int         nmatch = 0;
            int         i;

            for (i = n_skip; i < sslot.nvalues - n_skip; i++)
            {
                if (varonleft ?
                    DatumGetBool(FunctionCall2Coll(opproc,
                                                   sslot.stacoll,
                                                   sslot.values[i],
                                                   constval)) :
                    DatumGetBool(FunctionCall2Coll(opproc,
                                                   sslot.stacoll,
                                                   constval,
                                                   sslot.values[i])))
                    nmatch++;
            }
            result = ((double) nmatch) / ((double) (sslot.nvalues - 2 * n_skip));
        }
        else
            result = -1;
        free_attstatsslot(&sslot);
    }
    else
    {
        *hist_size = 0;
        result = -1;
    }

    return result;
}

/*
 *  ineq_histogram_selectivity  - Examine the histogram for scalarineqsel
 *
 * Determine the fraction of the variable's histogram population that
 * satisfies the inequality condition, ie, VAR < (or <=, >, >=) CONST.
 * The isgt and iseq flags distinguish which of the four cases apply.
 *
 * Returns -1 if there is no histogram (valid results will always be >= 0).
 *
 * Note that the result disregards both the most-common-values (if any) and
 * null entries.  The caller is expected to combine this result with
 * statistics for those portions of the column population.
 *
 * This is exported so that some other estimation functions can use it.
 */
double
ineq_histogram_selectivity(PlannerInfo *root,
                           VariableStatData *vardata,
                           FmgrInfo *opproc, bool isgt, bool iseq,
                           Datum constval, Oid consttype)
{
    double      hist_selec;
    AttStatsSlot sslot;

    hist_selec = -1.0;

    /*
     * Someday, ANALYZE might store more than one histogram per rel/att,
     * corresponding to more than one possible sort ordering defined for the
     * column type.  However, to make that work we will need to figure out
     * which staop to search for --- it's not necessarily the one we have at
     * hand!  (For example, we might have a '<=' operator rather than the '<'
     * operator that will appear in staop.)  For now, assume that whatever
     * appears in pg_statistic is sorted the same way our operator sorts, or
     * the reverse way if isgt is true.
     */
    if (HeapTupleIsValid(vardata->statsTuple) &&
        statistic_proc_security_check(vardata, opproc->fn_oid) &&
        get_attstatsslot(&sslot, vardata->statsTuple,
                         STATISTIC_KIND_HISTOGRAM, InvalidOid,
                         ATTSTATSSLOT_VALUES))
    {
        if (sslot.nvalues > 1)
        {
            /*
             * Use binary search to find the desired location, namely the
             * right end of the histogram bin containing the comparison value,
             * which is the leftmost entry for which the comparison operator
             * succeeds (if isgt) or fails (if !isgt).  (If the given operator
             * isn't actually sort-compatible with the histogram, you'll get
             * garbage results ... but probably not any more garbage-y than
             * you would have from the old linear search.)
             *
             * In this loop, we pay no attention to whether the operator iseq
             * or not; that detail will be mopped up below.  (We cannot tell,
             * anyway, whether the operator thinks the values are equal.)
             *
             * If the binary search accesses the first or last histogram
             * entry, we try to replace that endpoint with the true column min
             * or max as found by get_actual_variable_range().  This
             * ameliorates misestimates when the min or max is moving as a
             * result of changes since the last ANALYZE.  Note that this could
             * result in effectively including MCVs into the histogram that
             * weren't there before, but we don't try to correct for that.
             */
            double      histfrac;
            int         lobound = 0;    /* first possible slot to search */
            int         hibound = sslot.nvalues;    /* last+1 slot to search */
            bool        have_end = false;

            /*
             * If there are only two histogram entries, we'll want up-to-date
             * values for both.  (If there are more than two, we need at most
             * one of them to be updated, so we deal with that within the
             * loop.)
             */
            if (sslot.nvalues == 2)
                have_end = get_actual_variable_range(root,
                                                     vardata,
                                                     sslot.staop,
                                                     &sslot.values[0],
                                                     &sslot.values[1]);

            while (lobound < hibound)
            {
                int         probe = (lobound + hibound) / 2;
                bool        ltcmp;

                /*
                 * If we find ourselves about to compare to the first or last
                 * histogram entry, first try to replace it with the actual
                 * current min or max (unless we already did so above).
                 */
                if (probe == 0 && sslot.nvalues > 2)
                    have_end = get_actual_variable_range(root,
                                                         vardata,
                                                         sslot.staop,
                                                         &sslot.values[0],
                                                         NULL);
                else if (probe == sslot.nvalues - 1 && sslot.nvalues > 2)
                    have_end = get_actual_variable_range(root,
                                                         vardata,
                                                         sslot.staop,
                                                         NULL,
                                                         &sslot.values[probe]);

                ltcmp = DatumGetBool(FunctionCall2Coll(opproc,
                                                       sslot.stacoll,
                                                       sslot.values[probe],
                                                       constval));
                if (isgt)
                    ltcmp = !ltcmp;
                if (ltcmp)
                    lobound = probe + 1;
                else
                    hibound = probe;
            }

            if (lobound <= 0)
            {
                /*
                 * Constant is below lower histogram boundary.  More
                 * precisely, we have found that no entry in the histogram
                 * satisfies the inequality clause (if !isgt) or they all do
                 * (if isgt).  We estimate that that's true of the entire
                 * table, so set histfrac to 0.0 (which we'll flip to 1.0
                 * below, if isgt).
                 */
                histfrac = 0.0;
            }
            else if (lobound >= sslot.nvalues)
            {
                /*
                 * Inverse case: constant is above upper histogram boundary.
                 */
                histfrac = 1.0;
            }
            else
            {
                /* We have values[i-1] <= constant <= values[i]. */
                int         i = lobound;
                double      eq_selec = 0;
                double      val,
                            high,
                            low;
                double      binfrac;

                /*
                 * In the cases where we'll need it below, obtain an estimate
                 * of the selectivity of "x = constval".  We use a calculation
                 * similar to what var_eq_const() does for a non-MCV constant,
                 * ie, estimate that all distinct non-MCV values occur equally
                 * often.  But multiplication by "1.0 - sumcommon - nullfrac"
                 * will be done by our caller, so we shouldn't do that here.
                 * Therefore we can't try to clamp the estimate by reference
                 * to the least common MCV; the result would be too small.
                 *
                 * Note: since this is effectively assuming that constval
                 * isn't an MCV, it's logically dubious if constval in fact is
                 * one.  But we have to apply *some* correction for equality,
                 * and anyway we cannot tell if constval is an MCV, since we
                 * don't have a suitable equality operator at hand.
                 */
                if (i == 1 || isgt == iseq)
                {
                    double      otherdistinct;
                    bool        isdefault;
                    AttStatsSlot mcvslot;

                    /* Get estimated number of distinct values */
                    otherdistinct = get_variable_numdistinct(vardata,
                                                             &isdefault);

                    /* Subtract off the number of known MCVs */
                    if (get_attstatsslot(&mcvslot, vardata->statsTuple,
                                         STATISTIC_KIND_MCV, InvalidOid,
                                         ATTSTATSSLOT_NUMBERS))
                    {
                        otherdistinct -= mcvslot.nnumbers;
                        free_attstatsslot(&mcvslot);
                    }

                    /* If result doesn't seem sane, leave eq_selec at 0 */
                    if (otherdistinct > 1)
                        eq_selec = 1.0 / otherdistinct;
                }

                /*
                 * Convert the constant and the two nearest bin boundary
                 * values to a uniform comparison scale, and do a linear
                 * interpolation within this bin.
                 */
                if (convert_to_scalar(constval, consttype, sslot.stacoll,
                                      &val,
                                      sslot.values[i - 1], sslot.values[i],
                                      vardata->vartype,
                                      &low, &high))
                {
                    if (high <= low)
                    {
                        /* cope if bin boundaries appear identical */
                        binfrac = 0.5;
                    }
                    else if (val <= low)
                        binfrac = 0.0;
                    else if (val >= high)
                        binfrac = 1.0;
                    else
                    {
                        binfrac = (val - low) / (high - low);

                        /*
                         * Watch out for the possibility that we got a NaN or
                         * Infinity from the division.  This can happen
                         * despite the previous checks, if for example "low"
                         * is -Infinity.
                         */
                        if (isnan(binfrac) ||
                            binfrac < 0.0 || binfrac > 1.0)
                            binfrac = 0.5;
                    }
                }
                else
                {
                    /*
                     * Ideally we'd produce an error here, on the grounds that
                     * the given operator shouldn't have scalarXXsel
                     * registered as its selectivity func unless we can deal
                     * with its operand types.  But currently, all manner of
                     * stuff is invoking scalarXXsel, so give a default
                     * estimate until that can be fixed.
                     */
                    binfrac = 0.5;
                }

                /*
                 * Now, compute the overall selectivity across the values
                 * represented by the histogram.  We have i-1 full bins and
                 * binfrac partial bin below the constant.
                 */
                histfrac = (double) (i - 1) + binfrac;
                histfrac /= (double) (sslot.nvalues - 1);

                /*
                 * At this point, histfrac is an estimate of the fraction of
                 * the population represented by the histogram that satisfies
                 * "x <= constval".  Somewhat remarkably, this statement is
                 * true regardless of which operator we were doing the probes
                 * with, so long as convert_to_scalar() delivers reasonable
                 * results.  If the probe constant is equal to some histogram
                 * entry, we would have considered the bin to the left of that
                 * entry if probing with "<" or ">=", or the bin to the right
                 * if probing with "<=" or ">"; but binfrac would have come
                 * out as 1.0 in the first case and 0.0 in the second, leading
                 * to the same histfrac in either case.  For probe constants
                 * between histogram entries, we find the same bin and get the
                 * same estimate with any operator.
                 *
                 * The fact that the estimate corresponds to "x <= constval"
                 * and not "x < constval" is because of the way that ANALYZE
                 * constructs the histogram: each entry is, effectively, the
                 * rightmost value in its sample bucket.  So selectivity
                 * values that are exact multiples of 1/(histogram_size-1)
                 * should be understood as estimates including a histogram
                 * entry plus everything to its left.
                 *
                 * However, that breaks down for the first histogram entry,
                 * which necessarily is the leftmost value in its sample
                 * bucket.  That means the first histogram bin is slightly
                 * narrower than the rest, by an amount equal to eq_selec.
                 * Another way to say that is that we want "x <= leftmost" to
                 * be estimated as eq_selec not zero.  So, if we're dealing
                 * with the first bin (i==1), rescale to make that true while
                 * adjusting the rest of that bin linearly.
                 */
                if (i == 1)
                    histfrac += eq_selec * (1.0 - binfrac);

                /*
                 * "x <= constval" is good if we want an estimate for "<=" or
                 * ">", but if we are estimating for "<" or ">=", we now need
                 * to decrease the estimate by eq_selec.
                 */
                if (isgt == iseq)
                    histfrac -= eq_selec;
            }

            /*
             * Now the estimate is finished for "<" and "<=" cases.  If we are
             * estimating for ">" or ">=", flip it.
             */
            hist_selec = isgt ? (1.0 - histfrac) : histfrac;

            /*
             * The histogram boundaries are only approximate to begin with,
             * and may well be out of date anyway.  Therefore, don't believe
             * extremely small or large selectivity estimates --- unless we
             * got actual current endpoint values from the table, in which
             * case just do the usual sanity clamp.  Somewhat arbitrarily, we
             * set the cutoff for other cases at a hundredth of the histogram
             * resolution.
             */
            if (have_end)
                CLAMP_PROBABILITY(hist_selec);
            else
            {
                double      cutoff = 0.01 / (double) (sslot.nvalues - 1);

                if (hist_selec < cutoff)
                    hist_selec = cutoff;
                else if (hist_selec > 1.0 - cutoff)
                    hist_selec = 1.0 - cutoff;
            }
        }

        free_attstatsslot(&sslot);
    }

    return hist_selec;
}

/*
 * Common wrapper function for the selectivity estimators that simply
 * invoke scalarineqsel().
 */
static Datum
scalarineqsel_wrapper(PG_FUNCTION_ARGS, bool isgt, bool iseq)
{
    PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
    Oid         operator = PG_GETARG_OID(1);
    List       *args = (List *) PG_GETARG_POINTER(2);
    int         varRelid = PG_GETARG_INT32(3);
    VariableStatData vardata;
    Node       *other;
    bool        varonleft;
    Datum       constval;
    Oid         consttype;
    double      selec;

    /*
     * If expression is not variable op something or something op variable,
     * then punt and return a default estimate.
     */
    if (!get_restriction_variable(root, args, varRelid,
                                  &vardata, &other, &varonleft))
        PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);

    /*
     * Can't do anything useful if the something is not a constant, either.
     */
    if (!IsA(other, Const))
    {
        ReleaseVariableStats(vardata);
        PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
    }

    /*
     * If the constant is NULL, assume operator is strict and return zero, ie,
     * operator will never return TRUE.
     */
    if (((Const *) other)->constisnull)
    {
        ReleaseVariableStats(vardata);
        PG_RETURN_FLOAT8(0.0);
    }
    constval = ((Const *) other)->constvalue;
    consttype = ((Const *) other)->consttype;

    /*
     * Force the var to be on the left to simplify logic in scalarineqsel.
     */
    if (!varonleft)
    {
        operator = get_commutator(operator);
        if (!operator)
        {
            /* Use default selectivity (should we raise an error instead?) */
            ReleaseVariableStats(vardata);
            PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
        }
        isgt = !isgt;
    }

    /* The rest of the work is done by scalarineqsel(). */
    selec = scalarineqsel(root, operator, isgt, iseq,
                          &vardata, constval, consttype);

    ReleaseVariableStats(vardata);

    PG_RETURN_FLOAT8((float8) selec);
}

/*
 *      scalarltsel     - Selectivity of "<" for scalars.
 */
Datum
scalarltsel(PG_FUNCTION_ARGS)
{
    return scalarineqsel_wrapper(fcinfo, false, false);
}

/*
 *      scalarlesel     - Selectivity of "<=" for scalars.
 */
Datum
scalarlesel(PG_FUNCTION_ARGS)
{
    return scalarineqsel_wrapper(fcinfo, false, true);
}

/*
 *      scalargtsel     - Selectivity of ">" for scalars.
 */
Datum
scalargtsel(PG_FUNCTION_ARGS)
{
    return scalarineqsel_wrapper(fcinfo, true, false);
}

/*
 *      scalargesel     - Selectivity of ">=" for scalars.
 */
Datum
scalargesel(PG_FUNCTION_ARGS)
{
    return scalarineqsel_wrapper(fcinfo, true, true);
}

/*
 *      boolvarsel      - Selectivity of Boolean variable.
 *
 * This can actually be called on any boolean-valued expression.  If it
 * involves only Vars of the specified relation, and if there are statistics
 * about the Var or expression (the latter is possible if it's indexed) then
 * we'll produce a real estimate; otherwise it's just a default.
 */
Selectivity
boolvarsel(PlannerInfo *root, Node *arg, int varRelid)
{
    VariableStatData vardata;
    double      selec;

    examine_variable(root, arg, varRelid, &vardata);
    if (HeapTupleIsValid(vardata.statsTuple))
    {
        /*
         * A boolean variable V is equivalent to the clause V = 't', so we
         * compute the selectivity as if that is what we have.
         */
        selec = var_eq_const(&vardata, BooleanEqualOperator,
                             BoolGetDatum(true), false, true, false);
    }
    else
    {
        /* Otherwise, the default estimate is 0.5 */
        selec = 0.5;
    }
    ReleaseVariableStats(vardata);
    return selec;
}

/*
 *      booltestsel     - Selectivity of BooleanTest Node.
 */
Selectivity
booltestsel(PlannerInfo *root, BoolTestType booltesttype, Node *arg,
            int varRelid, JoinType jointype, SpecialJoinInfo *sjinfo)
{
    VariableStatData vardata;
    double      selec;

    examine_variable(root, arg, varRelid, &vardata);

    if (HeapTupleIsValid(vardata.statsTuple))
    {
        Form_pg_statistic stats;
        double      freq_null;
        AttStatsSlot sslot;

        stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
        freq_null = stats->stanullfrac;

        if (get_attstatsslot(&sslot, vardata.statsTuple,
                             STATISTIC_KIND_MCV, InvalidOid,
                             ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS)
            && sslot.nnumbers > 0)
        {
            double      freq_true;
            double      freq_false;

            /*
             * Get first MCV frequency and derive frequency for true.
             */
            if (DatumGetBool(sslot.values[0]))
                freq_true = sslot.numbers[0];
            else
                freq_true = 1.0 - sslot.numbers[0] - freq_null;

            /*
             * Next derive frequency for false. Then use these as appropriate
             * to derive frequency for each case.
             */
            freq_false = 1.0 - freq_true - freq_null;

            switch (booltesttype)
            {
                case IS_UNKNOWN:
                    /* select only NULL values */
                    selec = freq_null;
                    break;
                case IS_NOT_UNKNOWN:
                    /* select non-NULL values */
                    selec = 1.0 - freq_null;
                    break;
                case IS_TRUE:
                    /* select only TRUE values */
                    selec = freq_true;
                    break;
                case IS_NOT_TRUE:
                    /* select non-TRUE values */
                    selec = 1.0 - freq_true;
                    break;
                case IS_FALSE:
                    /* select only FALSE values */
                    selec = freq_false;
                    break;
                case IS_NOT_FALSE:
                    /* select non-FALSE values */
                    selec = 1.0 - freq_false;
                    break;
                default:
                    elog(ERROR, "unrecognized booltesttype: %d",
                         (int) booltesttype);
                    selec = 0.0;    /* Keep compiler quiet */
                    break;
            }

            free_attstatsslot(&sslot);
        }
        else
        {
            /*
             * No most-common-value info available. Still have null fraction
             * information, so use it for IS [NOT] UNKNOWN. Otherwise adjust
             * for null fraction and assume a 50-50 split of TRUE and FALSE.
             */
            switch (booltesttype)
            {
                case IS_UNKNOWN:
                    /* select only NULL values */
                    selec = freq_null;
                    break;
                case IS_NOT_UNKNOWN:
                    /* select non-NULL values */
                    selec = 1.0 - freq_null;
                    break;
                case IS_TRUE:
                case IS_FALSE:
                    /* Assume we select half of the non-NULL values */
                    selec = (1.0 - freq_null) / 2.0;
                    break;
                case IS_NOT_TRUE:
                case IS_NOT_FALSE:
                    /* Assume we select NULLs plus half of the non-NULLs */
                    /* equiv. to freq_null + (1.0 - freq_null) / 2.0 */
                    selec = (freq_null + 1.0) / 2.0;
                    break;
                default:
                    elog(ERROR, "unrecognized booltesttype: %d",
                         (int) booltesttype);
                    selec = 0.0;    /* Keep compiler quiet */
                    break;
            }
        }
    }
    else
    {
        /*
         * If we can't get variable statistics for the argument, perhaps
         * clause_selectivity can do something with it.  We ignore the
         * possibility of a NULL value when using clause_selectivity, and just
         * assume the value is either TRUE or FALSE.
         */
        switch (booltesttype)
        {
            case IS_UNKNOWN:
                selec = DEFAULT_UNK_SEL;
                break;
            case IS_NOT_UNKNOWN:
                selec = DEFAULT_NOT_UNK_SEL;
                break;
            case IS_TRUE:
            case IS_NOT_FALSE:
                selec = (double) clause_selectivity(root, arg,
                                                    varRelid,
                                                    jointype, sjinfo);
                break;
            case IS_FALSE:
            case IS_NOT_TRUE:
                selec = 1.0 - (double) clause_selectivity(root, arg,
                                                          varRelid,
                                                          jointype, sjinfo);
                break;
            default:
                elog(ERROR, "unrecognized booltesttype: %d",
                     (int) booltesttype);
                selec = 0.0;    /* Keep compiler quiet */
                break;
        }
    }

    ReleaseVariableStats(vardata);

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(selec);

    return (Selectivity) selec;
}

/*
 *      nulltestsel     - Selectivity of NullTest Node.
 */
Selectivity
nulltestsel(PlannerInfo *root, NullTestType nulltesttype, Node *arg,
            int varRelid, JoinType jointype, SpecialJoinInfo *sjinfo)
{
    VariableStatData vardata;
    double      selec;

    examine_variable(root, arg, varRelid, &vardata);

    if (HeapTupleIsValid(vardata.statsTuple))
    {
        Form_pg_statistic stats;
        double      freq_null;

        stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
        freq_null = stats->stanullfrac;

        switch (nulltesttype)
        {
            case IS_NULL:

                /*
                 * Use freq_null directly.
                 */
                selec = freq_null;
                break;
            case IS_NOT_NULL:

                /*
                 * Select not unknown (not null) values. Calculate from
                 * freq_null.
                 */
                selec = 1.0 - freq_null;
                break;
            default:
                elog(ERROR, "unrecognized nulltesttype: %d",
                     (int) nulltesttype);
                return (Selectivity) 0; /* keep compiler quiet */
        }
    }
    else if (vardata.var && IsA(vardata.var, Var) &&
             ((Var *) vardata.var)->varattno < 0)
    {
        /*
         * There are no stats for system columns, but we know they are never
         * NULL.
         */
        selec = (nulltesttype == IS_NULL) ? 0.0 : 1.0;
    }
    else
    {
        /*
         * No ANALYZE stats available, so make a guess
         */
        switch (nulltesttype)
        {
            case IS_NULL:
                selec = DEFAULT_UNK_SEL;
                break;
            case IS_NOT_NULL:
                selec = DEFAULT_NOT_UNK_SEL;
                break;
            default:
                elog(ERROR, "unrecognized nulltesttype: %d",
                     (int) nulltesttype);
                return (Selectivity) 0; /* keep compiler quiet */
        }
    }

