gistproc.c

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/*-------------------------------------------------------------------------
 *
 * gistproc.c
 *    Support procedures for GiSTs over 2-D objects (boxes, polygons, circles,
 *    points).
 *
 * This gives R-tree behavior, with Guttman's poly-time split algorithm.
 *
 *
 * Portions Copyright (c) 1996-2024, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *  src/backend/access/gist/gistproc.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"
 
#include <math.h>
 
#include "access/gist.h"
#include "access/stratnum.h"
#include "utils/float.h"
#include "utils/fmgrprotos.h"
#include "utils/geo_decls.h"
#include "utils/sortsupport.h"
 
 
static bool gist_box_leaf_consistent(BOX *key, BOX *query,
                                     StrategyNumber strategy);
static bool rtree_internal_consistent(BOX *key, BOX *query,
                                      StrategyNumber strategy);
 
static uint64 point_zorder_internal(float4 x, float4 y);
static uint64 part_bits32_by2(uint32 x);
static uint32 ieee_float32_to_uint32(float f);
static int  gist_bbox_zorder_cmp(Datum a, Datum b, SortSupport ssup);
static Datum gist_bbox_zorder_abbrev_convert(Datum original, SortSupport ssup);
static bool gist_bbox_zorder_abbrev_abort(int memtupcount, SortSupport ssup);
 
 
/* Minimum accepted ratio of split */
#define LIMIT_RATIO 0.3
 
 
/**************************************************
 * Box ops
 **************************************************/
 
/*
 * Calculates union of two boxes, a and b. The result is stored in *n.
 */
static void
rt_box_union(BOX *n, const BOX *a, const BOX *b)
{
    n->high.x = float8_max(a->high.x, b->high.x);
    n->high.y = float8_max(a->high.y, b->high.y);
    n->low.x = float8_min(a->low.x, b->low.x);
    n->low.y = float8_min(a->low.y, b->low.y);
}
 
/*
 * Size of a BOX for penalty-calculation purposes.
 * The result can be +Infinity, but not NaN.
 */
static float8
size_box(const BOX *box)
{
    /*
     * Check for zero-width cases.  Note that we define the size of a zero-
     * by-infinity box as zero.  It's important to special-case this somehow,
     * as naively multiplying infinity by zero will produce NaN.
     *
     * The less-than cases should not happen, but if they do, say "zero".
     */
    if (float8_le(box->high.x, box->low.x) ||
        float8_le(box->high.y, box->low.y))
        return 0.0;
 
    /*
     * We treat NaN as larger than +Infinity, so any distance involving a NaN
     * and a non-NaN is infinite.  Note the previous check eliminated the
     * possibility that the low fields are NaNs.
     */
    if (isnan(box->high.x) || isnan(box->high.y))
        return get_float8_infinity();
    return float8_mul(float8_mi(box->high.x, box->low.x),
                      float8_mi(box->high.y, box->low.y));
}
 
/*
 * Return amount by which the union of the two boxes is larger than
 * the original BOX's area.  The result can be +Infinity, but not NaN.
 */
static float8
box_penalty(const BOX *original, const BOX *new)
{
    BOX         unionbox;
 
    rt_box_union(&unionbox, original, new);
    return float8_mi(size_box(&unionbox), size_box(original));
}
 
/*
 * The GiST Consistent method for boxes
 *
 * Should return false if for all data items x below entry,
 * the predicate x op query must be false, where op is the oper
 * corresponding to strategy in the pg_amop table.
 */
Datum
gist_box_consistent(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    BOX        *query = PG_GETARG_BOX_P(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid      subtype = PG_GETARG_OID(3); */
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
 
    /* All cases served by this function are exact */
    *recheck = false;
 
    if (DatumGetBoxP(entry->key) == NULL || query == NULL)
        PG_RETURN_BOOL(false);
 
    /*
     * if entry is not leaf, use rtree_internal_consistent, else use
     * gist_box_leaf_consistent
     */
    if (GIST_LEAF(entry))
        PG_RETURN_BOOL(gist_box_leaf_consistent(DatumGetBoxP(entry->key),
                                                query,
                                                strategy));
    else
        PG_RETURN_BOOL(rtree_internal_consistent(DatumGetBoxP(entry->key),
                                                 query,
                                                 strategy));
}
 
/*
 * Increase BOX b to include addon.
 */
static void
adjustBox(BOX *b, const BOX *addon)
{
    if (float8_lt(b->high.x, addon->high.x))
        b->high.x = addon->high.x;
    if (float8_gt(b->low.x, addon->low.x))
        b->low.x = addon->low.x;
    if (float8_lt(b->high.y, addon->high.y))
        b->high.y = addon->high.y;
    if (float8_gt(b->low.y, addon->low.y))
        b->low.y = addon->low.y;
}
 
/*
 * The GiST Union method for boxes
 *
 * returns the minimal bounding box that encloses all the entries in entryvec
 */
Datum
gist_box_union(PG_FUNCTION_ARGS)
{
    GistEntryVector *entryvec = (GistEntryVector *) PG_GETARG_POINTER(0);
    int        *sizep = (int *) PG_GETARG_POINTER(1);
    int         numranges,
                i;
    BOX        *cur,
               *pageunion;
 
    numranges = entryvec->n;
    pageunion = (BOX *) palloc(sizeof(BOX));
    cur = DatumGetBoxP(entryvec->vector[0].key);
    memcpy(pageunion, cur, sizeof(BOX));
 
    for (i = 1; i < numranges; i++)
    {
        cur = DatumGetBoxP(entryvec->vector[i].key);
        adjustBox(pageunion, cur);
    }
    *sizep = sizeof(BOX);
 
    PG_RETURN_POINTER