    ReleaseVariableStats(vardata);

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(selec);

    return (Selectivity) selec;
}

/*
 * strip_array_coercion - strip binary-compatible relabeling from an array expr
 *
 * For array values, the parser normally generates ArrayCoerceExpr conversions,
 * but it seems possible that RelabelType might show up.  Also, the planner
 * is not currently tense about collapsing stacked ArrayCoerceExpr nodes,
 * so we need to be ready to deal with more than one level.
 */
static Node *
strip_array_coercion(Node *node)
{
    for (;;)
    {
        if (node && IsA(node, ArrayCoerceExpr))
        {
            ArrayCoerceExpr *acoerce = (ArrayCoerceExpr *) node;

            /*
             * If the per-element expression is just a RelabelType on top of
             * CaseTestExpr, then we know it's a binary-compatible relabeling.
             */
            if (IsA(acoerce->elemexpr, RelabelType) &&
                IsA(((RelabelType *) acoerce->elemexpr)->arg, CaseTestExpr))
                node = (Node *) acoerce->arg;
            else
                break;
        }
        else if (node && IsA(node, RelabelType))
        {
            /* We don't really expect this case, but may as well cope */
            node = (Node *) ((RelabelType *) node)->arg;
        }
        else
            break;
    }
    return node;
}

/*
 *      scalararraysel      - Selectivity of ScalarArrayOpExpr Node.
 */
Selectivity
scalararraysel(PlannerInfo *root,
               ScalarArrayOpExpr *clause,
               bool is_join_clause,
               int varRelid,
               JoinType jointype,
               SpecialJoinInfo *sjinfo)
{
    Oid         operator = clause->opno;
    bool        useOr = clause->useOr;
    bool        isEquality = false;
    bool        isInequality = false;
    Node       *leftop;
    Node       *rightop;
    Oid         nominal_element_type;
    Oid         nominal_element_collation;
    TypeCacheEntry *typentry;
    RegProcedure oprsel;
    FmgrInfo    oprselproc;
    Selectivity s1;
    Selectivity s1disjoint;

    /* First, deconstruct the expression */
    Assert(list_length(clause->args) == 2);
    leftop = (Node *) linitial(clause->args);
    rightop = (Node *) lsecond(clause->args);

    /* aggressively reduce both sides to constants */
    leftop = estimate_expression_value(root, leftop);
    rightop = estimate_expression_value(root, rightop);

    /* get nominal (after relabeling) element type of rightop */
    nominal_element_type = get_base_element_type(exprType(rightop));
    if (!OidIsValid(nominal_element_type))
        return (Selectivity) 0.5;   /* probably shouldn't happen */
    /* get nominal collation, too, for generating constants */
    nominal_element_collation = exprCollation(rightop);

    /* look through any binary-compatible relabeling of rightop */
    rightop = strip_array_coercion(rightop);

    /*
     * Detect whether the operator is the default equality or inequality
     * operator of the array element type.
     */
    typentry = lookup_type_cache(nominal_element_type, TYPECACHE_EQ_OPR);
    if (OidIsValid(typentry->eq_opr))
    {
        if (operator == typentry->eq_opr)
            isEquality = true;
        else if (get_negator(operator) == typentry->eq_opr)
            isInequality = true;
    }

    /*
     * If it is equality or inequality, we might be able to estimate this as a
     * form of array containment; for instance "const = ANY(column)" can be
     * treated as "ARRAY[const] <@ column".  scalararraysel_containment tries
     * that, and returns the selectivity estimate if successful, or -1 if not.
     */
    if ((isEquality || isInequality) && !is_join_clause)
    {
        s1 = scalararraysel_containment(root, leftop, rightop,
                                        nominal_element_type,
                                        isEquality, useOr, varRelid);
        if (s1 >= 0.0)
            return s1;
    }

    /*
     * Look up the underlying operator's selectivity estimator. Punt if it
     * hasn't got one.
     */
    if (is_join_clause)
        oprsel = get_oprjoin(operator);
    else
        oprsel = get_oprrest(operator);
    if (!oprsel)
        return (Selectivity) 0.5;
    fmgr_info(oprsel, &oprselproc);

    /*
     * In the array-containment check above, we must only believe that an
     * operator is equality or inequality if it is the default btree equality
     * operator (or its negator) for the element type, since those are the
     * operators that array containment will use.  But in what follows, we can
     * be a little laxer, and also believe that any operators using eqsel() or
     * neqsel() as selectivity estimator act like equality or inequality.
     */
    if (oprsel == F_EQSEL || oprsel == F_EQJOINSEL)
        isEquality = true;
    else if (oprsel == F_NEQSEL || oprsel == F_NEQJOINSEL)
        isInequality = true;

    /*
     * We consider three cases:
     *
     * 1. rightop is an Array constant: deconstruct the array, apply the
     * operator's selectivity function for each array element, and merge the
     * results in the same way that clausesel.c does for AND/OR combinations.
     *
     * 2. rightop is an ARRAY[] construct: apply the operator's selectivity
     * function for each element of the ARRAY[] construct, and merge.
     *
     * 3. otherwise, make a guess ...
     */
    if (rightop && IsA(rightop, Const))
    {
        Datum       arraydatum = ((Const *) rightop)->constvalue;
        bool        arrayisnull = ((Const *) rightop)->constisnull;
        ArrayType  *arrayval;
        int16       elmlen;
        bool        elmbyval;
        char        elmalign;
        int         num_elems;
        Datum      *elem_values;
        bool       *elem_nulls;
        int         i;

        if (arrayisnull)        /* qual can't succeed if null array */
            return (Selectivity) 0.0;
        arrayval = DatumGetArrayTypeP(arraydatum);
        get_typlenbyvalalign(ARR_ELEMTYPE(arrayval),
                             &elmlen, &elmbyval, &elmalign);
        deconstruct_array(arrayval,
                          ARR_ELEMTYPE(arrayval),
                          elmlen, elmbyval, elmalign,
                          &elem_values, &elem_nulls, &num_elems);

        /*
         * For generic operators, we assume the probability of success is
         * independent for each array element.  But for "= ANY" or "<> ALL",
         * if the array elements are distinct (which'd typically be the case)
         * then the probabilities are disjoint, and we should just sum them.
         *
         * If we were being really tense we would try to confirm that the
         * elements are all distinct, but that would be expensive and it
         * doesn't seem to be worth the cycles; it would amount to penalizing
         * well-written queries in favor of poorly-written ones.  However, we
         * do protect ourselves a little bit by checking whether the
         * disjointness assumption leads to an impossible (out of range)
         * probability; if so, we fall back to the normal calculation.
         */
        s1 = s1disjoint = (useOr ? 0.0 : 1.0);

        for (i = 0; i < num_elems; i++)
        {
            List       *args;
            Selectivity s2;

            args = list_make2(leftop,
                              makeConst(nominal_element_type,
                                        -1,
                                        nominal_element_collation,
                                        elmlen,
                                        elem_values[i],
                                        elem_nulls[i],
                                        elmbyval));
            if (is_join_clause)
                s2 = DatumGetFloat8(FunctionCall5Coll(&oprselproc,
                                                      clause->inputcollid,
                                                      PointerGetDatum(root),
                                                      ObjectIdGetDatum(operator),
                                                      PointerGetDatum(args),
                                                      Int16GetDatum(jointype),
                                                      PointerGetDatum(sjinfo)));
            else
                s2 = DatumGetFloat8(FunctionCall4Coll(&oprselproc,
                                                      clause->inputcollid,
                                                      PointerGetDatum(root),
                                                      ObjectIdGetDatum(operator),
                                                      PointerGetDatum(args),
                                                      Int32GetDatum(varRelid)));

            if (useOr)
            {
                s1 = s1 + s2 - s1 * s2;
                if (isEquality)
                    s1disjoint += s2;
            }
            else
            {
                s1 = s1 * s2;
                if (isInequality)
                    s1disjoint += s2 - 1.0;
            }
        }

        /* accept disjoint-probability estimate if in range */
        if ((useOr ? isEquality : isInequality) &&
            s1disjoint >= 0.0 && s1disjoint <= 1.0)
            s1 = s1disjoint;
    }
    else if (rightop && IsA(rightop, ArrayExpr) &&
             !((ArrayExpr *) rightop)->multidims)
    {
        ArrayExpr  *arrayexpr = (ArrayExpr *) rightop;
        int16       elmlen;
        bool        elmbyval;
        ListCell   *l;

        get_typlenbyval(arrayexpr->element_typeid,
                        &elmlen, &elmbyval);

        /*
         * We use the assumption of disjoint probabilities here too, although
         * the odds of equal array elements are rather higher if the elements
         * are not all constants (which they won't be, else constant folding
         * would have reduced the ArrayExpr to a Const).  In this path it's
         * critical to have the sanity check on the s1disjoint estimate.
         */
        s1 = s1disjoint = (useOr ? 0.0 : 1.0);

        foreach(l, arrayexpr->elements)
        {
            Node       *elem = (Node *) lfirst(l);
            List       *args;
            Selectivity s2;

            /*
             * Theoretically, if elem isn't of nominal_element_type we should
             * insert a RelabelType, but it seems unlikely that any operator
             * estimation function would really care ...
             */
            args = list_make2(leftop, elem);
            if (is_join_clause)
                s2 = DatumGetFloat8(FunctionCall5Coll(&oprselproc,
                                                      clause->inputcollid,
                                                      PointerGetDatum(root),
                                                      ObjectIdGetDatum(operator),
                                                      PointerGetDatum(args),
                                                      Int16GetDatum(jointype),
                                                      PointerGetDatum(sjinfo)));
            else
                s2 = DatumGetFloat8(FunctionCall4Coll(&oprselproc,
                                                      clause->inputcollid,
                                                      PointerGetDatum(root),
                                                      ObjectIdGetDatum(operator),
                                                      PointerGetDatum(args),
                                                      Int32GetDatum(varRelid)));

            if (useOr)
            {
                s1 = s1 + s2 - s1 * s2;
                if (isEquality)
                    s1disjoint += s2;
            }
            else
            {
                s1 = s1 * s2;
                if (isInequality)
                    s1disjoint += s2 - 1.0;
            }
        }

        /* accept disjoint-probability estimate if in range */
        if ((useOr ? isEquality : isInequality) &&
            s1disjoint >= 0.0 && s1disjoint <= 1.0)
            s1 = s1disjoint;
    }
    else
    {
        CaseTestExpr *dummyexpr;
        List       *args;
        Selectivity s2;
        int         i;

        /*
         * We need a dummy rightop to pass to the operator selectivity
         * routine.  It can be pretty much anything that doesn't look like a
         * constant; CaseTestExpr is a convenient choice.
         */
        dummyexpr = makeNode(CaseTestExpr);
        dummyexpr->typeId = nominal_element_type;
        dummyexpr->typeMod = -1;
        dummyexpr->collation = clause->inputcollid;
        args = list_make2(leftop, dummyexpr);
        if (is_join_clause)
            s2 = DatumGetFloat8(FunctionCall5Coll(&oprselproc,
                                                  clause->inputcollid,
                                                  PointerGetDatum(root),
                                                  ObjectIdGetDatum(operator),
                                                  PointerGetDatum(args),
                                                  Int16GetDatum(jointype),
                                                  PointerGetDatum(sjinfo)));
        else
            s2 = DatumGetFloat8(FunctionCall4Coll(&oprselproc,
                                                  clause->inputcollid,
                                                  PointerGetDatum(root),
                                                  ObjectIdGetDatum(operator),
                                                  PointerGetDatum(args),
                                                  Int32GetDatum(varRelid)));
        s1 = useOr ? 0.0 : 1.0;

        /*
         * Arbitrarily assume 10 elements in the eventual array value (see
         * also estimate_array_length).  We don't risk an assumption of
         * disjoint probabilities here.
         */
        for (i = 0; i < 10; i++)
        {
            if (useOr)
                s1 = s1 + s2 - s1 * s2;
            else
                s1 = s1 * s2;
        }
    }

    /* result should be in range, but make sure... */
    CLAMP_PROBABILITY(s1);

    return s1;
}

/*
 * Estimate number of elements in the array yielded by an expression.
 *
 * It's important that this agree with scalararraysel.
 */
int
estimate_array_length(Node *arrayexpr)
{
    /* look through any binary-compatible relabeling of arrayexpr */
    arrayexpr = strip_array_coercion(arrayexpr);

    if (arrayexpr && IsA(arrayexpr, Const))
    {
        Datum       arraydatum = ((Const *) arrayexpr)->constvalue;
        bool        arrayisnull = ((Const *) arrayexpr)->constisnull;
        ArrayType  *arrayval;

        if (arrayisnull)
            return 0;
        arrayval = DatumGetArrayTypeP(arraydatum);
        return ArrayGetNItems(ARR_NDIM(arrayval), ARR_DIMS(arrayval));
    }
    else if (arrayexpr && IsA(arrayexpr, ArrayExpr) &&
             !((ArrayExpr *) arrayexpr)->multidims)
    {
        return list_length(((ArrayExpr *) arrayexpr)->elements);
    }
    else
    {
        /* default guess --- see also scalararraysel */
        return 10;
    }
}

/*
 *      rowcomparesel       - Selectivity of RowCompareExpr Node.
 *
 * We estimate RowCompare selectivity by considering just the first (high
 * order) columns, which makes it equivalent to an ordinary OpExpr.  While
 * this estimate could be refined by considering additional columns, it
 * seems unlikely that we could do a lot better without multi-column
 * statistics.
 */
Selectivity
rowcomparesel(PlannerInfo *root,
              RowCompareExpr *clause,
              int varRelid, JoinType jointype, SpecialJoinInfo *sjinfo)
{
    Selectivity s1;
    Oid         opno = linitial_oid(clause->opnos);
    Oid         inputcollid = linitial_oid(clause->inputcollids);
    List       *opargs;
    bool        is_join_clause;

    /* Build equivalent arg list for single operator */
    opargs = list_make2(linitial(clause->largs), linitial(clause->rargs));

    /*
     * Decide if it's a join clause.  This should match clausesel.c's
     * treat_as_join_clause(), except that we intentionally consider only the
     * leading columns and not the rest of the clause.
     */
    if (varRelid != 0)
    {
        /*
         * Caller is forcing restriction mode (eg, because we are examining an
         * inner indexscan qual).
         */
        is_join_clause = false;
    }
    else if (sjinfo == NULL)
    {
        /*
         * It must be a restriction clause, since it's being evaluated at a
         * scan node.
         */
        is_join_clause = false;
    }
    else
    {
        /*
         * Otherwise, it's a join if there's more than one relation used.
         */
        is_join_clause = (NumRelids((Node *) opargs) > 1);
    }

    if (is_join_clause)
    {
        /* Estimate selectivity for a join clause. */
        s1 = join_selectivity(root, opno,
                              opargs,
                              inputcollid,
                              jointype,
                              sjinfo);
    }
    else
    {
        /* Estimate selectivity for a restriction clause. */
        s1 = restriction_selectivity(root, opno,
                                     opargs,
                                     inputcollid,
                                     varRelid);
    }

    return s1;
}

/*
 *      eqjoinsel       - Join selectivity of "="
 */
Datum
eqjoinsel(PG_FUNCTION_ARGS)
{
    PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
    Oid         operator = PG_GETARG_OID(1);
    List       *args = (List *) PG_GETARG_POINTER(2);

#ifdef NOT_USED
    JoinType    jointype = (JoinType) PG_GETARG_INT16(3);
#endif
    SpecialJoinInfo *sjinfo = (SpecialJoinInfo *) PG_GETARG_POINTER(4);
    double      selec;
    double      selec_inner;
    VariableStatData vardata1;
    VariableStatData vardata2;
    double      nd1;
    double      nd2;
    bool        isdefault1;
    bool        isdefault2;
    Oid         opfuncoid;
    AttStatsSlot sslot1;
    AttStatsSlot sslot2;
    Form_pg_statistic stats1 = NULL;
    Form_pg_statistic stats2 = NULL;
    bool        have_mcvs1 = false;
    bool        have_mcvs2 = false;
    bool        join_is_reversed;
    RelOptInfo *inner_rel;

    get_join_variables(root, args, sjinfo,
                       &vardata1, &vardata2, &join_is_reversed);

    nd1 = get_variable_numdistinct(&vardata1, &isdefault1);
    nd2 = get_variable_numdistinct(&vardata2, &isdefault2);

    opfuncoid = get_opcode(operator);

    memset(&sslot1, 0, sizeof(sslot1));
    memset(&sslot2, 0, sizeof(sslot2));

    if (HeapTupleIsValid(vardata1.statsTuple))
    {
        /* note we allow use of nullfrac regardless of security check */
        stats1 = (Form_pg_statistic) GETSTRUCT(vardata1.statsTuple);
        if (statistic_proc_security_check(&vardata1, opfuncoid))
            have_mcvs1 = get_attstatsslot(&sslot1, vardata1.statsTuple,
                                          STATISTIC_KIND_MCV, InvalidOid,
                                          ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS);
    }

    if (HeapTupleIsValid(vardata2.statsTuple))
    {
        /* note we allow use of nullfrac regardless of security check */
        stats2 = (Form_pg_statistic) GETSTRUCT(vardata2.statsTuple);
        if (statistic_proc_security_check(&vardata2, opfuncoid))
            have_mcvs2 = get_attstatsslot(&sslot2, vardata2.statsTuple,
                                          STATISTIC_KIND_MCV, InvalidOid,
                                          ATTSTATSSLOT_VALUES | ATTSTATSSLOT_NUMBERS);
    }

    /* We need to compute the inner-join selectivity in all cases */
    selec_inner = eqjoinsel_inner(opfuncoid,
                                  &vardata1, &vardata2,
                                  nd1, nd2,
                                  isdefault1, isdefault2,
                                  &sslot1, &sslot2,
                                  stats1, stats2,
                                  have_mcvs1, have_mcvs2);

    switch (sjinfo->jointype)
    {
        case JOIN_INNER:
        case JOIN_LEFT:
        case JOIN_FULL:
            selec = selec_inner;
            break;
        case JOIN_SEMI:
        case JOIN_ANTI:

            /*
             * Look up the join's inner relation.  min_righthand is sufficient
             * information because neither SEMI nor ANTI joins permit any
             * reassociation into or out of their RHS, so the righthand will
             * always be exactly that set of rels.
             */
            inner_rel = find_join_input_rel(root, sjinfo->min_righthand);

            if (!join_is_reversed)
                selec = eqjoinsel_semi(opfuncoid,
                                       &vardata1, &vardata2,
                                       nd1, nd2,
                                       isdefault1, isdefault2,
                                       &sslot1, &sslot2,
                                       stats1, stats2,
                                       have_mcvs1, have_mcvs2,
                                       inner_rel);
            else
            {
                Oid         commop = get_commutator(operator);
                Oid         commopfuncoid = OidIsValid(commop) ? get_opcode(commop) : InvalidOid;

                selec = eqjoinsel_semi(commopfuncoid,
                                       &vardata2, &vardata1,
                                       nd2, nd1,
                                       isdefault2, isdefault1,
                                       &sslot2, &sslot1,
                                       stats2, stats1,
                                       have_mcvs2, have_mcvs1,
                                       inner_rel);
            }

            /*
             * We should never estimate the output of a semijoin to be more
             * rows than we estimate for an inner join with the same input
             * rels and join condition; it's obviously impossible for that to
             * happen.  The former estimate is N1 * Ssemi while the latter is
             * N1 * N2 * Sinner, so we may clamp Ssemi <= N2 * Sinner.  Doing
             * this is worthwhile because of the shakier estimation rules we
             * use in eqjoinsel_semi, particularly in cases where it has to
             * punt entirely.
             */
            selec = Min(selec, inner_rel->rows * selec_inner);
            break;
        default:
            /* other values not expected here */
            elog(ERROR, "unrecognized join type: %d",
                 (int) sjinfo->jointype);
            selec = 0;          /* keep compiler quiet */
            break;
    }

    free_attstatsslot(&sslot1);
    free_attstatsslot(&sslot2);

    ReleaseVariableStats(vardata1);
    ReleaseVariableStats(vardata2);

    CLAMP_PROBABILITY(selec);

    PG_RETURN_FLOAT8((float8) selec);
}

/*
 * eqjoinsel_inner --- eqjoinsel for normal inner join
 *
 * We also use this for LEFT/FULL outer joins; it's not presently clear
 * that it's worth trying to distinguish them here.
 */
static double
eqjoinsel_inner(Oid opfuncoid,
                VariableStatData *vardata1, VariableStatData *vardata2,
                double nd1, double nd2,
                bool isdefault1, bool isdefault2,
                AttStatsSlot *sslot1, AttStatsSlot *sslot2,
                Form_pg_statistic stats1, Form_pg_statistic stats2,
                bool have_mcvs1, bool have_mcvs2)
{
    double      selec;

    if (have_mcvs1 && have_mcvs2)
    {
        /*
         * We have most-common-value lists for both relations.  Run through
         * the lists to see which MCVs actually join to each other with the
         * given operator.  This allows us to determine the exact join
         * selectivity for the portion of the relations represented by the MCV
         * lists.  We still have to estimate for the remaining population, but
         * in a skewed distribution this gives us a big leg up in accuracy.
         * For motivation see the analysis in Y. Ioannidis and S.
         * Christodoulakis, "On the propagation of errors in the size of join
         * results", Technical Report 1018, Computer Science Dept., University
         * of Wisconsin, Madison, March 1991 (available from ftp.cs.wisc.edu).
         */
        FmgrInfo    eqproc;
        bool       *hasmatch1;
        bool       *hasmatch2;
        double      nullfrac1 = stats1->stanullfrac;
        double      nullfrac2 = stats2->stanullfrac;
        double      matchprodfreq,
                    matchfreq1,
                    matchfreq2,
                    unmatchfreq1,
                    unmatchfreq2,
                    otherfreq1,
                    otherfreq2,
                    totalsel1,
                    totalsel2;
        int         i,
                    nmatches;

        fmgr_info(opfuncoid, &eqproc);
        hasmatch1 = (bool *) palloc0(sslot1->nvalues * sizeof(bool));
        hasmatch2 = (bool *) palloc0(sslot2->nvalues * sizeof(bool));

        /*
         * Note we assume that each MCV will match at most one member of the
         * other MCV list.  If the operator isn't really equality, there could
         * be multiple matches --- but we don't look for them, both for speed
         * and because the math wouldn't add up...
         */
        matchprodfreq = 0.0;
        nmatches = 0;
        for (i = 0; i < sslot1->nvalues; i++)
        {
            int         j;

            for (j = 0; j < sslot2->nvalues; j++)
            {
                if (hasmatch2[j])
                    continue;
                if (DatumGetBool(FunctionCall2Coll(&eqproc,
                                                   sslot1->stacoll,
                                                   sslot1->values[i],
                                                   sslot2->values[j])))
                {
                    hasmatch1[i] = hasmatch2[j] = true;
                    matchprodfreq += sslot1->numbers[i] * sslot2->numbers[j];
                    nmatches++;
                    break;
                }
            }
        }
        CLAMP_PROBABILITY(matchprodfreq);
        /* Sum up frequencies of matched and unmatched MCVs */
        matchfreq1 = unmatchfreq1 = 0.0;
        for (i = 0; i < sslot1->nvalues; i++)
        {
            if (hasmatch1[i])
                matchfreq1 += sslot1->numbers[i];
            else
                unmatchfreq1 += sslot1->numbers[i];
        }
        CLAMP_PROBABILITY(matchfreq1);
        CLAMP_PROBABILITY(unmatchfreq1);
        matchfreq2 = unmatchfreq2 = 0.0;
        for (i = 0; i < sslot2->nvalues; i++)
        {
            if (hasmatch2[i])
                matchfreq2 += sslot2->numbers[i];
            else
                unmatchfreq2 += sslot2->numbers[i];
        }
        CLAMP_PROBABILITY(matchfreq2);
        CLAMP_PROBABILITY(unmatchfreq2);
        pfree(hasmatch1);
        pfree(hasmatch2);

        /*
         * Compute total frequency of non-null values that are not in the MCV
         * lists.
         */
        otherfreq1 = 1.0 - nullfrac1 - matchfreq1 - unmatchfreq1;
        otherfreq2 = 1.0 - nullfrac2 - matchfreq2 - unmatchfreq2;
        CLAMP_PROBABILITY(otherfreq1);
        CLAMP_PROBABILITY(otherfreq2);

        /*
         * We can estimate the total selectivity from the point of view of
         * relation 1 as: the known selectivity for matched MCVs, plus
         * unmatched MCVs that are assumed to match against random members of
         * relation 2's non-MCV population, plus non-MCV values that are
         * assumed to match against random members of relation 2's unmatched
         * MCVs plus non-MCV values.
         */
        totalsel1 = matchprodfreq;
        if (nd2 > sslot2->nvalues)
            totalsel1 += unmatchfreq1 * otherfreq2 / (nd2 - sslot2->nvalues);
        if (nd2 > nmatches)
            totalsel1 += otherfreq1 * (otherfreq2 + unmatchfreq2) /
                (nd2 - nmatches);
        /* Same estimate from the point of view of relation 2. */
        totalsel2 = matchprodfreq;
        if (nd1 > sslot1->nvalues)
            totalsel2 += unmatchfreq2 * otherfreq1 / (nd1 - sslot1->nvalues);
        if (nd1 > nmatches)
            totalsel2 += otherfreq2 * (otherfreq1 + unmatchfreq1) /
                (nd1 - nmatches);

        /*
         * Use the smaller of the two estimates.  This can be justified in
         * essentially the same terms as given below for the no-stats case: to
         * a first approximation, we are estimating from the point of view of
         * the relation with smaller nd.
         */
        selec = (totalsel1 < totalsel2) ? totalsel1 : totalsel2;
    }
    else
    {
        /*
         * We do not have MCV lists for both sides.  Estimate the join
         * selectivity as MIN(1/nd1,1/nd2)*(1-nullfrac1)*(1-nullfrac2). This
         * is plausible if we assume that the join operator is strict and the
         * non-null values are about equally distributed: a given non-null
         * tuple of rel1 will join to either zero or N2*(1-nullfrac2)/nd2 rows
         * of rel2, so total join rows are at most
         * N1*(1-nullfrac1)*N2*(1-nullfrac2)/nd2 giving a join selectivity of
         * not more than (1-nullfrac1)*(1-nullfrac2)/nd2. By the same logic it
         * is not more than (1-nullfrac1)*(1-nullfrac2)/nd1, so the expression
         * with MIN() is an upper bound.  Using the MIN() means we estimate
         * from the point of view of the relation with smaller nd (since the
         * larger nd is determining the MIN).  It is reasonable to assume that
         * most tuples in this rel will have join partners, so the bound is
         * probably reasonably tight and should be taken as-is.
         *
         * XXX Can we be smarter if we have an MCV list for just one side? It
         * seems that if we assume equal distribution for the other side, we
         * end up with the same answer anyway.
         */
        double      nullfrac1 = stats1 ? stats1->stanullfrac : 0.0;
        double      nullfrac2 = stats2 ? stats2->stanullfrac : 0.0;

        selec = (1.0 - nullfrac1) * (1.0 - nullfrac2);
        if (nd1 > nd2)
            selec /= nd1;
        else
            selec /= nd2;
    }

    return selec;
}

/*
 * eqjoinsel_semi --- eqjoinsel for semi join
 *
 * (Also used for anti join, which we are supposed to estimate the same way.)
 * Caller has ensured that vardata1 is the LHS variable.
 * Unlike eqjoinsel_inner, we have to cope with opfuncoid being InvalidOid.
 */
static double
eqjoinsel_semi(Oid opfuncoid,
               VariableStatData *vardata1, VariableStatData *vardata2,
               double nd1, double nd2,
               bool isdefault1, bool isdefault2,
               AttStatsSlot *sslot1, AttStatsSlot *sslot2,
               Form_pg_statistic stats1, Form_pg_statistic stats2,
               bool have_mcvs1, bool have_mcvs2,
               RelOptInfo *inner_rel)
{
    double      selec;

    /*
     * We clamp nd2 to be not more than what we estimate the inner relation's
     * size to be.  This is intuitively somewhat reasonable since obviously
     * there can't be more than that many distinct values coming from the
     * inner rel.  The reason for the asymmetry (ie, that we don't clamp nd1
     * likewise) is that this is the only pathway by which restriction clauses
     * applied to the inner rel will affect the join result size estimate,
     * since set_joinrel_size_estimates will multiply SEMI/ANTI selectivity by
     * only the outer rel's size.  If we clamped nd1 we'd be double-counting
     * the selectivity of outer-rel restrictions.
     *
     * We can apply this clamping both with respect to the base relation from
     * which the join variable comes (if there is just one), and to the
     * immediate inner input relation of the current join.
     *
     * If we clamp, we can treat nd2 as being a non-default estimate; it's not
     * great, maybe, but it didn't come out of nowhere either.  This is most
     * helpful when the inner relation is empty and consequently has no stats.
     */
    if (vardata2->rel)
    {
        if (nd2 >= vardata2->rel->rows)
        {
            nd2 = vardata2->rel->rows;
            isdefault2 = false;
        }
    }
    if (nd2 >= inner_rel->rows)
    {
        nd2 = inner_rel->rows;
        isdefault2 = false;
    }

    if (have_mcvs1 && have_mcvs2 && OidIsValid(opfuncoid))
    {
        /*
         * We have most-common-value lists for both relations.  Run through
         * the lists to see which MCVs actually join to each other with the
         * given operator.  This allows us to determine the exact join
         * selectivity for the portion of the relations represented by the MCV
         * lists.  We still have to estimate for the remaining population, but
         * in a skewed distribution this gives us a big leg up in accuracy.
         */
        FmgrInfo    eqproc;
        bool       *hasmatch1;
        bool       *hasmatch2;
        double      nullfrac1 = stats1->stanullfrac;
        double      matchfreq1,
                    uncertainfrac,
                    uncertain;
        int         i,
                    nmatches,
                    clamped_nvalues2;

        /*
         * The clamping above could have resulted in nd2 being less than
         * sslot2->nvalues; in which case, we assume that precisely the nd2
         * most common values in the relation will appear in the join input,
         * and so compare to only the first nd2 members of the MCV list.  Of
         * course this is frequently wrong, but it's the best bet we can make.
         */
        clamped_nvalues2 = Min(sslot2->nvalues, nd2);

        fmgr_info(opfuncoid, &eqproc);
        hasmatch1 = (bool *) palloc0(sslot1->nvalues * sizeof(bool));
        hasmatch2 = (bool *) palloc0(clamped_nvalues2 * sizeof(bool));

        /*
         * Note we assume that each MCV will match at most one member of the
         * other MCV list.  If the operator isn't really equality, there could
         * be multiple matches --- but we don't look for them, both for speed
         * and because the math wouldn't add up...
         */
        nmatches = 0;
        for (i = 0; i < sslot1->nvalues; i++)
        {
            int         j;

            for (j = 0; j < clamped_nvalues2; j++)
            {
                if (hasmatch2[j])
                    continue;
                if (DatumGetBool(FunctionCall2Coll(&eqproc,
                                                   sslot1->stacoll,
                                                   sslot1->values[i],
                                                   sslot2->values[j])))
                {
                    hasmatch1[i] = hasmatch2[j] = true;
                    nmatches++;
                    break;
                }
            }
        }
        /* Sum up frequencies of matched MCVs */
        matchfreq1 = 0.0;
        for (i = 0; i < sslot1->nvalues; i++)
        {
            if (hasmatch1[i])
                matchfreq1 += sslot1->numbers[i];
        }
        CLAMP_PROBABILITY(matchfreq1);
        pfree(hasmatch1);
        pfree(hasmatch2);

        /*
         * Now we need to estimate the fraction of relation 1 that has at
         * least one join partner.  We know for certain that the matched MCVs
         * do, so that gives us a lower bound, but we're really in the dark
         * about everything else.  Our crude approach is: if nd1 <= nd2 then
         * assume all non-null rel1 rows have join partners, else assume for
         * the uncertain rows that a fraction nd2/nd1 have join partners. We
         * can discount the known-matched MCVs from the distinct-values counts
         * before doing the division.
         *
         * Crude as the above is, it's completely useless if we don't have
         * reliable ndistinct values for both sides.  Hence, if either nd1 or
         * nd2 is default, punt and assume half of the uncertain rows have
         * join partners.
         */
        if (!isdefault1 && !isdefault2)
        {
            nd1 -= nmatches;
            nd2 -= nmatches;
            if (nd1 <= nd2 || nd2 < 0)
                uncertainfrac = 1.0;
            else
                uncertainfrac = nd2 / nd1;
        }
        else
            uncertainfrac = 0.5;
        uncertain = 1.0 - matchfreq1 - nullfrac1;
        CLAMP_PROBABILITY(uncertain);
        selec = matchfreq1 + uncertainfrac * uncertain;
    }
    else
    {
        /*
         * Without MCV lists for both sides, we can only use the heuristic
         * about nd1 vs nd2.
         */
        double      nullfrac1 = stats1 ? stats1->stanullfrac : 0.0;

        if (!isdefault1 && !isdefault2)
        {
            if (nd1 <= nd2 || nd2 < 0)
                selec = 1.0 - nullfrac1;
            else
                selec = (nd2 / nd1) * (1.0 - nullfrac1);
        }
        else
            selec = 0.5 * (1.0 - nullfrac1);
    }

    return selec;
}

/*
 *      neqjoinsel      - Join selectivity of "!="
 */
Datum
neqjoinsel(PG_FUNCTION_ARGS)
{
    PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
    Oid         operator = PG_GETARG_OID(1);
    List       *args = (List *) PG_GETARG_POINTER(2);
    JoinType    jointype = (JoinType) PG_GETARG_INT16(3);
    SpecialJoinInfo *sjinfo = (SpecialJoinInfo *) PG_GETARG_POINTER(4);
    float8      result;

    if (jointype == JOIN_SEMI || jointype == JOIN_ANTI)
    {
        /*
         * For semi-joins, if there is more than one distinct value in the RHS
         * relation then every non-null LHS row must find a row to join since
         * it can only be equal to one of them.  We'll assume that there is
         * always more than one distinct RHS value for the sake of stability,
         * though in theory we could have special cases for empty RHS
         * (selectivity = 0) and single-distinct-value RHS (selectivity =
         * fraction of LHS that has the same value as the single RHS value).
         *
         * For anti-joins, if we use the same assumption that there is more
         * than one distinct key in the RHS relation, then every non-null LHS
         * row must be suppressed by the anti-join.
         *
         * So either way, the selectivity estimate should be 1 - nullfrac.
         */
        VariableStatData leftvar;
        VariableStatData rightvar;
        bool        reversed;
        HeapTuple   statsTuple;
        double      nullfrac;

        get_join_variables(root, args, sjinfo, &leftvar, &rightvar, &reversed);
        statsTuple = reversed ? rightvar.statsTuple : leftvar.statsTuple;
        if (HeapTupleIsValid(statsTuple))
            nullfrac = ((Form_pg_statistic) GETSTRUCT(statsTuple))->stanullfrac;
        else
            nullfrac = 0.0;
        ReleaseVariableStats(leftvar);
        ReleaseVariableStats(rightvar);

        result = 1.0 - nullfrac;
    }
    else
    {
        /*
         * We want 1 - eqjoinsel() where the equality operator is the one
         * associated with this != operator, that is, its negator.
         */
        Oid         eqop = get_negator(operator);

        if (eqop)
        {
            result = DatumGetFloat8(DirectFunctionCall5(eqjoinsel,
                                                        PointerGetDatum(root),
                                                        ObjectIdGetDatum(eqop),
                                                        PointerGetDatum(args),
                                                        Int16GetDatum(jointype),
                                                        PointerGetDatum(sjinfo)));
        }
        else
        {
            /* Use default selectivity (should we raise an error instead?) */
            result = DEFAULT_EQ_SEL;
        }
        result = 1.0 - result;
    }

    PG_RETURN_FLOAT8(result);
}

/*
 *      scalarltjoinsel - Join selectivity of "<" for scalars
 */
Datum
scalarltjoinsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}

/*
 *      scalarlejoinsel - Join selectivity of "<=" for scalars
 */
Datum
scalarlejoinsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}

/*
 *      scalargtjoinsel - Join selectivity of ">" for scalars
 */
Datum
scalargtjoinsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}

/*
 *      scalargejoinsel - Join selectivity of ">=" for scalars
 */
Datum
scalargejoinsel(PG_FUNCTION_ARGS)
{
    PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}


/*
 * mergejoinscansel         - Scan selectivity of merge join.
 *
 * A merge join will stop as soon as it exhausts either input stream.
 * Therefore, if we can estimate the ranges of both input variables,
 * we can estimate how much of the input will actually be read.  This
 * can have a considerable impact on the cost when using indexscans.
 *
 * Also, we can estimate how much of each input has to be read before the
 * first join pair is found, which will affect the join's startup time.
 *
 * clause should be a clause already known to be mergejoinable.  opfamily,
 * strategy, and nulls_first specify the sort ordering being used.
 *
 * The outputs are:
 *      *leftstart is set to the fraction of the left-hand variable expected
 *       to be scanned before the first join pair is found (0 to 1).
 *      *leftend is set to the fraction of the left-hand variable expected
 *       to be scanned before the join terminates (0 to 1).
 *      *rightstart, *rightend similarly for the right-hand variable.
 */
void
mergejoinscansel(PlannerInfo *root, Node *clause,
                 Oid opfamily, int strategy, bool nulls_first,
                 Selectivity *leftstart, Selectivity *leftend,
                 Selectivity *rightstart, Selectivity *rightend)
{
    Node       *left,
               *right;
    VariableStatData leftvar,
                rightvar;
    int         op_strategy;
    Oid         op_lefttype;
    Oid         op_righttype;
    Oid         opno,
                lsortop,
                rsortop,
                lstatop,
                rstatop,
                ltop,
                leop,
                revltop,
                revleop;
    bool        isgt;
    Datum       leftmin,
                leftmax,
                rightmin,
                rightmax;
    double      selec;

    /* Set default results if we can't figure anything out. */
    /* XXX should default "start" fraction be a bit more than 0? */
    *leftstart = *rightstart = 0.0;
    *leftend = *rightend = 1.0;

    /* Deconstruct the merge clause */
    if (!is_opclause(clause))
        return;                 /* shouldn't happen */
    opno = ((OpExpr *) clause)->opno;
    left = get_leftop((Expr *) clause);
    right = get_rightop((Expr *) clause);
    if (!right)
        return;                 /* shouldn't happen */

    /* Look for stats for the inputs */
    examine_variable(root, left, 0, &leftvar);
    examine_variable(root, right, 0, &rightvar);

    /* Extract the operator's declared left/right datatypes */
    get_op_opfamily_properties(opno, opfamily, false,
                               &op_strategy,
                               &op_lefttype,
                               &op_righttype);
    Assert(op_strategy == BTEqualStrategyNumber);

    /*
     * Look up the various operators we need.  If we don't find them all, it
     * probably means the opfamily is broken, but we just fail silently.
     *
     * Note: we expect that pg_statistic histograms will be sorted by the '<'
     * operator, regardless of which sort direction we are considering.
     */
    switch (strategy)
    {
        case BTLessStrategyNumber:
            isgt = false;
            if (op_lefttype == op_righttype)
            {
                /* easy case */
                ltop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTLessStrategyNumber);
                leop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTLessEqualStrategyNumber);
                lsortop = ltop;
                rsortop = ltop;
                lstatop = lsortop;
                rstatop = rsortop;
                revltop = ltop;
                revleop = leop;
            }
            else
            {
                ltop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTLessStrategyNumber);
                leop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTLessEqualStrategyNumber);
                lsortop = get_opfamily_member(opfamily,
                                              op_lefttype, op_lefttype,
                                              BTLessStrategyNumber);
                rsortop = get_opfamily_member(opfamily,
                                              op_righttype, op_righttype,
                                              BTLessStrategyNumber);
                lstatop = lsortop;
                rstatop = rsortop;
                revltop = get_opfamily_member(opfamily,
                                              op_righttype, op_lefttype,
                                              BTLessStrategyNumber);
                revleop = get_opfamily_member(opfamily,
                                              op_righttype, op_lefttype,
                                              BTLessEqualStrategyNumber);
            }
            break;
        case BTGreaterStrategyNumber:
            /* descending-order case */
            isgt = true;
            if (op_lefttype == op_righttype)
            {
                /* easy case */
                ltop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTGreaterStrategyNumber);
                leop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTGreaterEqualStrategyNumber);
                lsortop = ltop;
                rsortop = ltop;
                lstatop = get_opfamily_member(opfamily,
                                              op_lefttype, op_lefttype,
                                              BTLessStrategyNumber);
                rstatop = lstatop;
                revltop = ltop;
                revleop = leop;
            }
            else
            {
                ltop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTGreaterStrategyNumber);
                leop = get_opfamily_member(opfamily,
                                           op_lefttype, op_righttype,
                                           BTGreaterEqualStrategyNumber);
                lsortop = get_opfamily_member(opfamily,
                                              op_lefttype, op_lefttype,
                                              BTGreaterStrategyNumber);
                rsortop = get_opfamily_member(opfamily,
                                              op_righttype, op_righttype,
                                              BTGreaterStrategyNumber);
                lstatop = get_opfamily_member(opfamily,
                                              op_lefttype, op_lefttype,
                                              BTLessStrategyNumber);
                rstatop = get_opfamily_member(opfamily,
                                              op_righttype, op_righttype,
                                              BTLessStrategyNumber);
                revltop = get_opfamily_member(opfamily,
                                              op_righttype, op_lefttype,
                                              BTGreaterStrategyNumber);
                revleop = get_opfamily_member(opfamily,
                                              op_righttype, op_lefttype,
                                              BTGreaterEqualStrategyNumber);
            }
            break;
        default:
            goto fail;          /* shouldn't get here */
    }

    if (!OidIsValid(lsortop) ||
        !OidIsValid(rsortop) ||
        !OidIsValid(lstatop) ||
        !OidIsValid(rstatop) ||
        !OidIsValid(ltop) ||
        !OidIsValid(leop) ||
        !OidIsValid(revltop) ||
        !OidIsValid(revleop))
        goto fail;              /* insufficient info in catalogs */

    /* Try to get ranges of both inputs */
    if (!isgt)
    {
        if (!get_variable_range(root, &leftvar, lstatop,
                                &leftmin, &leftmax))
            goto fail;          /* no range available from stats */
        if (!get_variable_range(root, &rightvar, rstatop,
                                &rightmin, &rightmax))
            goto fail;          /* no range available from stats */
    }
    else
    {
        /* need to swap the max and min */
        if (!get_variable_range(root, &leftvar, lstatop,
                                &leftmax, &leftmin))
            goto fail;          /* no range available from stats */
        if (!get_variable_range(root, &rightvar, rstatop,
                                &rightmax, &rightmin))
            goto fail;          /* no range available from stats */
    }

    /*
     * Now, the fraction of the left variable that will be scanned is the
     * fraction that's <= the right-side maximum value.  But only believe
     * non-default estimates, else stick with our 1.0.
     */
    selec = scalarineqsel(root, leop, isgt, true, &leftvar,
                          rightmax, op_righttype);
    if (selec != DEFAULT_INEQ_SEL)
        *leftend = selec;

    /* And similarly for the right variable. */
    selec = scalarineqsel(root, revleop, isgt, true, &rightvar,
                          leftmax, op_lefttype);
    if (selec != DEFAULT_INEQ_SEL)
        *rightend = selec;

    /*
     * Only one of the two "end" fractions can really be less than 1.0;
     * believe the smaller estimate and reset the other one to exactly 1.0. If
     * we get exactly equal estimates (as can easily happen with self-joins),
     * believe neither.
     */
    if (*leftend > *rightend)
        *leftend = 1.0;
    else if (*leftend < *rightend)
        *rightend = 1.0;
    else
        *leftend = *rightend = 1.0;

    /*
     * Also, the fraction of the left variable that will be scanned before the
     * first join pair is found is the fraction that's < the right-side
     * minimum value.  But only believe non-default estimates, else stick with
     * our own default.
     */
    selec = scalarineqsel(root, ltop, isgt, false, &leftvar,
                          rightmin, op_righttype);
    if (selec != DEFAULT_INEQ_SEL)
        *leftstart = selec;

    /* And similarly for the right variable. */
    selec = scalarineqsel(root, revltop, isgt, false, &rightvar,
                          leftmin, op_lefttype);
    if (selec != DEFAULT_INEQ_SEL)
        *rightstart = selec;

    /*
     * Only one of the two "start" fractions can really be more than zero;
     * believe the larger estimate and reset the other one to exactly 0.0. If
     * we get exactly equal estimates (as can easily happen with self-joins),
     * believe neither.
     */
    if (*leftstart < *rightstart)
        *leftstart = 0.0;
    else if (*leftstart > *rightstart)
        *rightstart = 0.0;
    else
        *leftstart = *rightstart = 0.0;

    /*
     * If the sort order is nulls-first, we're going to have to skip over any
     * nulls too.  These would not have been counted by scalarineqsel, and we
     * can safely add in this fraction regardless of whether we believe
     * scalarineqsel's results or not.  But be sure to clamp the sum to 1.0!
     */
    if (nulls_first)
    {
        Form_pg_statistic stats;

        if (HeapTupleIsValid(leftvar.statsTuple))
        {
            stats = (Form_pg_statistic) GETSTRUCT(leftvar.statsTuple);
            *leftstart += stats->stanullfrac;
            CLAMP_PROBABILITY(*leftstart);
            *leftend += stats->stanullfrac;
            CLAMP_PROBABILITY(*leftend);
        }
        if (HeapTupleIsValid(rightvar.statsTuple))
        {
            stats = (Form_pg_statistic) GETSTRUCT(rightvar.statsTuple);
            *rightstart += stats->stanullfrac;
            CLAMP_PROBABILITY(*rightstart);
            *rightend += stats->stanullfrac;
            CLAMP_PROBABILITY(*rightend);
        }
    }

    /* Disbelieve start >= end, just in case that can happen */
    if (*leftstart >= *leftend)
    {
        *leftstart = 0.0;
        *leftend = 1.0;
    }
    if (*rightstart >= *rightend)
    {
        *rightstart = 0.0;
        *rightend = 1.0;
    }

fail:
    ReleaseVariableStats(leftvar);
    ReleaseVariableStats(rightvar);
}


/*
 * Helper routine for estimate_num_groups: add an item to a list of
 * GroupVarInfos, but only if it's not known equal to any of the existing
 * entries.
 */
typedef struct
{
    Node       *var;            /* might be an expression, not just a Var */
    RelOptInfo *rel;            /* relation it belongs to */
    double      ndistinct;      /* # distinct values */
} GroupVarInfo;

static List *
add_unique_group_var(PlannerInfo *root, List *varinfos,
                     Node *var, VariableStatData *vardata)
{
    GroupVarInfo *varinfo;
    double      ndistinct;
    bool        isdefault;
    ListCell   *lc;

    ndistinct = get_variable_numdistinct(vardata, &isdefault);

    foreach(lc, varinfos)
    {
        varinfo = (GroupVarInfo *) lfirst(lc);

        /* Drop exact duplicates */
        if (equal(var, varinfo->var))
            return varinfos;

        /*
         * Drop known-equal vars, but only if they belong to different
         * relations (see comments for estimate_num_groups)
         */
        if (vardata->rel != varinfo->rel &&
            exprs_known_equal(root, var, varinfo->var))
        {
            if (varinfo->ndistinct <= ndistinct)
            {
                /* Keep older item, forget new one */
                return varinfos;
            }
            else
            {
                /* Delete the older item */
                varinfos = foreach_delete_current(varinfos, lc);
            }
        }
    }

    varinfo = (GroupVarInfo *) palloc(sizeof(GroupVarInfo));

    varinfo->var = var;
    varinfo->rel = vardata->rel;
    varinfo->ndistinct = ndistinct;
    varinfos = lappend(varinfos, varinfo);
    return varinfos;
}

/*
 * estimate_num_groups      - Estimate number of groups in a grouped query
 *
 * Given a query having a GROUP BY clause, estimate how many groups there
 * will be --- ie, the number of distinct combinations of the GROUP BY
 * expressions.
 *
 * This routine is also used to estimate the number of rows emitted by
 * a DISTINCT filtering step; that is an isomorphic problem.  (Note:
 * actually, we only use it for DISTINCT when there's no grouping or
 * aggregation ahead of the DISTINCT.)
 *
 * Inputs:
 *  root - the query
 *  groupExprs - list of expressions being grouped by
 *  input_rows - number of rows estimated to arrive at the group/unique
 *      filter step
 *  pgset - NULL, or a List** pointing to a grouping set to filter the
 *      groupExprs against
 *
 * Given the lack of any cross-correlation statistics in the system, it's
 * impossible to do anything really trustworthy with GROUP BY conditions
 * involving multiple Vars.  We should however avoid assuming the worst
 * case (all possible cross-product terms actually appear as groups) since
 * very often the grouped-by Vars are highly correlated.  Our current approach
 * is as follows:
 *  1.  Expressions yielding boolean are assumed to contribute two groups,
 *      independently of their content, and are ignored in the subsequent
 *      steps.  This is mainly because tests like "col IS NULL" break the
 *      heuristic used in step 2 especially badly.
 *  2.  Reduce the given expressions to a list of unique Vars used.  For
 *      example, GROUP BY a, a + b is treated the same as GROUP BY a, b.
 *      It is clearly correct not to count the same Var more than once.
 *      It is also reasonable to treat f(x) the same as x: f() cannot
 *      increase the number of distinct values (unless it is volatile,
 *      which we consider unlikely for grouping), but it probably won't
 *      reduce the number of distinct values much either.
 *      As a special case, if a GROUP BY expression can be matched to an
 *      expressional index for which we have statistics, then we treat the
 *      whole expression as though it were just a Var.
 *  3.  If the list contains Vars of different relations that are known equal
 *      due to equivalence classes, then drop all but one of the Vars from each
 *      known-equal set, keeping the one with smallest estimated # of values
 *      (since the extra values of the others can't appear in joined rows).
 *      Note the reason we only consider Vars of different relations is that
 *      if we considered ones of the same rel, we'd be double-counting the
 *      restriction selectivity of the equality in the next step.
 *  4.  For Vars within a single source rel, we multiply together the numbers
 *      of values, clamp to the number of rows in the rel (divided by 10 if
 *      more than one Var), and then multiply by a factor based on the
 *      selectivity of the restriction clauses for that rel.  When there's
 *      more than one Var, the initial product is probably too high (it's the
 *      worst case) but clamping to a fraction of the rel's rows seems to be a
 *      helpful heuristic for not letting the estimate get out of hand.  (The
 *      factor of 10 is derived from pre-Postgres-7.4 practice.)  The factor
 *      we multiply by to adjust for the restriction selectivity assumes that
 *      the restriction clauses are independent of the grouping, which may not
 *      be a valid assumption, but it's hard to do better.
 *  5.  If there are Vars from multiple rels, we repeat step 4 for each such
 *      rel, and multiply the results together.
 * Note that rels not containing grouped Vars are ignored completely, as are
 * join clauses.  Such rels cannot increase the number of groups, and we
 * assume such clauses do not reduce the number either (somewhat bogus,
 * but we don't have the info to do better).
 */
double
estimate_num_groups(PlannerInfo *root, List *groupExprs, double input_rows,
                    List **pgset)
{
    List       *varinfos = NIL;
    double      srf_multiplier = 1.0;
    double      numdistinct;
    ListCell   *l;
    int         i;

    /*
     * We don't ever want to return an estimate of zero groups, as that tends
     * to lead to division-by-zero and other unpleasantness.  The input_rows
     * estimate is usually already at least 1, but clamp it just in case it
     * isn't.
     */
    input_rows = clamp_row_est(input_rows);

    /*
     * If no grouping columns, there's exactly one group.  (This can't happen
     * for normal cases with GROUP BY or DISTINCT, but it is possible for
     * corner cases with set operations.)
     */
    if (groupExprs == NIL || (pgset && list_length(*pgset) < 1))
        return 1.0;

    /*
     * Count groups derived from boolean grouping expressions.  For other
     * expressions, find the unique Vars used, treating an expression as a Var
     * if we can find stats for it.  For each one, record the statistical
     * estimate of number of distinct values (total in its table, without
     * regard for filtering).
     */
    numdistinct = 1.0;

    i = 0;
    foreach(l, groupExprs)
    {
        Node       *groupexpr = (Node *) lfirst(l);
        double      this_srf_multiplier;
        VariableStatData vardata;
        List       *varshere;
        ListCell   *l2;

        /* is expression in this grouping set? */
        if (pgset && !list_member_int(*pgset, i++))
            continue;

        /*
         * Set-returning functions in grouping columns are a bit problematic.
         * The code below will effectively ignore their SRF nature and come up
         * with a numdistinct estimate as though they were scalar functions.
         * We compensate by scaling up the end result by the largest SRF
         * rowcount estimate.  (This will be an overestimate if the SRF
         * produces multiple copies of any output value, but it seems best to
         * assume the SRF's outputs are distinct.  In any case, it's probably
         * pointless to worry too much about this without much better
         * estimates for SRF output rowcounts than we have today.)
         */
        this_srf_multiplier = expression_returns_set_rows(root, groupexpr);
        if (srf_multiplier < this_srf_multiplier)
            srf_multiplier = this_srf_multiplier;

        /* Short-circuit for expressions returning boolean */
        if (exprType(groupexpr) == BOOLOID)
        {
            numdistinct *= 2.0;
            continue;
        }

        /*
         * If examine_variable is able to deduce anything about the GROUP BY
         * expression, treat it as a single variable even if it's really more
         * complicated.
         */
        examine_variable(root, groupexpr, 0, &vardata);
        if (HeapTupleIsValid(vardata.statsTuple) || vardata.isunique)
        {
            varinfos = add_unique_group_var(root, varinfos,
                                            groupexpr, &vardata);
            ReleaseVariableStats(vardata);
            continue;
        }
        ReleaseVariableStats(vardata);

        /*
         * Else pull out the component Vars.  Handle PlaceHolderVars by
         * recursing into their arguments (effectively assuming that the
         * PlaceHolderVar doesn't change the number of groups, which boils
         * down to ignoring the possible addition of nulls to the result set).
         */
        varshere = pull_var_clause(groupexpr,
                                   PVC_RECURSE_AGGREGATES |
                                   PVC_RECURSE_WINDOWFUNCS |
                                   PVC_RECURSE_PLACEHOLDERS);

        /*
         * If we find any variable-free GROUP BY item, then either it is a
         * constant (and we can ignore it) or it contains a volatile function;
         * in the latter case we punt and assume that each input row will
         * yield a distinct group.
         */
        if (varshere == NIL)
        {
            if (contain_volatile_functions(groupexpr))
                return input_rows;
            continue;
        }

        /*
         * Else add variables to varinfos list
         */
        foreach(l2, varshere)
        {
            Node       *var = (Node *) lfirst(l2);

            examine_variable(root, var, 0, &vardata);
            varinfos = add_unique_group_var(root, varinfos, var, &vardata);
            ReleaseVariableStats(vardata);
        }
    }

    /*
     * If now no Vars, we must have an all-constant or all-boolean GROUP BY
     * list.
     */
    if (varinfos == NIL)
    {
        /* Apply SRF multiplier as we would do in the long path */
        numdistinct *= srf_multiplier;
        /* Round off */
        numdistinct = ceil(numdistinct);
        /* Guard against out-of-range answers */
        if (numdistinct > input_rows)
            numdistinct = input_rows;
        if (numdistinct < 1.0)
            numdistinct = 1.0;
        return numdistinct;
    }

    /*
     * Group Vars by relation and estimate total numdistinct.
     *
     * For each iteration of the outer loop, we process the frontmost Var in
     * varinfos, plus all other Vars in the same relation.  We remove these
     * Vars from the newvarinfos list for the next iteration. This is the
     * easiest way to group Vars of same rel together.
     */
    do
    {
        GroupVarInfo *varinfo1 = (GroupVarInfo *) linitial(varinfos);
        RelOptInfo *rel = varinfo1->rel;
        double      reldistinct = 1;
        double      relmaxndistinct = reldistinct;
        int         relvarcount = 0;
        List       *newvarinfos = NIL;
        List       *relvarinfos = NIL;

        /*
         * Split the list of varinfos in two - one for the current rel, one
         * for remaining Vars on other rels.
         */
        relvarinfos = lappend(relvarinfos, varinfo1);
        for_each_cell(l, varinfos, list_second_cell(varinfos))
        {
            GroupVarInfo *varinfo2 = (GroupVarInfo *) lfirst(l);

            if (varinfo2->rel == varinfo1->rel)
            {
                /* varinfos on current rel */
                relvarinfos = lappend(relvarinfos, varinfo2);
            }
            else
            {
                /* not time to process varinfo2 yet */
                newvarinfos = lappend(newvarinfos, varinfo2);
            }
        }

        /*
         * Get the numdistinct estimate for the Vars of this rel.  We
         * iteratively search for multivariate n-distinct with maximum number
         * of vars; assuming that each var group is independent of the others,
         * we multiply them together.  Any remaining relvarinfos after no more
         * multivariate matches are found are assumed independent too, so
         * their individual ndistinct estimates are multiplied also.
         *
         * While iterating, count how many separate numdistinct values we
         * apply.  We apply a fudge factor below, but only if we multiplied
         * more than one such values.
         */
        while (relvarinfos)
        {
            double      mvndistinct;

            if (estimate_multivariate_ndistinct(root, rel, &relvarinfos,
                                                &mvndistinct))
            {
                reldistinct *= mvndistinct;
                if (relmaxndistinct < mvndistinct)
                    relmaxndistinct = mvndistinct;
                relvarcount++;
            }
            else
            {
                foreach(l, relvarinfos)
                {
                    GroupVarInfo *varinfo2 = (GroupVarInfo *) lfirst(l);

                    reldistinct *= varinfo2->ndistinct;
                    if (relmaxndistinct < varinfo2->ndistinct)
                        relmaxndistinct = varinfo2->ndistinct;
                    relvarcount++;
                }

                /* we're done with this relation */
                relvarinfos = NIL;
            }
        }

        /*
         * Sanity check --- don't divide by zero if empty relation.
         */
        Assert(IS_SIMPLE_REL(rel));
        if (rel->tuples > 0)
        {
            /*
             * Clamp to size of rel, or size of rel / 10 if multiple Vars. The
             * fudge factor is because the Vars are probably correlated but we
             * don't know by how much.  We should never clamp to less than the
             * largest ndistinct value for any of the Vars, though, since
             * there will surely be at least that many groups.
             */
            double      clamp = rel->tuples;

            if (relvarcount > 1)
            {
                clamp *= 0.1;
                if (clamp < relmaxndistinct)
                {
                    clamp = relmaxndistinct;
                    /* for sanity in case some ndistinct is too large: */
                    if (clamp > rel->tuples)
                        clamp = rel->tuples;
                }
            }
            if (reldistinct > clamp)
                reldistinct = clamp;

            /*
             * Update the estimate based on the restriction selectivity,
             * guarding against division by zero when reldistinct is zero.
             * Also skip this if we know that we are returning all rows.
             */
            if (reldistinct > 0 && rel->rows < rel->tuples)
            {
                /*
                 * Given a table containing N rows with n distinct values in a
                 * uniform distribution, if we select p rows at random then
                 * the expected number of distinct values selected is
                 *
                 * n * (1 - product((N-N/n-i)/(N-i), i=0..p-1))
                 *
                 * = n * (1 - (N-N/n)! / (N-N/n-p)! * (N-p)! / N!)
                 *
                 * See "Approximating block accesses in database
                 * organizations", S. B. Yao, Communications of the ACM,
                 * Volume 20 Issue 4, April 1977 Pages 260-261.
                 *
                 * Alternatively, re-arranging the terms from the factorials,
                 * this may be written as
                 *
                 * n * (1 - product((N-p-i)/(N-i), i=0..N/n-1))
                 *
                 * This form of the formula is more efficient to compute in
                 * the common case where p is larger than N/n.  Additionally,
                 * as pointed out by Dell'Era, if i << N for all terms in the
                 * product, it can be approximated by
                 *
                 * n * (1 - ((N-p)/N)^(N/n))
                 *
                 * See "Expected distinct values when selecting from a bag
                 * without replacement", Alberto Dell'Era,
                 * http://www.adellera.it/investigations/distinct_balls/.
                 *
                 * The condition i << N is equivalent to n >> 1, so this is a
                 * good approximation when the number of distinct values in
                 * the table is large.  It turns out that this formula also
                 * works well even when n is small.
                 */
                reldistinct *=
                    (1 - pow((rel->tuples - rel->rows) / rel->tuples,
                             rel->tuples / reldistinct));
            }
            reldistinct = clamp_row_est(reldistinct);

            /*
             * Update estimate of total distinct groups.
             */
            numdistinct *= reldistinct;
        }

        varinfos = newvarinfos;
    } while (varinfos != NIL);

    /* Now we can account for the effects of any SRFs */
    numdistinct *= srf_multiplier;

    /* Round off */
    numdistinct = ceil(numdistinct);

    /* Guard against out-of-range answers */
    if (numdistinct > input_rows)
        numdistinct = input_rows;
    if (numdistinct < 1.0)
        numdistinct = 1.0;

    return numdistinct;
}

/*
 * Estimate hash bucket statistics when the specified expression is used
 * as a hash key for the given number of buckets.
 *
 * This attempts to determine two values:
 *
 * 1. The frequency of the most common value of the expression (returns
 * zero into *mcv_freq if we can't get that).
 *
 * 2. The "bucketsize fraction", ie, average number of entries in a bucket
 * divided by total tuples in relation.
 *
 * XXX This is really pretty bogus since we're effectively assuming that the
 * distribution of hash keys will be the same after applying restriction
 * clauses as it was in the underlying relation.  However, we are not nearly
 * smart enough to figure out how the restrict clauses might change the
 * distribution, so this will have to do for now.
 *
 * We are passed the number of buckets the executor will use for the given
 * input relation.  If the data were perfectly distributed, with the same
 * number of tuples going into each available bucket, then the bucketsize
 * fraction would be 1/nbuckets.  But this happy state of affairs will occur
 * only if (a) there are at least nbuckets distinct data values, and (b)
 * we have a not-too-skewed data distribution.  Otherwise the buckets will
 * be nonuniformly occupied.  If the other relation in the join has a key
 * distribution similar to this one's, then the most-loaded buckets are
 * exactly those that will be probed most often.  Therefore, the "average"
 * bucket size for costing purposes should really be taken as something close
 * to the "worst case" bucket size.  We try to estimate this by adjusting the
 * fraction if there are too few distinct data values, and then scaling up
 * by the ratio of the most common value's frequency to the average frequency.
 *
 * If no statistics are available, use a default estimate of 0.1.  This will
 * discourage use of a hash rather strongly if the inner relation is large,
 * which is what we want.  We do not want to hash unless we know that the
 * inner rel is well-dispersed (or the alternatives seem much worse).
 *
 * The caller should also check that the mcv_freq is not so large that the
 * most common value would by itself require an impractically large bucket.
 * In a hash join, the executor can split buckets if they get too big, but
 * obviously that doesn't help for a bucket that contains many duplicates of
 * the same value.
 */
void
estimate_hash_bucket_stats(PlannerInfo *root, Node *hashkey, double nbuckets,
                           Selectivity *mcv_freq,
                           Selectivity *bucketsize_frac)
{
    VariableStatData vardata;
    double      estfract,
                ndistinct,
                stanullfrac,
                avgfreq;
    bool        isdefault;
    AttStatsSlot sslot;

    examine_variable(root, hashkey, 0, &vardata);

    /* Look up the frequency of the most common value, if available */
    *mcv_freq = 0.0;

    if (HeapTupleIsValid(vardata.statsTuple))
    {
        if (get_attstatsslot(&sslot, vardata.statsTuple,
                             STATISTIC_KIND_MCV, InvalidOid,
                             ATTSTATSSLOT_NUMBERS))
        {
            /*
             * The first MCV stat is for the most common value.
             */
            if (sslot.nnumbers > 0)
                *mcv_freq = sslot.numbers[0];
            free_attstatsslot(&sslot);
        }
    }

    /* Get number of distinct values */
    ndistinct = get_variable_numdistinct(&vardata, &isdefault);

    /*
     * If ndistinct isn't real, punt.  We normally return 0.1, but if the
     * mcv_freq is known to be even higher than that, use it instead.
     */
    if (isdefault)
    {
        *bucketsize_frac = (Selectivity) Max(0.1, *mcv_freq);
        ReleaseVariableStats(vardata);
        return;
    }

    /* Get fraction that are null */
    if (HeapTupleIsValid(vardata.statsTuple))
    {
        Form_pg_statistic stats;

        stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);
        stanullfrac = stats->stanullfrac;
    }
    else
        stanullfrac = 0.0;

    /* Compute avg freq of all distinct data values in raw relation */
    avgfreq = (1.0 - stanullfrac) / ndistinct;

    /*
     * Adjust ndistinct to account for restriction clauses.  Observe we are
     * assuming that the data distribution is affected uniformly by the
     * restriction clauses!
     *
     * XXX Possibly better way, but much more expensive: multiply by
     * selectivity of rel's restriction clauses that mention the target Var.
     */
    if (vardata.rel && vardata.rel->tuples > 0)
    {
        ndistinct *= vardata.rel->rows / vardata.rel->tuples;
        ndistinct = clamp_row_est(ndistinct);
    }

    /*
     * Initial estimate of bucketsize fraction is 1/nbuckets as long as the
     * number of buckets is less than the expected number of distinct values;
     * otherwise it is 1/ndistinct.
     */
    if (ndistinct > nbuckets)
        estfract = 1.0 / nbuckets;
    else
        estfract = 1.0 / ndistinct;

    /*
     * Adjust estimated bucketsize upward to account for skewed distribution.
     */
    if (avgfreq > 0.0 && *mcv_freq > avgfreq)
        estfract *= *mcv_freq / avgfreq;

    /*
     * Clamp bucketsize to sane range (the above adjustment could easily
     * produce an out-of-range result).  We set the lower bound a little above
     * zero, since zero isn't a very sane result.
     */
    if (estfract < 1.0e-6)
        estfract = 1.0e-6;
    else if (estfract > 1.0)
        estfract = 1.0;

    *bucketsize_frac = (Selectivity) estfract;

    ReleaseVariableStats(vardata);
}

/*
 * estimate_hashagg_tablesize
 *    estimate the number of bytes that a hash aggregate hashtable will
 *    require based on the agg_costs, path width and number of groups.
 *
 * We return the result as "double" to forestall any possible overflow
 * problem in the multiplication by dNumGroups.
 *
 * XXX this may be over-estimating the size now that hashagg knows to omit
 * unneeded columns from the hashtable.  Also for mixed-mode grouping sets,
 * grouping columns not in the hashed set are counted here even though hashagg
 * won't store them.  Is this a problem?
 */
double
estimate_hashagg_tablesize(Path *path, const AggClauseCosts *agg_costs,
                           double dNumGroups)
{
    Size        hashentrysize;

    /* Estimate per-hash-entry space at tuple width... */
    hashentrysize = MAXALIGN(path->pathtarget->width) +
        MAXALIGN(SizeofMinimalTupleHeader);

    /* plus space for pass-by-ref transition values... */
    hashentrysize += agg_costs->transitionSpace;
    /* plus the per-hash-entry overhead */
    hashentrysize += hash_agg_entry_size(agg_costs->numAggs);

    /*
     * Note that this disregards the effect of fill-factor and growth policy
     * of the hash table.  That's probably ok, given that the default
     * fill-factor is relatively high.  It'd be hard to meaningfully factor in
     * "double-in-size" growth policies here.
     */
    return hashentrysize * dNumGroups;
}


/*-------------------------------------------------------------------------
 *
 * Support routines
 *
 *-------------------------------------------------------------------------
 */

/*
 * Find applicable ndistinct statistics for the given list of VarInfos (which
 * must all belong to the given rel), and update *ndistinct to the estimate of
 * the MVNDistinctItem that best matches.  If a match it found, *varinfos is
 * updated to remove the list of matched varinfos.
 *
 * Varinfos that aren't for simple Vars are ignored.
 *
 * Return true if we're able to find a match, false otherwise.
 */
static bool
estimate_multivariate_ndistinct(PlannerInfo *root, RelOptInfo *rel,
                                List **varinfos, double *ndistinct)
{
    ListCell   *lc;
    Bitmapset  *attnums = NULL;
    int         nmatches;
    Oid         statOid = InvalidOid;
    MVNDistinct *stats;
    Bitmapset  *matched = NULL;

    /* bail out immediately if the table has no extended statistics */
    if (!rel->statlist)
        return false;

    /* Determine the attnums we're looking for */
    foreach(lc, *varinfos)
    {
        GroupVarInfo *varinfo = (GroupVarInfo *) lfirst(lc);
        AttrNumber  attnum;

        Assert(varinfo->rel == rel);

        if (!IsA(varinfo->var, Var))
            continue;

        attnum = ((Var *) varinfo->var)->varattno;

        if (!AttrNumberIsForUserDefinedAttr(attnum))
            continue;

        attnums = bms_add_member(attnums, attnum);
    }

    /* look for the ndistinct statistics matching the most vars */
    nmatches = 1;               /* we require at least two matches */
    foreach(lc, rel->statlist)
    {
        StatisticExtInfo *info = (StatisticExtInfo *) lfirst(lc);
        Bitmapset  *shared;
        int         nshared;

        /* skip statistics of other kinds */
        if (info->kind != STATS_EXT_NDISTINCT)
            continue;

        /* compute attnums shared by the vars and the statistics object */
        shared = bms_intersect(info->keys, attnums);
        nshared = bms_num_members(shared);

        /*
         * Does this statistics object match more columns than the currently
         * best object?  If so, use this one instead.
         *
         * XXX This should break ties using name of the object, or something
         * like that, to make the outcome stable.
         */
        if (nshared > nmatches)
        {
            statOid = info->statOid;
            nmatches = nshared;
            matched = shared;
        }
    }

    /* No match? */
    if (statOid == InvalidOid)
        return false;
    Assert(nmatches > 1 && matched != NULL);

    stats = statext_ndistinct_load(statOid);

    /*
     * If we have a match, search it for the specific item that matches (there
     * must be one), and construct the output values.
     */
    if (stats)
    {
        int         i;
        List       *newlist = NIL;
        MVNDistinctItem *item = NULL;

        /* Find the specific item that exactly matches the combination */
        for (i = 0; i < stats->nitems; i++)
        {
            MVNDistinctItem *tmpitem = &stats->items[i];

            if (bms_subset_compare(tmpitem->attrs, matched) == BMS_EQUAL)
            {
                item = tmpitem;
                break;
            }
        }

        /* make sure we found an item */
        if (!item)
            elog(ERROR, "corrupt MVNDistinct entry");

        /* Form the output varinfo list, keeping only unmatched ones */
        foreach(lc, *varinfos)
        {
            GroupVarInfo *varinfo = (GroupVarInfo *) lfirst(lc);
            AttrNumber  attnum;

            if (!IsA(varinfo->var, Var))
            {
                newlist = lappend(newlist, varinfo);
                continue;
            }

            attnum = ((Var *) varinfo->var)->varattno;

            if (!AttrNumberIsForUserDefinedAttr(attnum))
                continue;

            if (!bms_is_member(attnum, matched))
                newlist = lappend(newlist, varinfo);
        }

        *varinfos = newlist;
        *ndistinct = item->ndistinct;
        return true;
    }

    return false;
}

/*
 * convert_to_scalar
 *    Convert non-NULL values of the indicated types to the comparison
 *    scale needed by scalarineqsel().
 *    Returns "true" if successful.
 *
 * XXX this routine is a hack: ideally we should look up the conversion
 * subroutines in pg_type.
 *
 * All numeric datatypes are simply converted to their equivalent
 * "double" values.  (NUMERIC values that are outside the range of "double"
 * are clamped to +/- HUGE_VAL.)
 *
 * String datatypes are converted by convert_string_to_scalar(),
 * which is explained below.  The reason why this routine deals with
 * three values at a time, not just one, is that we need it for strings.
 *
 * The bytea datatype is just enough different from strings that it has
 * to be treated separately.
 *
 * The several datatypes representing absolute times are all converted
 * to Timestamp, which is actually a double, and then we just use that
 * double value.  Note this will give correct results even for the "special"
 * values of Timestamp, since those are chosen to compare correctly;
 * see timestamp_cmp.
 *
 * The several datatypes representing relative times (intervals) are all
 * converted to measurements expressed in seconds.
 */
static bool
convert_to_scalar(Datum value, Oid valuetypid, Oid collid, double *scaledvalue,
                  Datum lobound, Datum hibound, Oid boundstypid,
                  double *scaledlobound, double *scaledhibound)
{
    bool        failure = false;

    /*
     * Both the valuetypid and the boundstypid should exactly match the
     * declared input type(s) of the operator we are invoked for.  However,
     * extensions might try to use scalarineqsel as estimator for operators
     * with input type(s) we don't handle here; in such cases, we want to
     * return false, not fail.  In any case, we mustn't assume that valuetypid
     * and boundstypid are identical.
     *
     * XXX The histogram we are interpolating between points of could belong
     * to a column that's only binary-compatible with the declared type. In
     * essence we are assuming that the semantics of binary-compatible types
     * are enough alike that we can use a histogram generated with one type's
     * operators to estimate selectivity for the other's.  This is outright
     * wrong in some cases --- in particular signed versus unsigned
     * interpretation could trip us up.  But it's useful enough in the
     * majority of cases that we do it anyway.  Should think about more
     * rigorous ways to do it.
     */
    switch (valuetypid)
    {
            /*
             * Built-in numeric types
             */
        case BOOLOID:
        case INT2OID:
        case INT4OID:
        case INT8OID:
        case FLOAT4OID:
        case FLOAT8OID:
        case NUMERICOID:
        case OIDOID:
        case REGPROCOID:
        case REGPROCEDUREOID:
        case REGOPEROID:
        case REGOPERATOROID:
        case REGCLASSOID:
        case REGTYPEOID:
        case REGCONFIGOID:
        case REGDICTIONARYOID:
        case REGROLEOID:
        case REGNAMESPACEOID:
            *scaledvalue = convert_numeric_to_scalar(value, valuetypid,
                                                     &failure);
            *scaledlobound = convert_numeric_to_scalar(lobound, boundstypid,
                                                       &failure);
            *scaledhibound = convert_numeric_to_scalar(hibound, boundstypid,
                                                       &failure);
            return !failure;

            /*
             * Built-in string types
             */
        case CHAROID:
        case BPCHAROID:
        case VARCHAROID:
        case TEXTOID:
        case NAMEOID:
            {
                char       *valstr = convert_string_datum(value, valuetypid,
                                                          collid, &failure);
                char       *lostr = convert_string_datum(lobound, boundstypid,
                                                         collid, &failure);
                char       *histr = convert_string_datum(hibound, boundstypid,
                                                         collid, &failure);

                /*
                 * Bail out if any of the values is not of string type.  We
                 * might leak converted strings for the other value(s), but
                 * that's not worth troubling over.
                 */
                if (failure)
                    return false;

                convert_string_to_scalar(valstr, scaledvalue,
                                         lostr, scaledlobound,
                                         histr, scaledhibound);
                pfree(valstr);
                pfree(lostr);
                pfree(histr);
                return true;
            }

            /*
             * Built-in bytea type
             */
        case BYTEAOID:
            {
                /* We only support bytea vs bytea comparison */
                if (boundstypid != BYTEAOID)
                    return false;
                convert_bytea_to_scalar(value, scaledvalue,
                                        lobound, scaledlobound,
                                        hibound, scaledhibound);
                return true;
            }

            /*
             * Built-in time types
             */
        case TIMESTAMPOID:
        case TIMESTAMPTZOID:
        case DATEOID:
        case INTERVALOID:
        case TIMEOID:
        case TIMETZOID:
            *scaledvalue = convert_timevalue_to_scalar(value, valuetypid,
                                                       &failure);
            *scaledlobound = convert_timevalue_to_scalar(lobound, boundstypid,
                                                         &failure);
            *scaledhibound = convert_timevalue_to_scalar(hibound, boundstypid,
                                                         &failure);
            return !failure;

            /*
             * Built-in network types
             */
        case INETOID:
        case CIDROID:
        case MACADDROID:
        case MACADDR8OID:
            *scaledvalue = convert_network_to_scalar(value, valuetypid,
                                                     &failure);
            *scaledlobound = convert_network_to_scalar(lobound, boundstypid,
                                                       &failure);
            *scaledhibound = convert_network_to_scalar(hibound, boundstypid,
                                                       &failure);
            return !failure;
    }
    /* Don't know how to convert */
    *scaledvalue = *scaledlobound = *scaledhibound = 0;
    return false;
}

/*
 * Do convert_to_scalar()'s work for any numeric data type.
 *
 * On failure (e.g., unsupported typid), set *failure to true;
 * otherwise, that variable is not changed.
 */
static double
convert_numeric_to_scalar(Datum value, Oid typid, bool *failure)
{
    switch (typid)
    {
        case BOOLOID:
            return (double) DatumGetBool(value);
        case INT2OID:
            return (double) DatumGetInt16(value);
        case INT4OID:
            return (double) DatumGetInt32(value);
        case INT8OID:
            return (double) DatumGetInt64(value);
        case FLOAT4OID:
            return (double) DatumGetFloat4(value);
        case FLOAT8OID:
            return (double) DatumGetFloat8(value);
        case NUMERICOID:
            /* Note: out-of-range values will be clamped to +-HUGE_VAL */
            return (double)
                DatumGetFloat8(DirectFunctionCall1(numeric_float8_no_overflow,
                                                   value));
        case OIDOID:
        case REGPROCOID:
        case REGPROCEDUREOID:
        case REGOPEROID:
        case REGOPERATOROID:
        case REGCLASSOID:
        case REGTYPEOID:
        case REGCONFIGOID:
        case REGDICTIONARYOID:
        case REGROLEOID:
        case REGNAMESPACEOID:
            /* we can treat OIDs as integers... */
            return (double) DatumGetObjectId(value);
    }

    *failure = true;
    return 0;
}

/*
 * Do convert_to_scalar()'s work for any character-string data type.
 *
 * String datatypes are converted to a scale that ranges from 0 to 1,
 * where we visualize the bytes of the string as fractional digits.
 *
 * We do not want the base to be 256, however, since that tends to
 * generate inflated selectivity estimates; few databases will have
 * occurrences of all 256 possible byte values at each position.
 * Instead, use the smallest and largest byte values seen in the bounds
 * as the estimated range for each byte, after some fudging to deal with
 * the fact that we probably aren't going to see the full range that way.
 *
 * An additional refinement is that we discard any common prefix of the
 * three strings before computing the scaled values.  This allows us to
 * "zoom in" when we encounter a narrow data range.  An example is a phone
 * number database where all the values begin with the same area code.
 * (Actually, the bounds will be adjacent histogram-bin-boundary values,
 * so this is more likely to happen than you might think.)
 */
static void
convert_string_to_scalar(char *value,
                         double *scaledvalue,
                         char *lobound,
                         double *scaledlobound,
                         char *hibound,
                         double *scaledhibound)
{
    int         rangelo,
                rangehi;
    char       *sptr;

    rangelo = rangehi = (unsigned char) hibound[0];
    for (sptr = lobound; *sptr; sptr++)
    {
        if (rangelo > (unsigned char) *sptr)
            rangelo = (unsigned char) *sptr;
        if (rangehi < (unsigned char) *sptr)
            rangehi = (unsigned char) *sptr;
    }
    for (sptr = hibound; *sptr; sptr++)
    {
        if (rangelo > (unsigned char) *sptr)
            rangelo = (unsigned char) *sptr;
        if (rangehi < (unsigned char) *sptr)
            rangehi = (unsigned char) *sptr;
    }
    /* If range includes any upper-case ASCII chars, make it include all */
    if (rangelo <= 'Z' && rangehi >= 'A')
    {
        if (rangelo > 'A')
            rangelo = 'A';
        if (rangehi < 'Z')
            rangehi = 'Z';
    }
    /* Ditto lower-case */
    if (rangelo <= 'z' && rangehi >= 'a')
    {
        if (rangelo > 'a')
            rangelo = 'a';
        if (rangehi < 'z')
            rangehi = 'z';
    }
    /* Ditto digits */
    if (rangelo <= '9' && rangehi >= '0')
    {
        if (rangelo > '0')
            rangelo = '0';
        if (rangehi < '9')
            rangehi = '9';
    }

    /*
     * If range includes less than 10 chars, assume we have not got enough
     * data, and make it include regular ASCII set.
     */
    if (rangehi - rangelo < 9)
    {
        rangelo = ' ';
        rangehi = 127;
    }

    /*
     * Now strip any common prefix of the three strings.
     */
    while (*lobound)
    {
        if (*lobound != *hibound || *lobound != *value)
            break;
        lobound++, hibound++, value++;
    }

    /*
     * Now we can do the conversions.
     */
    *scaledvalue = convert_one_string_to_scalar(value, rangelo, rangehi);
    *scaledlobound = convert_one_string_to_scalar(lobound, rangelo, rangehi);
    *scaledhibound = convert_one_string_to_scalar(hibound, rangelo, rangehi);
}

static double
convert_one_string_to_scalar(char *value, int rangelo, int rangehi)
{
    int         slen = strlen(value);
    double      num,
                denom,
                base;

    if (slen <= 0)
        return 0.0;             /* empty string has scalar value 0 */

    /*
     * There seems little point in considering more than a dozen bytes from
     * the string.  Since base is at least 10, that will give us nominal
     * resolution of at least 12 decimal digits, which is surely far more
     * precision than this estimation technique has got anyway (especially in
     * non-C locales).  Also, even with the maximum possible base of 256, this
     * ensures denom cannot grow larger than 256^13 = 2.03e31, which will not
     * overflow on any known machine.
     */
    if (slen > 12)
        slen = 12;

    /* Convert initial characters to fraction */
    base = rangehi - rangelo + 1;
    num = 0.0;
    denom = base;
    while (slen-- > 0)
    {
        int         ch = (unsigned char) *value++;

        if (ch < rangelo)
            ch = rangelo - 1;
        else if (ch > rangehi)
            ch = rangehi + 1;
        num += ((double) (ch - rangelo)) / denom;
        denom *= base;
    }

    return num;
}

/*
 * Convert a string-type Datum into a palloc'd, null-terminated string.
 *
 * On failure (e.g., unsupported typid), set *failure to true;
 * otherwise, that variable is not changed.  (We'll return NULL on failure.)
 *
 * When using a non-C locale, we must pass the string through strxfrm()
 * before continuing, so as to generate correct locale-specific results.
 */
static char *
convert_string_datum(Datum value, Oid typid, Oid collid, bool *failure)
{
    char       *val;

    switch (typid)
    {
        case CHAROID:
            val = (char *) palloc(2);
            val[0] = DatumGetChar(value);
            val[1] = '\0';
            break;
        case BPCHAROID:
        case VARCHAROID:
        case TEXTOID:
            val = TextDatumGetCString(value);
            break;
        case NAMEOID:
            {
                NameData   *nm = (NameData *) DatumGetPointer(value);

                val = pstrdup(NameStr(*nm));
                break;
            }
        default:
            *failure = true;
            return NULL;
    }

    if (!lc_collate_is_c(collid))
    {
        char       *xfrmstr;
        size_t      xfrmlen;
        size_t      xfrmlen2 PG_USED_FOR_ASSERTS_ONLY;

        /*
         * XXX: We could guess at a suitable output buffer size and only call
         * strxfrm twice if our guess is too small.
         *
         * XXX: strxfrm doesn't support UTF-8 encoding on Win32, it can return
         * bogus data or set an error. This is not really a problem unless it
         * crashes since it will only give an estimation error and nothing
         * fatal.
         */
        xfrmlen = strxfrm(NULL, val, 0);
#ifdef WIN32

        /*
         * On Windows, strxfrm returns INT_MAX when an error occurs. Instead
         * of trying to allocate this much memory (and fail), just return the
         * original string unmodified as if we were in the C locale.
         */
        if (xfrmlen == INT_MAX)
            return val;
#endif
        xfrmstr = (char *) palloc(xfrmlen + 1);
        xfrmlen2 = strxfrm(xfrmstr, val, xfrmlen + 1);

        /*
         * Some systems (e.g., glibc) can return a smaller value from the
         * second call than the first; thus the Assert must be <= not ==.
         */
        Assert(xfrmlen2 <= xfrmlen);
        pfree(val);
        val = xfrmstr;
    }

    return val;
}

/*
 * Do convert_to_scalar()'s work for any bytea data type.
 *
 * Very similar to convert_string_to_scalar except we can't assume
 * null-termination and therefore pass explicit lengths around.
 *
 * Also, assumptions about likely "normal" ranges of characters have been
 * removed - a data range of 0..255 is always used, for now.  (Perhaps
 * someday we will add information about actual byte data range to
 * pg_statistic.)
 */
static void
convert_bytea_to_scalar(Datum value,
                        double *scaledvalue,
                        Datum lobound,
                        double *scaledlobound,
                        Datum hibound,
                        double *scaledhibound)
{
    bytea      *valuep = DatumGetByteaPP(value);
    bytea      *loboundp = DatumGetByteaPP(lobound);
    bytea      *hiboundp = DatumGetByteaPP(hibound);
    int         rangelo,
                rangehi,
                valuelen = VARSIZE_ANY_EXHDR(valuep),
                loboundlen = VARSIZE_ANY_EXHDR(loboundp),
                hiboundlen = VARSIZE_ANY_EXHDR(hiboundp),
                i,
                minlen;
    unsigned char *valstr = (unsigned char *) VARDATA_ANY(valuep);
    unsigned char *lostr = (unsigned char *) VARDATA_ANY(loboundp);
    unsigned char *histr = (unsigned char *) VARDATA_ANY(hiboundp);

    /*
     * Assume bytea data is uniformly distributed across all byte values.
     */
    rangelo = 0;
    rangehi = 255;

    /*
     * Now strip any common prefix of the three strings.
     */
    minlen = Min(Min(valuelen, loboundlen), hiboundlen);
    for (i = 0; i < minlen; i++)
    {
        if (*lostr != *histr || *lostr != *valstr)
            break;
        lostr++, histr++, valstr++;
        loboundlen--, hiboundlen--, valuelen--;
    }

    /*
     * Now we can do the conversions.
     */
    *scaledvalue = convert_one_bytea_to_scalar(valstr, valuelen, rangelo, rangehi);
    *scaledlobound = convert_one_bytea_to_scalar(lostr, loboundlen, rangelo, rangehi);
    *scaledhibound = convert_one_bytea_to_scalar(histr, hiboundlen, rangelo, rangehi);
}

static double
convert_one_bytea_to_scalar(unsigned char *value, int valuelen,
                            int rangelo, int rangehi)
{
    double      num,
                denom,
                base;

    if (valuelen <= 0)
        return 0.0;             /* empty string has scalar value 0 */

    /*
     * Since base is 256, need not consider more than about 10 chars (even
     * this many seems like overkill)
     */
    if (valuelen > 10)
        valuelen = 10;

    /* Convert initial characters to fraction */
    base = rangehi - rangelo + 1;
    num = 0.0;
    denom = base;
    while (valuelen-- > 0)
    {
        int         ch = *value++;

        if (ch < rangelo)
            ch = rangelo - 1;
        else if (ch > rangehi)
            ch = rangehi + 1;
        num += ((double) (ch - rangelo)) / denom;
        denom *= base;
    }

    return num;
}

/*
 * Do convert_to_scalar()'s work for any timevalue data type.
 *
 * On failure (e.g., unsupported typid), set *failure to true;
 * otherwise, that variable is not changed.
 */
static double
convert_timevalue_to_scalar(Datum value, Oid typid, bool *failure)
{
    switch (typid)
    {
        case TIMESTAMPOID:
            return DatumGetTimestamp(value);
        case TIMESTAMPTZOID:
            return DatumGetTimestampTz(value);
        case DATEOID:
            return date2timestamp_no_overflow(DatumGetDateADT(value));
        case INTERVALOID:
            {
                Interval   *interval = DatumGetIntervalP(value);

                /*
                 * Convert the month part of Interval to days using assumed
                 * average month length of 365.25/12.0 days.  Not too
                 * accurate, but plenty good enough for our purposes.
                 */
                return interval->time + interval->day * (double) USECS_PER_DAY +
                    interval->month * ((DAYS_PER_YEAR / (double) MONTHS_PER_YEAR) * USECS_PER_DAY);
            }
        case TIMEOID:
            return DatumGetTimeADT(value);
        case TIMETZOID:
            {
                TimeTzADT  *timetz = DatumGetTimeTzADTP(value);

                /* use GMT-equivalent time */
                return (double) (timetz->time + (timetz->zone * 1000000.0));
            }
    }

    *failure = true;
    return 0;
}


/*
 * get_restriction_variable
 *      Examine the args of a restriction clause to see if it's of the
 *      form (variable op pseudoconstant) or (pseudoconstant op variable),
 *      where "variable" could be either a Var or an expression in vars of a
 *      single relation.  If so, extract information about the variable,
 *      and also indicate which side it was on and the other argument.
 *
 * Inputs:
 *  root: the planner info
 *  args: clause argument list
 *  varRelid: see specs for restriction selectivity functions
 *
 * Outputs: (these are valid only if true is returned)
 *  *vardata: gets information about variable (see examine_variable)
 *  *other: gets other clause argument, aggressively reduced to a constant
 *  *varonleft: set true if variable is on the left, false if on the right
 *
 * Returns true if a variable is identified, otherwise false.
 *
 * Note: if there are Vars on both sides of the clause, we must fail, because
 * callers are expecting that the other side will act like a pseudoconstant.
 */
bool
get_restriction_variable(PlannerInfo *root, List *args, int varRelid,
                         VariableStatData *vardata, Node **other,
                         bool *varonleft)
{
    Node       *left,
               *right;
    VariableStatData rdata;

    /* Fail if not a binary opclause (probably shouldn't happen) */
    if (list_length(args) != 2)
        return false;

    left = (Node *) linitial(args);
    right = (Node *) lsecond(args);

    /*
     * Examine both sides.  Note that when varRelid is nonzero, Vars of other
     * relations will be treated as pseudoconstants.
     */
    examine_variable(root, left, varRelid, vardata);
    examine_variable(root, right, varRelid, &rdata);

    /*
     * If one side is a variable and the other not, we win.
     */
    if (vardata->rel && rdata.rel == NULL)
    {
        *varonleft = true;
        *other = estimate_expression_value(root, rdata.var);
        /* Assume we need no ReleaseVariableStats(rdata) here */
        return true;
    }

    if (vardata->rel == NULL && rdata.rel)
    {
        *varonleft = false;
        *other = estimate_expression_value(root, vardata->var);
        /* Assume we need no ReleaseVariableStats(*vardata) here */
        *vardata = rdata;
        return true;
    }

    /* Oops, clause has wrong structure (probably var op var) */
    ReleaseVariableStats(*vardata);
    ReleaseVariableStats(rdata);

    return false;
}

/*
 * get_join_variables
 *      Apply examine_variable() to each side of a join clause.
 *      Also, attempt to identify whether the join clause has the same
 *      or reversed sense compared to the SpecialJoinInfo.
 *
 * We consider the join clause "normal" if it is "lhs_var OP rhs_var",
 * or "reversed" if it is "rhs_var OP lhs_var".  In complicated cases
 * where we can't tell for sure, we default to assuming it's normal.
 */
void
get_join_variables(PlannerInfo *root, List *args, SpecialJoinInfo *sjinfo,
                   VariableStatData *vardata1, VariableStatData *vardata2,
                   bool *join_is_reversed)
{
    Node       *left,
               *right;

    if (list_length(args) != 2)
        elog(ERROR, "join operator should take two arguments");

    left = (Node *) linitial(args);
    right = (Node *) lsecond(args);

    examine_variable(root, left, 0, vardata1);
    examine_variable(root, right, 0, vardata2);

    if (vardata1->rel &&
        bms_is_subset(vardata1->rel->relids, sjinfo->syn_righthand))
        *join_is_reversed = true;   /* var1 is on RHS */
    else if (vardata2->rel &&
             bms_is_subset(vardata2->rel->relids, sjinfo->syn_lefthand))
        *join_is_reversed = true;   /* var2 is on LHS */
    else
        *join_is_reversed = false;
}

/*
 * examine_variable
 *      Try to look up statistical data about an expression.
 *      Fill in a VariableStatData struct to describe the expression.
 *
 * Inputs:
 *  root: the planner info
 *  node: the expression tree to examine
 *  varRelid: see specs for restriction selectivity functions
 *
 * Outputs: *vardata is filled as follows:
 *  var: the input expression (with any binary relabeling stripped, if
 *      it is or contains a variable; but otherwise the type is preserved)
 *  rel: RelOptInfo for relation containing variable; NULL if expression
 *      contains no Vars (NOTE this could point to a RelOptInfo of a
 *      subquery, not one in the current query).
 *  statsTuple: the pg_statistic entry for the variable, if one exists;
 *      otherwise NULL.
 *  freefunc: pointer to a function to release statsTuple with.
 *  vartype: exposed type of the expression; this should always match
 *      the declared input type of the operator we are estimating for.
 *  atttype, atttypmod: actual type/typmod of the "var" expression.  This is
 *      commonly the same as the exposed type of the variable argument,
 *      but can be different in binary-compatible-type cases.
 *  isunique: true if we were able to match the var to a unique index or a
 *      single-column DISTINCT clause, implying its values are unique for
 *      this query.  (Caution: this should be trusted for statistical
 *      purposes only, since we do not check indimmediate nor verify that
 *      the exact same definition of equality applies.)
 *  acl_ok: true if current user has permission to read the column(s)
 *      underlying the pg_statistic entry.  This is consulted by
 *      statistic_proc_security_check().
 *
 * Caller is responsible for doing ReleaseVariableStats() before exiting.
 */
void
examine_variable(PlannerInfo *root, Node *node, int varRelid,
                 VariableStatData *vardata)
{
    Node       *basenode;
    Relids      varnos;
    RelOptInfo *onerel;

    /* Make sure we don't return dangling pointers in vardata */
    MemSet(vardata, 0, sizeof(VariableStatData));

    /* Save the exposed type of the expression */
    vardata->vartype = exprType(node);

    /* Look inside any binary-compatible relabeling */

    if (IsA(node, RelabelType))
        basenode = (Node *) ((RelabelType *) node)->arg;
    else
        basenode = node;

    /* Fast path for a simple Var */

    if (IsA(basenode, Var) &&
        (varRelid == 0 || varRelid == ((Var *) basenode)->varno))
    {
        Var        *var = (Var *) basenode;

        /* Set up result fields other than the stats tuple */
        vardata->var = basenode;    /* return Var without relabeling */
        vardata->rel = find_base_rel(root, var->varno);
        vardata->atttype = var->vartype;
        vardata->atttypmod = var->vartypmod;
        vardata->isunique = has_unique_index(vardata->rel, var->varattno);

        /* Try to locate some stats */
        examine_simple_variable(root, var, vardata);

        return;
    }

    /*
     * Okay, it's a more complicated expression.  Determine variable
     * membership.  Note that when varRelid isn't zero, only vars of that
     * relation are considered "real" vars.
     */
    varnos = pull_varnos(basenode);

    onerel = NULL;

    switch (bms_membership(varnos))
    {
        case BMS_EMPTY_SET:
            /* No Vars at all ... must be pseudo-constant clause */
            break;
        case BMS_SINGLETON:
            if (varRelid == 0 || bms_is_member(varRelid, varnos))
            {
                onerel = find_base_rel(root,
                                       (varRelid ? varRelid : bms_singleton_member(varnos)));
                vardata->rel = onerel;
                node = basenode;    /* strip any relabeling */
            }
            /* else treat it as a constant */
            break;
        case BMS_MULTIPLE:
            if (varRelid == 0)
            {
                /* treat it as a variable of a join relation */
                vardata->rel = find_join_rel(root, varnos);
                node = basenode;    /* strip any relabeling */
            }
            else if (bms_is_member(varRelid, varnos))
            {
                /* ignore the vars belonging to other relations */
                vardata->rel = find_base_rel(root, varRelid);
                node = basenode;    /* strip any relabeling */
                /* note: no point in expressional-index search here */
            }
            /* else treat it as a constant */
            break;
    }

    bms_free(varnos);

    vardata->var = node;
    vardata->atttype = exprType(node);
    vardata->atttypmod = exprTypmod(node);

    if (onerel)
    {
        /*
         * We have an expression in vars of a single relation.  Try to match
         * it to expressional index columns, in hopes of finding some
         * statistics.
         *
         * Note that we consider all index columns including INCLUDE columns,
         * since there could be stats for such columns.  But the test for
         * uniqueness needs to be warier.
         *
         * XXX it's conceivable that there are multiple matches with different
         * index opfamilies; if so, we need to pick one that matches the
         * operator we are estimating for.  FIXME later.
         */
        ListCell   *ilist;

        foreach(ilist, onerel->indexlist)
        {
            IndexOptInfo *index = (IndexOptInfo *) lfirst(ilist);
            ListCell   *indexpr_item;
            int         pos;

            indexpr_item = list_head(index->indexprs);
            if (indexpr_item == NULL)
                continue;       /* no expressions here... */

            for (pos = 0; pos < index->ncolumns; pos++)
            {
                if (index->indexkeys[pos] == 0)
                {
                    Node       *indexkey;

                    if (indexpr_item == NULL)
                        elog(ERROR, "too few entries in indexprs list");
                    indexkey = (Node *) lfirst(indexpr_item);
                    if (indexkey && IsA(indexkey, RelabelType))
                        indexkey = (Node *) ((RelabelType *) indexkey)->arg;
                    if (equal(node, indexkey))
                    {
                        /*
                         * Found a match ... is it a unique index? Tests here
                         * should match has_unique_index().
                         */
                        if (index->unique &&
                            index->nkeycolumns == 1 &&
                            pos == 0 &&
                            (index->indpred == NIL || index->predOK))
                            vardata->isunique = true;

                        /*
                         * Has it got stats?  We only consider stats for
                         * non-partial indexes, since partial indexes probably
                         * don't reflect whole-relation statistics; the above
                         * check for uniqueness is the only info we take from
                         * a partial index.
                         *
                         * An index stats hook, however, must make its own
                         * decisions about what to do with partial indexes.
                         */
                        if (get_index_stats_hook &&
                            (*get_index_stats_hook) (root, index->indexoid,
                                                     pos + 1, vardata))
                        {
                            /*
                             * The hook took control of acquiring a stats
                             * tuple.  If it did supply a tuple, it'd better
                             * have supplied a freefunc.
                             */
                            if (HeapTupleIsValid(vardata->statsTuple) &&
                                !vardata->freefunc)
                                elog(ERROR, "no function provided to release variable stats with");
                        }
                        else if (index->indpred == NIL)
                        {
                            vardata->statsTuple =
                                SearchSysCache3(STATRELATTINH,
                                                ObjectIdGetDatum(index->indexoid),
                                                Int16GetDatum(pos + 1),
                                                BoolGetDatum(false));
                            vardata->freefunc = ReleaseSysCache;

                            if (HeapTupleIsValid(vardata->statsTuple))
                            {
                                /* Get index's table for permission check */
                                RangeTblEntry *rte;
                                Oid         userid;

                                rte = planner_rt_fetch(index->rel->relid, root);
                                Assert(rte->rtekind == RTE_RELATION);

                                /*
                                 * Use checkAsUser if it's set, in case we're
                                 * accessing the table via a view.
                                 */
                                userid = rte->checkAsUser ? rte->checkAsUser : GetUserId();

                                /*
                                 * For simplicity, we insist on the whole
                                 * table being selectable, rather than trying
                                 * to identify which column(s) the index
                                 * depends on.  Also require all rows to be
                                 * selectable --- there must be no
                                 * securityQuals from security barrier views
                                 * or RLS policies.
                                 */
                                vardata->acl_ok =
                                    rte->securityQuals == NIL &&
                                    (pg_class_aclcheck(rte->relid, userid,
                                                       ACL_SELECT) == ACLCHECK_OK);

                                /*
                                 * If the user doesn't have permissions to
                                 * access an inheritance child relation, check
                                 * the permissions of the table actually
                                 * mentioned in the query, since most likely
                                 * the user does have that permission.  Note
                                 * that whole-table select privilege on the
                                 * parent doesn't quite guarantee that the
                                 * user could read all columns of the child.
                                 * But in practice it's unlikely that any
                                 * interesting security violation could result
                                 * from allowing access to the expression
                                 * index's stats, so we allow it anyway.  See
                                 * similar code in examine_simple_variable()
                                 * for additional comments.
                                 */
                                if (!vardata->acl_ok &&
                                    root->append_rel_array != NULL)
                                {
                                    AppendRelInfo *appinfo;
                                    Index       varno = index->rel->relid;

                                    appinfo = root->append_rel_array[varno];
                                    while (appinfo &&
                                           planner_rt_fetch(appinfo->parent_relid,
                                                            root)->rtekind == RTE_RELATION)
                                    {
                                        varno = appinfo->parent_relid;
                                        appinfo = root->append_rel_array[varno];
                                    }
                                    if (varno != index->rel->relid)
                                    {
                                        /* Repeat access check on this rel */
                                        rte = planner_rt_fetch(varno, root);
                                        Assert(rte->rtekind == RTE_RELATION);

                                        userid = rte->checkAsUser ? rte->checkAsUser : GetUserId();

                                        vardata->acl_ok =
                                            rte->securityQuals == NIL &&
                                            (pg_class_aclcheck(rte->relid,
                                                               userid,
                                                               ACL_SELECT) == ACLCHECK_OK);
                                    }
                                }
                            }
                            else
                            {
                                /* suppress leakproofness checks later */
                                vardata->acl_ok = true;
                            }
                        }
                        if (vardata->statsTuple)
                            break;
                    }
                    indexpr_item = lnext(index->indexprs, indexpr_item);
                }
            }
            if (vardata->statsTuple)
                break;
        }
    }
}

/*
 * examine_simple_variable
 *      Handle a simple Var for examine_variable
 *
 * This is split out as a subroutine so that we can recurse to deal with
 * Vars referencing subqueries.
 *
 * We already filled in all the fields of *vardata except for the stats tuple.
 */
static void
examine_simple_variable(PlannerInfo *root, Var *var,
                        VariableStatData *vardata)
{
    RangeTblEntry *rte = root->simple_rte_array[var->varno];

    Assert(IsA(rte, RangeTblEntry));

    if (get_relation_stats_hook &&
        (*get_relation_stats_hook) (root, rte, var->varattno, vardata))
    {
        /*
         * The hook took control of acquiring a stats tuple.  If it did supply
         * a tuple, it'd better have supplied a freefunc.
         */
        if (HeapTupleIsValid(vardata->statsTuple) &&
            !vardata->freefunc)
            elog(ERROR, "no function provided to release variable stats with");
    }
    else if (rte->rtekind == RTE_RELATION)
    {
        /*
         * Plain table or parent of an inheritance appendrel, so look up the
         * column in pg_statistic
         */
        vardata->statsTuple = SearchSysCache3(STATRELATTINH,
                                              ObjectIdGetDatum(rte->relid),
                                              Int16GetDatum(var->varattno),
                                              BoolGetDatum(rte->inh));
        vardata->freefunc = ReleaseSysCache;

        if (HeapTupleIsValid(vardata->statsTuple))
        {
            Oid         userid;

            /*
             * Check if user has permission to read this column.  We require
             * all rows to be accessible, so there must be no securityQuals
             * from security barrier views or RLS policies.  Use checkAsUser
             * if it's set, in case we're accessing the table via a view.
             */
            userid = rte->checkAsUser ? rte->checkAsUser : GetUserId();

            vardata->acl_ok =
                rte->securityQuals == NIL &&
                ((pg_class_aclcheck(rte->relid, userid,
                                    ACL_SELECT) == ACLCHECK_OK) ||
                 (pg_attribute_aclcheck(rte->relid, var->varattno, userid,
                                        ACL_SELECT) == ACLCHECK_OK));

            /*
             * If the user doesn't have permissions to access an inheritance
             * child relation or specifically this attribute, check the
             * permissions of the table/column actually mentioned in the
             * query, since most likely the user does have that permission
             * (else the query will fail at runtime), and if the user can read
             * the column there then he can get the values of the child table
             * too.  To do that, we must find out which of the root parent's
             * attributes the child relation's attribute corresponds to.
             */
            if (!vardata->acl_ok && var->varattno > 0 &&
                root->append_rel_array != NULL)
            {
                AppendRelInfo *appinfo;
                Index       varno = var->varno;
                int         varattno = var->varattno;
                bool        found = false;

                appinfo = root->append_rel_array[varno];

                /*
                 * Partitions are mapped to their immediate parent, not the
                 * root parent, so must be ready to walk up multiple
                 * AppendRelInfos.  But stop if we hit a parent that is not
                 * RTE_RELATION --- that's a flattened UNION ALL subquery, not
                 * an inheritance parent.
                 */
                while (appinfo &&
                       planner_rt_fetch(appinfo->parent_relid,
                                        root)->rtekind == RTE_RELATION)
                {
                    int         parent_varattno;

                    found = false;
                    if (varattno <= 0 || varattno > appinfo->num_child_cols)
                        break;  /* safety check */
                    parent_varattno = appinfo->parent_colnos[varattno - 1];
                    if (parent_varattno == 0)
                        break;  /* Var is local to child */

                    varno = appinfo->parent_relid;
                    varattno = parent_varattno;
                    found = true;

                    /* If the parent is itself a child, continue up. */
                    appinfo = root->append_rel_array[varno];
                }

                /*
                 * In rare cases, the Var may be local to the child table, in
                 * which case, we've got to live with having no access to this
                 * column's stats.
                 */
                if (!found)
                    return;

                /* Repeat the access check on this parent rel & column */
                rte = planner_rt_fetch(varno, root);
                Assert(rte->rtekind == RTE_RELATION);

                userid = rte->checkAsUser ? rte->checkAsUser : GetUserId();

                vardata->acl_ok =
                    rte->securityQuals == NIL &&
                    ((pg_class_aclcheck(rte->relid, userid,
                                        ACL_SELECT) == ACLCHECK_OK) ||
                     (pg_attribute_aclcheck(rte->relid, varattno, userid,
                                            ACL_SELECT) == ACLCHECK_OK));
            }
        }
        else
        {
            /* suppress any possible leakproofness checks later */
            vardata->acl_ok = true;
        }
    }
    else if (rte->rtekind == RTE_SUBQUERY && !rte->inh)
    {
        /*
         * Plain subquery (not one that was converted to an appendrel).
         */
        Query      *subquery = rte->subquery;
        RelOptInfo *rel;
        TargetEntry *ste;

        /*
         * Punt if it's a whole-row var rather than a plain column reference.
         */
        if (var->varattno == InvalidAttrNumber)
            return;

        /*
         * Punt if subquery uses set operations or GROUP BY, as these will
         * mash underlying columns' stats beyond recognition.  (Set ops are
         * particularly nasty; if we forged ahead, we would return stats
         * relevant to only the leftmost subselect...)  DISTINCT is also
         * problematic, but we check that later because there is a possibility
         * of learning something even with it.
         */
        if (subquery->setOperations ||
            subquery->groupClause)
            return;

        /*
         * OK, fetch RelOptInfo for subquery.  Note that we don't change the
         * rel returned in vardata, since caller expects it to be a rel of the
         * caller's query level.  Because we might already be recursing, we
         * can't use that rel pointer either, but have to look up the Var's
         * rel afresh.
         */
        rel = find_base_rel(root, var->varno);

        /* If the subquery hasn't been planned yet, we have to punt */
        if (rel->subroot == NULL)
            return;
        Assert(IsA(rel->subroot, PlannerInfo));

        /*
         * Switch our attention to the subquery as mangled by the planner. It
         * was okay to look at the pre-planning version for the tests above,
         * but now we need a Var that will refer to the subroot's live
         * RelOptInfos.  For instance, if any subquery pullup happened during
         * planning, Vars in the targetlist might have gotten replaced, and we
         * need to see the replacement expressions.
         */
        subquery = rel->subroot->parse;
        Assert(IsA(subquery, Query));

        /* Get the subquery output expression referenced by the upper Var */
        ste = get_tle_by_resno(subquery->targetList, var->varattno);
        if (ste == NULL || ste->resjunk)
            elog(ERROR, "subquery %s does not have attribute %d",
                 rte->eref->aliasname, var->varattno);
        var = (Var *) ste->expr;

        /*
         * If subquery uses DISTINCT, we can't make use of any stats for the
         * variable ... but, if it's the only DISTINCT column, we are entitled
         * to consider it unique.  We do the test this way so that it works
         * for cases involving DISTINCT ON.
         */
        if (subquery->distinctClause)
        {
            if (list_length(subquery->distinctClause) == 1 &&
                targetIsInSortList(ste, InvalidOid, subquery->distinctClause))
                vardata->isunique = true;
            /* cannot go further */
            return;
        }

        /*
         * If the sub-query originated from a view with the security_barrier
         * attribute, we must not look at the variable's statistics, though it
         * seems all right to notice the existence of a DISTINCT clause. So
         * stop here.
         *
         * This is probably a harsher restriction than necessary; it's
         * certainly OK for the selectivity estimator (which is a C function,
         * and therefore omnipotent anyway) to look at the statistics.  But
         * many selectivity estimators will happily *invoke the operator
         * function* to try to work out a good estimate - and that's not OK.
         * So for now, don't dig down for stats.
         */
        if (rte->security_barrier)
            return;

        /* Can only handle a simple Var of subquery's query level */
        if (var && IsA(var, Var) &&
            var->varlevelsup == 0)
        {
            /*
             * OK, recurse into the subquery.  Note that the original setting
             * of vardata->isunique (which will surely be false) is left
             * unchanged in this situation.  That's what we want, since even
             * if the underlying column is unique, the subquery may have
             * joined to other tables in a way that creates duplicates.
             */
            examine_simple_variable(rel->subroot, var, vardata);
        }
    }
    else
    {
        /*
         * Otherwise, the Var comes from a FUNCTION, VALUES, or CTE RTE.  (We
         * won't see RTE_JOIN here because join alias Vars have already been
         * flattened.)  There's not much we can do with function outputs, but
         * maybe someday try to be smarter about VALUES and/or CTEs.
         */
    }
}

/*
 * Check whether it is permitted to call func_oid passing some of the
 * pg_statistic data in vardata.  We allow this either if the user has SELECT
 * privileges on the table or column underlying the pg_statistic data or if
 * the function is marked leak-proof.
 */
bool
statistic_proc_security_check(VariableStatData *vardata, Oid func_oid)
{
    if (vardata->acl_ok)
        return true;

    if (!OidIsValid(func_oid))
        return false;

    if (get_func_leakproof(func_oid))
        return true;

    ereport(DEBUG2,
            (errmsg_internal("not using statistics because function \"%s\" is not leak-proof",
                             get_func_name(func_oid))));
    return false;
}

/*
 * get_variable_numdistinct
 *    Estimate the number of distinct values of a variable.
 *
 * vardata: results of examine_variable
 * *isdefault: set to true if the result is a default rather than based on
 * anything meaningful.
 *
 * NB: be careful to produce a positive integral result, since callers may
 * compare the result to exact integer counts, or might divide by it.
 */
double
get_variable_numdistinct(VariableStatData *vardata, bool *isdefault)
{
    double      stadistinct;
    double      stanullfrac = 0.0;
    double      ntuples;

    *isdefault = false;

    /*
     * Determine the stadistinct value to use.  There are cases where we can
     * get an estimate even without a pg_statistic entry, or can get a better
     * value than is in pg_statistic.  Grab stanullfrac too if we can find it
     * (otherwise, assume no nulls, for lack of any better idea).
     */
    if (HeapTupleIsValid(vardata->statsTuple))
    {
        /* Use the pg_statistic entry */
        Form_pg_statistic stats;

        stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);
        stadistinct = stats->stadistinct;
        stanullfrac = stats->stanullfrac;
    }
    else if (vardata->vartype == BOOLOID)
    {
        /*
         * Special-case boolean columns: presumably, two distinct values.
         *
         * Are there any other datatypes we should wire in special estimates
         * for?
         */
        stadistinct = 2.0;
    }
    else if (vardata->rel && vardata->rel->rtekind == RTE_VALUES)
    {
        /*
         * If the Var represents a column of a VALUES RTE, assume it's unique.
         * This could of course be very wrong, but it should tend to be true
         * in well-written queries.  We could consider examining the VALUES'
         * contents to get some real statistics; but that only works if the
         * entries are all constants, and it would be pretty expensive anyway.
         */
        stadistinct = -1.0;     /* unique (and all non null) */
    }
    else
    {
        /*
         * We don't keep statistics for system columns, but in some cases we
         * can infer distinctness anyway.
         */
        if (vardata->var && IsA(vardata->var, Var))
        {
            switch (((Var *) vardata->var)->varattno)
            {
                case SelfItemPointerAttributeNumber:
                    stadistinct = -1.0; /* unique (and all non null) */
                    break;
                case TableOidAttributeNumber:
                    stadistinct = 1.0;  /* only 1 value */
                    break;
                default:
                    stadistinct = 0.0;  /* means "unknown" */
                    break;
            }
        }
        else
            stadistinct = 0.0;  /* means "unknown" */

        /*
         * XXX consider using estimate_num_groups on expressions?
         */
    }

    /*
     * If there is a unique index or DISTINCT clause for the variable, assume
     * it is unique no matter what pg_statistic says; the statistics could be
     * out of date, or we might have found a partial unique index that proves
     * the var is unique for this query.  However, we'd better still believe
     * the null-fraction statistic.
     */
    if (vardata->isunique)
        stadistinct = -1.0 * (1.0 - stanullfrac);

    /*
     * If we had an absolute estimate, use that.
     */
    if (stadistinct > 0.0)
        return clamp_row_est(stadistinct);

    /*
     * Otherwise we need to get the relation size; punt if not available.
     */
    if (vardata->rel == NULL)
    {
        *isdefault = true;
        return DEFAULT_NUM_DISTINCT;
    }
    ntuples = vardata->rel->tuples;
    if (ntuples <= 0.0)
    {
        *isdefault = true;
        return DEFAULT_NUM_DISTINCT;
    }

    /*
     * If we had a relative estimate, use that.
     */
    if (stadistinct < 0.0)
        return clamp_row_est(-stadistinct * ntuples);

    /*
     * With no data, estimate ndistinct = ntuples if the table is small, else
     * use default.  We use DEFAULT_NUM_DISTINCT as the cutoff for "small" so
     * that the behavior isn't discontinuous.
     */
    if (ntuples < DEFAULT_NUM_DISTINCT)
        return clamp_row_est(ntuples);

    *isdefault = true;
    return DEFAULT_NUM_DISTINCT;
}

/*
 * get_variable_range
 *      Estimate the minimum and maximum value of the specified variable.
 *      If successful, store values in *min and *max, and return true.
 *      If no data available, return false.
 *
 * sortop is the "<" comparison operator to use.  This should generally
 * be "<" not ">", as only the former is likely to be found in pg_statistic.
 */
static bool
get_variable_range(PlannerInfo *root, VariableStatData *vardata, Oid sortop,
                   Datum *min, Datum *max)
{
    Datum       tmin = 0;
    Datum       tmax = 0;
    bool        have_data = false;
    int16       typLen;
    bool        typByVal;
    Oid         opfuncoid;
    AttStatsSlot sslot;
    int         i;

    /*
     * XXX It's very tempting to try to use the actual column min and max, if
     * we can get them relatively-cheaply with an index probe.  However, since
     * this function is called many times during join planning, that could
     * have unpleasant effects on planning speed.  Need more investigation
     * before enabling this.
     */
#ifdef NOT_USED
    if (get_actual_variable_range(root, vardata, sortop, min, max))
        return true;
#endif

    if (!HeapTupleIsValid(vardata->statsTuple))
    {
        /* no stats available, so default result */
        return false;
    }

    /*
     * If we can't apply the sortop to the stats data, just fail.  In
     * principle, if there's a histogram and no MCVs, we could return the
     * histogram endpoints without ever applying the sortop ... but it's
     * probably not worth trying, because whatever the caller wants to do with
     * the endpoints would likely fail the security check too.
     */
    if (!statistic_proc_security_check(vardata,
                                       (opfuncoid = get_opcode(sortop))))
        return false;

    get_typlenbyval(vardata->atttype, &typLen, &typByVal);

    /*
     * If there is a histogram, grab the first and last values.
     *
     * If there is a histogram that is sorted with some other operator than
     * the one we want, fail --- this suggests that there is data we can't
     * use.
     */
    if (get_attstatsslot(&sslot, vardata->statsTuple,
                         STATISTIC_KIND_HISTOGRAM, sortop,
                         ATTSTATSSLOT_VALUES))
    {
        if (sslot.nvalues > 0)
        {
            tmin = datumCopy(sslot.values[0], typByVal, typLen);
            tmax = datumCopy(sslot.values[sslot.nvalues - 1], typByVal, typLen);
            have_data = true;
        }
        free_attstatsslot(&sslot);
    }
    else if (get_attstatsslot(&sslot, vardata->statsTuple,
                              STATISTIC_KIND_HISTOGRAM, InvalidOid,
                              0))
    {
        free_attstatsslot(&sslot);
        return false;
    }

    /*
     * If we have most-common-values info, look for extreme MCVs.  This is
     * needed even if we also have a histogram, since the histogram excludes
     * the MCVs.  However, usually the MCVs will not be the extreme values, so
     * avoid unnecessary data copying.
     */
    if (get_attstatsslot(&sslot, vardata->statsTuple,
                         STATISTIC_KIND_MCV, InvalidOid,
                         ATTSTATSSLOT_VALUES))
    {
        bool        tmin_is_mcv = false;
        bool        tmax_is_mcv = false;
        FmgrInfo    opproc;

        fmgr_info(opfuncoid, &opproc);

        for (i = 0; i < sslot.nvalues; i++)
        {
            if (!have_data)
            {
                tmin = tmax = sslot.values[i];
                tmin_is_mcv = tmax_is_mcv = have_data = true;
                continue;
            }
            if (DatumGetBool(FunctionCall2Coll(&opproc,
                                               sslot.stacoll,
                                               sslot.values[i], tmin)))
            {
                tmin = sslot.values[i];
                tmin_is_mcv = true;
            }
            if (DatumGetBool(FunctionCall2Coll(&opproc,
                                               sslot.stacoll,
                                               tmax, sslot.values[i])))
            {
                tmax = sslot.values[i];
                tmax_is_mcv = true;
            }
        }
        if (tmin_is_mcv)
            tmin = datumCopy(tmin, typByVal, typLen);
        if (tmax_is_mcv)
            tmax = datumCopy(tmax, typByVal, typLen);
        free_attstatsslot(&sslot);
    }

    *min = tmin;
    *max = tmax;
    return have_data;
}


/*
 * get_actual_variable_range
 *      Attempt to identify the current *actual* minimum and/or maximum
 *      of the specified variable, by looking for a suitable btree index
 *      and fetching its low and/or high values.
 *      If successful, store values in *min and *max, and return true.
 *      (Either pointer can be NULL if that endpoint isn't needed.)
 *      If no data available, return false.
 *
 * sortop is the "<" comparison operator to use.
 */
static bool
get_actual_variable_range(PlannerInfo *root, VariableStatData *vardata,
                          Oid sortop,
                          Datum *min, Datum *max)
{
    bool        have_data = false;
    RelOptInfo *rel = vardata->rel;
    RangeTblEntry *rte;
    ListCell   *lc;

    /* No hope if no relation or it doesn't have indexes */
    if (rel == NULL || rel->indexlist == NIL)
        return false;
    /* If it has indexes it must be a plain relation */
    rte = root->simple_rte_array[rel->relid];
    Assert(rte->rtekind == RTE_RELATION);

    /* Search through the indexes to see if any match our problem */
    foreach(lc, rel->indexlist)
    {
        IndexOptInfo *index = (IndexOptInfo *) lfirst(lc);
        ScanDirection indexscandir;

        /* Ignore non-btree indexes */
        if (index->relam != BTREE_AM_OID)
            continue;

        /*
         * Ignore partial indexes --- we only want stats that cover the entire
         * relation.
         */
        if (index->indpred != NIL)
            continue;

        /*
         * The index list might include hypothetical indexes inserted by a
         * get_relation_info hook --- don't try to access them.
         */
        if (index->hypothetical)
            continue;

        /*
         * The first index column must match the desired variable and sort
         * operator --- but we can use a descending-order index.
         */
        if (!match_index_to_operand(vardata->var, 0, index))
            continue;
        switch (get_op_opfamily_strategy(sortop, index->sortopfamily[0]))
        {
            case BTLessStrategyNumber:
                if (index->reverse_sort[0])
                    indexscandir = BackwardScanDirection;
                else
                    indexscandir = ForwardScanDirection;
                break;
            case BTGreaterStrategyNumber:
                if (index->reverse_sort[0])
                    indexscandir = ForwardScanDirection;
                else
                    indexscandir = BackwardScanDirection;
                break;
            default:
                /* index doesn't match the sortop */
                continue;
        }

        /*
         * Found a suitable index to extract data from.  Set up some data that
         * can be used by both invocations of get_actual_variable_endpoint.
         */
        {
            MemoryContext tmpcontext;
            MemoryContext oldcontext;
            Relation    heapRel;
            Relation    indexRel;
            TupleTableSlot *slot;
            int16       typLen;
            bool        typByVal;
            ScanKeyData scankeys[1];

            /* Make sure any cruft gets recycled when we're done */
            tmpcontext = AllocSetContextCreate(CurrentMemoryContext,
                                               "get_actual_variable_range workspace",
                                               ALLOCSET_DEFAULT_SIZES);
            oldcontext = MemoryContextSwitchTo(tmpcontext);

            /*
             * Open the table and index so we can read from them.  We should
             * already have some type of lock on each.
             */
            heapRel = table_open(rte->relid, NoLock);
            indexRel = index_open(index->indexoid, NoLock);

            /* build some stuff needed for indexscan execution */
            slot = table_slot_create(heapRel, NULL);
            get_typlenbyval(vardata->atttype, &typLen, &typByVal);

            /* set up an IS NOT NULL scan key so that we ignore nulls */
            ScanKeyEntryInitialize(&scankeys[0],
                                   SK_ISNULL | SK_SEARCHNOTNULL,
                                   1,   /* index col to scan */
                                   InvalidStrategy, /* no strategy */
                                   InvalidOid,  /* no strategy subtype */
                                   InvalidOid,  /* no collation */
                                   InvalidOid,  /* no reg proc for this */
                                   (Datum) 0);  /* constant */

            /* If min is requested ... */
            if (min)
            {
                have_data = get_actual_variable_endpoint(heapRel,
                                                         indexRel,
                                                         indexscandir,
                                                         scankeys,
                                                         typLen,
                                                         typByVal,
                                                         slot,
                                                         oldcontext,
                                                         min);
            }
            else
            {
                /* If min not requested, assume index is nonempty */
                have_data = true;
            }

            /* If max is requested, and we didn't find the index is empty */
            if (max && have_data)
            {
                /* scan in the opposite direction; all else is the same */
                have_data = get_actual_variable_endpoint(heapRel,
                                                         indexRel,
                                                         -indexscandir,
                                                         scankeys,
                                                         typLen,
                                                         typByVal,
                                                         slot,
                                                         oldcontext,
                                                         max);
            }

            /* Clean everything up */
            ExecDropSingleTupleTableSlot(slot);

            index_close(indexRel, NoLock);
            table_close(heapRel, NoLock);

            MemoryContextSwitchTo(oldcontext);
            MemoryContextDelete(tmpcontext);

            /* And we're done */
            break;
        }
    }

    return have_data;
}

/*
 * Get one endpoint datum (min or max depending on indexscandir) from the
 * specified index.  Return true if successful, false if index is empty.
 * On success, endpoint value is stored to *endpointDatum (and copied into
 * outercontext).
 *
 * scankeys is a 1-element scankey array set up to reject nulls.
 * typLen/typByVal describe the datatype of the index's first column.
 * tableslot is a slot suitable to hold table tuples, in case we need
 * to probe the heap.
 * (We could compute these values locally, but that would mean computing them
 * twice when get_actual_variable_range needs both the min and the max.)
 */
static bool
get_actual_variable_endpoint(Relation heapRel,
                             Relation indexRel,
                             ScanDirection indexscandir,
                             ScanKey scankeys,
                             int16 typLen,
                             bool typByVal,
                             TupleTableSlot *tableslot,
                             MemoryContext outercontext,
                             Datum *endpointDatum)
{
    bool        have_data = false;
    SnapshotData SnapshotNonVacuumable;
    IndexScanDesc index_scan;
    Buffer      vmbuffer = InvalidBuffer;
    ItemPointer tid;
    Datum       values[INDEX_MAX_KEYS];
    bool        isnull[INDEX_MAX_KEYS];
    MemoryContext oldcontext;

    /*
     * We use the index-only-scan machinery for this.  With mostly-static
     * tables that's a win because it avoids a heap visit.  It's also a win
     * for dynamic data, but the reason is less obvious; read on for details.
     *
     * In principle, we should scan the index with our current active
     * snapshot, which is the best approximation we've got to what the query
     * will see when executed.  But that won't be exact if a new snap is taken
     * before running the query, and it can be very expensive if a lot of
     * recently-dead or uncommitted rows exist at the beginning or end of the
     * index (because we'll laboriously fetch each one and reject it).
     * Instead, we use SnapshotNonVacuumable.  That will accept recently-dead
     * and uncommitted rows as well as normal visible rows.  On the other
     * hand, it will reject known-dead rows, and thus not give a bogus answer
     * when the extreme value has been deleted (unless the deletion was quite
     * recent); that case motivates not using SnapshotAny here.
     *
     * A crucial point here is that SnapshotNonVacuumable, with
     * RecentGlobalXmin as horizon, yields the inverse of the condition that
     * the indexscan will use to decide that index entries are killable (see
     * heap_hot_search_buffer()).  Therefore, if the snapshot rejects a tuple
     * (or more precisely, all tuples of a HOT chain) and we have to continue
     * scanning past it, we know that the indexscan will mark that index entry
     * killed.  That means that the next get_actual_variable_endpoint() call
     * will not have to re-consider that index entry.  In this way we avoid
     * repetitive work when this function is used a lot during planning.
     *
     * But using SnapshotNonVacuumable creates a hazard of its own.  In a
     * recently-created index, some index entries may point at "broken" HOT
     * chains in which not all the tuple versions contain data matching the
     * index entry.  The live tuple version(s) certainly do match the index,
     * but SnapshotNonVacuumable can accept recently-dead tuple versions that
     * don't match.  Hence, if we took data from the selected heap tuple, we
     * might get a bogus answer that's not close to the index extremal value,
     * or could even be NULL.  We avoid this hazard because we take the data
     * from the index entry not the heap.
     */
    InitNonVacuumableSnapshot(SnapshotNonVacuumable, RecentGlobalXmin);

    index_scan = index_beginscan(heapRel, indexRel,
                                 &SnapshotNonVacuumable,
                                 1, 0);
    /* Set it up for index-only scan */
    index_scan->xs_want_itup = true;
    index_rescan(index_scan, scankeys, 1, NULL, 0);

    /* Fetch first/next tuple in specified direction */
    while ((tid = index_getnext_tid(index_scan, indexscandir)) != NULL)
    {
        if (!VM_ALL_VISIBLE(heapRel,
                            ItemPointerGetBlockNumber(tid),
                            &vmbuffer))
        {
            /* Rats, we have to visit the heap to check visibility */
            if (!index_fetch_heap(index_scan, tableslot))
                continue;       /* no visible tuple, try next index entry */

            /* We don't actually need the heap tuple for anything */
            ExecClearTuple(tableslot);

            /*
             * We don't care whether there's more than one visible tuple in
             * the HOT chain; if any are visible, that's good enough.
             */
        }

        /*
         * We expect that btree will return data in IndexTuple not HeapTuple
         * format.  It's not lossy either.
         */
        if (!index_scan->xs_itup)
            elog(ERROR, "no data returned for index-only scan");
        if (index_scan->xs_recheck)
            elog(ERROR, "unexpected recheck indication from btree");

        /* OK to deconstruct the index tuple */
        index_deform_tuple(index_scan->xs_itup,
                           index_scan->xs_itupdesc,
                           values, isnull);

        /* Shouldn't have got a null, but be careful */
        if (isnull[0])
            elog(ERROR, "found unexpected null value in index \"%s\"",
                 RelationGetRelationName(indexRel));

        /* Copy the index column value out to caller's context */
        oldcontext = MemoryContextSwitchTo(outercontext);
        *endpointDatum = datumCopy(values[0], typByVal, typLen);
        MemoryContextSwitchTo(oldcontext);
        have_data = true;
        break;
    }

    if (vmbuffer != InvalidBuffer)
        ReleaseBuffer(vmbuffer);
    index_endscan(index_scan);

    return have_data;
}

/*
 * find_join_input_rel
 *      Look up the input relation for a join.
 *
 * We assume that the input relation's RelOptInfo must have been constructed
 * already.
 */
static RelOptInfo *
find_join_input_rel(PlannerInfo *root, Relids relids)
{
    RelOptInfo *rel = NULL;

    switch (bms_membership(relids))
    {
        case BMS_EMPTY_SET:
            /* should not happen */
            break;
        case BMS_SINGLETON:
            rel = find_base_rel(root, bms_singleton_member(relids));
            break;
        case BMS_MULTIPLE:
            rel = find_join_rel(root, relids);
            break;
    }

    if (rel == NULL)
        elog(ERROR, "could not find RelOptInfo for given relids");

    return rel;
}


/*-------------------------------------------------------------------------
 *
 * Index cost estimation functions
 *
 *-------------------------------------------------------------------------
 */

/*
 * Extract the actual indexquals (as RestrictInfos) from an IndexClause list
 */
List *
get_quals_from_indexclauses(List *indexclauses)
{
    List       *result = NIL;
    ListCell   *lc;

    foreach(lc, indexclauses)
    {
        IndexClause *iclause = lfirst_node(IndexClause, lc);
        ListCell   *lc2;

        foreach(lc2, iclause->indexquals)
        {
            RestrictInfo *rinfo = lfirst_node(RestrictInfo, lc2);

            result = lappend(result, rinfo);
        }
    }
    return result;
}

/*
 * Compute the total evaluation cost of the comparison operands in a list
 * of index qual expressions.  Since we know these will be evaluated just
 * once per scan, there's no need to distinguish startup from per-row cost.
 *
 * This can be used either on the result of get_quals_from_indexclauses(),
 * or directly on an indexorderbys list.  In both cases, we expect that the
 * index key expression is on the left side of binary clauses.
 */
Cost
index_other_operands_eval_cost(PlannerInfo *root, List *indexquals)
{
    Cost        qual_arg_cost = 0;
    ListCell   *lc;

    foreach(lc, indexquals)
    {
        Expr       *clause = (Expr *) lfirst(lc);
        Node       *other_operand;
        QualCost    index_qual_cost;

        /*
         * Index quals will have RestrictInfos, indexorderbys won't.  Look
         * through RestrictInfo if present.
         */
        if (IsA(clause, RestrictInfo))
            clause = ((RestrictInfo *) clause)->clause;

        if (IsA(clause, OpExpr))
        {
            OpExpr     *op = (OpExpr *) clause;

            other_operand = (Node *) lsecond(op->args);
        }
        else if (IsA(clause, RowCompareExpr))
        {
            RowCompareExpr *rc = (RowCompareExpr *) clause;

            other_operand = (Node *) rc->rargs;
        }
        else if (IsA(clause, ScalarArrayOpExpr))
        {
            ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) clause;

            other_operand = (Node *) lsecond(saop->args);
        }
        else if (IsA(clause, NullTest))
        {
            other_operand = NULL;
        }
        else
        {
            elog(ERROR, "unsupported indexqual type: %d",
                 (int) nodeTag(clause));
            other_operand = NULL;   /* keep compiler quiet */
        }

        cost_qual_eval_node(&index_qual_cost, other_operand, root);
        qual_arg_cost += index_qual_cost.startup + index_qual_cost.per_tuple;
    }
    return qual_arg_cost;
}

void
genericcostestimate(PlannerInfo *root,
                    IndexPath *path,
                    double loop_count,
                    GenericCosts *costs)
{
    IndexOptInfo *index = path->indexinfo;
    List       *indexQuals = get_quals_from_indexclauses(path->indexclauses);
    List       *indexOrderBys = path->indexorderbys;
    Cost        indexStartupCost;
    Cost        indexTotalCost;
    Selectivity indexSelectivity;
    double      indexCorrelation;
    double      numIndexPages;
    double      numIndexTuples;
    double      spc_random_page_cost;
    double      num_sa_scans;
    double      num_outer_scans;
    double      num_scans;
    double      qual_op_cost;
    double      qual_arg_cost;
    List       *selectivityQuals;
    ListCell   *l;

    /*
     * If the index is partial, AND the index predicate with the explicitly
     * given indexquals to produce a more accurate idea of the index
     * selectivity.
     */
    selectivityQuals = add_predicate_to_index_quals(index, indexQuals);

    /*
     * Check for ScalarArrayOpExpr index quals, and estimate the number of
     * index scans that will be performed.
     */
    num_sa_scans = 1;
    foreach(l, indexQuals)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(l);

        if (IsA(rinfo->clause, ScalarArrayOpExpr))
        {
            ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) rinfo->clause;
            int         alength = estimate_array_length(lsecond(saop->args));

            if (alength > 1)
                num_sa_scans *= alength;
        }
    }

    /* Estimate the fraction of main-table tuples that will be visited */
    indexSelectivity = clauselist_selectivity(root, selectivityQuals,
                                              index->rel->relid,
                                              JOIN_INNER,
                                              NULL);

    /*
     * If caller didn't give us an estimate, estimate the number of index
     * tuples that will be visited.  We do it in this rather peculiar-looking
     * way in order to get the right answer for partial indexes.
     */
    numIndexTuples = costs->numIndexTuples;
    if (numIndexTuples <= 0.0)
    {
        numIndexTuples = indexSelectivity * index->rel->tuples;

        /*
         * The above calculation counts all the tuples visited across all
         * scans induced by ScalarArrayOpExpr nodes.  We want to consider the
         * average per-indexscan number, so adjust.  This is a handy place to
         * round to integer, too.  (If caller supplied tuple estimate, it's
         * responsible for handling these considerations.)
         */
        numIndexTuples = rint(numIndexTuples / num_sa_scans);
    }

    /*
     * We can bound the number of tuples by the index size in any case. Also,
     * always estimate at least one tuple is touched, even when
     * indexSelectivity estimate is tiny.
     */
    if (numIndexTuples > index->tuples)
        numIndexTuples = index->tuples;
    if (numIndexTuples < 1.0)
        numIndexTuples = 1.0;

    /*
     * Estimate the number of index pages that will be retrieved.
     *
     * We use the simplistic method of taking a pro-rata fraction of the total
     * number of index pages.  In effect, this counts only leaf pages and not
     * any overhead such as index metapage or upper tree levels.
     *
     * In practice access to upper index levels is often nearly free because
     * those tend to stay in cache under load; moreover, the cost involved is
     * highly dependent on index type.  We therefore ignore such costs here
     * and leave it to the caller to add a suitable charge if needed.
     */
    if (index->pages > 1 && index->tuples > 1)
        numIndexPages = ceil(numIndexTuples * index->pages / index->tuples);
    else
        numIndexPages = 1.0;

    /* fetch estimated page cost for tablespace containing index */
    get_tablespace_page_costs(index->reltablespace,
                              &spc_random_page_cost,
                              NULL);

    /*
     * Now compute the disk access costs.
     *
     * The above calculations are all per-index-scan.  However, if we are in a
     * nestloop inner scan, we can expect the scan to be repeated (with
     * different search keys) for each row of the outer relation.  Likewise,
     * ScalarArrayOpExpr quals result in multiple index scans.  This creates
     * the potential for cache effects to reduce the number of disk page
     * fetches needed.  We want to estimate the average per-scan I/O cost in
     * the presence of caching.
     *
     * We use the Mackert-Lohman formula (see costsize.c for details) to
     * estimate the total number of page fetches that occur.  While this
     * wasn't what it was designed for, it seems a reasonable model anyway.
     * Note that we are counting pages not tuples anymore, so we take N = T =
     * index size, as if there were one "tuple" per page.
     */
    num_outer_scans = loop_count;
    num_scans = num_sa_scans * num_outer_scans;

    if (num_scans > 1)
    {
        double      pages_fetched;

        /* total page fetches ignoring cache effects */
        pages_fetched = numIndexPages * num_scans;

        /* use Mackert and Lohman formula to adjust for cache effects */
        pages_fetched = index_pages_fetched(pages_fetched,
                                            index->pages,
                                            (double) index->pages,
                                            root);

        /*
         * Now compute the total disk access cost, and then report a pro-rated
         * share for each outer scan.  (Don't pro-rate for ScalarArrayOpExpr,
         * since that's internal to the indexscan.)
         */
        indexTotalCost = (pages_fetched * spc_random_page_cost)
            / num_outer_scans;
    }
    else
    {
        /*
         * For a single index scan, we just charge spc_random_page_cost per
         * page touched.
         */
        indexTotalCost = numIndexPages * spc_random_page_cost;
    }

    /*
     * CPU cost: any complex expressions in the indexquals will need to be
     * evaluated once at the start of the scan to reduce them to runtime keys
     * to pass to the index AM (see nodeIndexscan.c).  We model the per-tuple
     * CPU costs as cpu_index_tuple_cost plus one cpu_operator_cost per
     * indexqual operator.  Because we have numIndexTuples as a per-scan
     * number, we have to multiply by num_sa_scans to get the correct result
     * for ScalarArrayOpExpr cases.  Similarly add in costs for any index
     * ORDER BY expressions.
     *
     * Note: this neglects the possible costs of rechecking lossy operators.
     * Detecting that that might be needed seems more expensive than it's
     * worth, though, considering all the other inaccuracies here ...
     */
    qual_arg_cost = index_other_operands_eval_cost(root, indexQuals) +
        index_other_operands_eval_cost(root, indexOrderBys);
    qual_op_cost = cpu_operator_cost *
        (list_length(indexQuals) + list_length(indexOrderBys));

    indexStartupCost = qual_arg_cost;
    indexTotalCost += qual_arg_cost;
    indexTotalCost += numIndexTuples * num_sa_scans * (cpu_index_tuple_cost + qual_op_cost);

    /*
     * Generic assumption about index correlation: there isn't any.
     */
    indexCorrelation = 0.0;

    /*
     * Return everything to caller.
     */
    costs->indexStartupCost = indexStartupCost;
    costs->indexTotalCost = indexTotalCost;
    costs->indexSelectivity = indexSelectivity;
    costs->indexCorrelation = indexCorrelation;
    costs->numIndexPages = numIndexPages;
    costs->numIndexTuples = numIndexTuples;
    costs->spc_random_page_cost = spc_random_page_cost;
    costs->num_sa_scans = num_sa_scans;
}

/*
 * If the index is partial, add its predicate to the given qual list.
 *
 * ANDing the index predicate with the explicitly given indexquals produces
 * a more accurate idea of the index's selectivity.  However, we need to be
 * careful not to insert redundant clauses, because clauselist_selectivity()
 * is easily fooled into computing a too-low selectivity estimate.  Our
 * approach is to add only the predicate clause(s) that cannot be proven to
 * be implied by the given indexquals.  This successfully handles cases such
 * as a qual "x = 42" used with a partial index "WHERE x >= 40 AND x < 50".
 * There are many other cases where we won't detect redundancy, leading to a
 * too-low selectivity estimate, which will bias the system in favor of using
 * partial indexes where possible.  That is not necessarily bad though.
 *
 * Note that indexQuals contains RestrictInfo nodes while the indpred
 * does not, so the output list will be mixed.  This is OK for both
 * predicate_implied_by() and clauselist_selectivity(), but might be
 * problematic if the result were passed to other things.
 */
List *
add_predicate_to_index_quals(IndexOptInfo *index, List *indexQuals)
{
    List       *predExtraQuals = NIL;
    ListCell   *lc;

    if (index->indpred == NIL)
        return indexQuals;

    foreach(lc, index->indpred)
    {
        Node       *predQual = (Node *) lfirst(lc);
        List       *oneQual = list_make1(predQual);

        if (!predicate_implied_by(oneQual, indexQuals, false))
            predExtraQuals = list_concat(predExtraQuals, oneQual);
    }
    return list_concat(predExtraQuals, indexQuals);
}


void
btcostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
               Cost *indexStartupCost, Cost *indexTotalCost,
               Selectivity *indexSelectivity, double *indexCorrelation,
               double *indexPages)
{
    IndexOptInfo *index = path->indexinfo;
    GenericCosts costs;
    Oid         relid;
    AttrNumber  colnum;
    VariableStatData vardata;
    double      numIndexTuples;
    Cost        descentCost;
    List       *indexBoundQuals;
    int         indexcol;
    bool        eqQualHere;
    bool        found_saop;
    bool        found_is_null_op;
    double      num_sa_scans;
    ListCell   *lc;

    /*
     * For a btree scan, only leading '=' quals plus inequality quals for the
     * immediately next attribute contribute to index selectivity (these are
     * the "boundary quals" that determine the starting and stopping points of
     * the index scan).  Additional quals can suppress visits to the heap, so
     * it's OK to count them in indexSelectivity, but they should not count
     * for estimating numIndexTuples.  So we must examine the given indexquals
     * to find out which ones count as boundary quals.  We rely on the
     * knowledge that they are given in index column order.
     *
     * For a RowCompareExpr, we consider only the first column, just as
     * rowcomparesel() does.
     *
     * If there's a ScalarArrayOpExpr in the quals, we'll actually perform N
     * index scans not one, but the ScalarArrayOpExpr's operator can be
     * considered to act the same as it normally does.
     */
    indexBoundQuals = NIL;
    indexcol = 0;
    eqQualHere = false;
    found_saop = false;
    found_is_null_op = false;
    num_sa_scans = 1;
    foreach(lc, path->indexclauses)
    {
        IndexClause *iclause = lfirst_node(IndexClause, lc);
        ListCell   *lc2;

        if (indexcol != iclause->indexcol)
        {
            /* Beginning of a new column's quals */
            if (!eqQualHere)
                break;          /* done if no '=' qual for indexcol */
            eqQualHere = false;
            indexcol++;
            if (indexcol != iclause->indexcol)
                break;          /* no quals at all for indexcol */
        }

        /* Examine each indexqual associated with this index clause */
        foreach(lc2, iclause->indexquals)
        {
            RestrictInfo *rinfo = lfirst_node(RestrictInfo, lc2);
            Expr       *clause = rinfo->clause;
            Oid         clause_op = InvalidOid;
            int         op_strategy;

            if (IsA(clause, OpExpr))
            {
                OpExpr     *op = (OpExpr *) clause;

                clause_op = op->opno;
            }
            else if (IsA(clause, RowCompareExpr))
            {
                RowCompareExpr *rc = (RowCompareExpr *) clause;

                clause_op = linitial_oid(rc->opnos);
            }
            else if (IsA(clause, ScalarArrayOpExpr))
            {
                ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) clause;
                Node       *other_operand = (Node *) lsecond(saop->args);
                int         alength = estimate_array_length(other_operand);

                clause_op = saop->opno;
                found_saop = true;
                /* count number of SA scans induced by indexBoundQuals only */
                if (alength > 1)
                    num_sa_scans *= alength;
            }
            else if (IsA(clause, NullTest))
            {
                NullTest   *nt = (NullTest *) clause;

                if (nt->nulltesttype == IS_NULL)
                {
                    found_is_null_op = true;
                    /* IS NULL is like = for selectivity purposes */
                    eqQualHere = true;
                }
            }
            else
                elog(ERROR, "unsupported indexqual type: %d",
                     (int) nodeTag(clause));

            /* check for equality operator */
            if (OidIsValid(clause_op))
            {
                op_strategy = get_op_opfamily_strategy(clause_op,
                                                       index->opfamily[indexcol]);
                Assert(op_strategy != 0);   /* not a member of opfamily?? */
                if (op_strategy == BTEqualStrategyNumber)
                    eqQualHere = true;
            }

            indexBoundQuals = lappend(indexBoundQuals, rinfo);
        }
    }

    /*
     * If index is unique and we found an '=' clause for each column, we can
     * just assume numIndexTuples = 1 and skip the expensive
     * clauselist_selectivity calculations.  However, a ScalarArrayOp or
     * NullTest invalidates that theory, even though it sets eqQualHere.
     */
    if (index->unique &&
        indexcol == index->nkeycolumns - 1 &&
        eqQualHere &&
        !found_saop &&
        !found_is_null_op)
        numIndexTuples = 1.0;
    else
    {
        List       *selectivityQuals;
        Selectivity btreeSelectivity;

        /*
         * If the index is partial, AND the index predicate with the
         * index-bound quals to produce a more accurate idea of the number of
         * rows covered by the bound conditions.
         */
        selectivityQuals = add_predicate_to_index_quals(index, indexBoundQuals);

        btreeSelectivity = clauselist_selectivity(root, selectivityQuals,
                                                  index->rel->relid,
                                                  JOIN_INNER,
                                                  NULL);
        numIndexTuples = btreeSelectivity * index->rel->tuples;

        /*
         * As in genericcostestimate(), we have to adjust for any
         * ScalarArrayOpExpr quals included in indexBoundQuals, and then round
         * to integer.
         */
        numIndexTuples = rint(numIndexTuples / num_sa_scans);
    }

    /*
     * Now do generic index cost estimation.
     */
    MemSet(&costs, 0, sizeof(costs));
    costs.numIndexTuples = numIndexTuples;

    genericcostestimate(root, path, loop_count, &costs);

    /*
     * Add a CPU-cost component to represent the costs of initial btree
     * descent.  We don't charge any I/O cost for touching upper btree levels,
     * since they tend to stay in cache, but we still have to do about log2(N)
     * comparisons to descend a btree of N leaf tuples.  We charge one
     * cpu_operator_cost per comparison.
     *
     * If there are ScalarArrayOpExprs, charge this once per SA scan.  The
     * ones after the first one are not startup cost so far as the overall
     * plan is concerned, so add them only to "total" cost.
     */
    if (index->tuples > 1)      /* avoid computing log(0) */
    {
        descentCost = ceil(log(index->tuples) / log(2.0)) * cpu_operator_cost;
        costs.indexStartupCost += descentCost;
        costs.indexTotalCost += costs.num_sa_scans * descentCost;
    }

    /*
     * Even though we're not charging I/O cost for touching upper btree pages,
     * it's still reasonable to charge some CPU cost per page descended
     * through.  Moreover, if we had no such charge at all, bloated indexes
     * would appear to have the same search cost as unbloated ones, at least
     * in cases where only a single leaf page is expected to be visited.  This
     * cost is somewhat arbitrarily set at 50x cpu_operator_cost per page
     * touched.  The number of such pages is btree tree height plus one (ie,
     * we charge for the leaf page too).  As above, charge once per SA scan.
     */
    descentCost = (index->tree_height + 1) * 50.0 * cpu_operator_cost;
    costs.indexStartupCost += descentCost;
    costs.indexTotalCost += costs.num_sa_scans * descentCost;

    /*
     * If we can get an estimate of the first column's ordering correlation C
     * from pg_statistic, estimate the index correlation as C for a
     * single-column index, or C * 0.75 for multiple columns. (The idea here
     * is that multiple columns dilute the importance of the first column's
     * ordering, but don't negate it entirely.  Before 8.0 we divided the
     * correlation by the number of columns, but that seems too strong.)
     */
    MemSet(&vardata, 0, sizeof(vardata));

    if (index->indexkeys[0] != 0)
    {
        /* Simple variable --- look to stats for the underlying table */
        RangeTblEntry *rte = planner_rt_fetch(index->rel->relid, root);

        Assert(rte->rtekind == RTE_RELATION);
        relid = rte->relid;
        Assert(relid != InvalidOid);
        colnum = index->indexkeys[0];

        if (get_relation_stats_hook &&
            (*get_relation_stats_hook) (root, rte, colnum, &vardata))
        {
            /*
             * The hook took control of acquiring a stats tuple.  If it did
             * supply a tuple, it'd better have supplied a freefunc.
             */
            if (HeapTupleIsValid(vardata.statsTuple) &&
                !vardata.freefunc)
                elog(ERROR, "no function provided to release variable stats with");
        }
        else
        {
            vardata.statsTuple = SearchSysCache3(STATRELATTINH,
                                                 ObjectIdGetDatum(relid),
                                                 Int16GetDatum(colnum),
                                                 BoolGetDatum(rte->inh));
            vardata.freefunc = ReleaseSysCache;
        }
    }
    else
    {
        /* Expression --- maybe there are stats for the index itself */
        relid = index->indexoid;
        colnum = 1;

        if (get_index_stats_hook &&
            (*get_index_stats_hook) (root, relid, colnum, &vardata))
        {
            /*
             * The hook took control of acquiring a stats tuple.  If it did
             * supply a tuple, it'd better have supplied a freefunc.
             */
            if (HeapTupleIsValid(vardata.statsTuple) &&
                !vardata.freefunc)
                elog(ERROR, "no function provided to release variable stats with");
        }
        else
        {
            vardata.statsTuple = SearchSysCache3(STATRELATTINH,
                                                 ObjectIdGetDatum(relid),
                                                 Int16GetDatum(colnum),
                                                 BoolGetDatum(false));
            vardata.freefunc = ReleaseSysCache;
        }
    }

    if (HeapTupleIsValid(vardata.statsTuple))
    {
        Oid         sortop;
        AttStatsSlot sslot;

        sortop = get_opfamily_member(index->opfamily[0],
                                     index->opcintype[0],
                                     index->opcintype[0],
                                     BTLessStrategyNumber);
        if (OidIsValid(sortop) &&
            get_attstatsslot(&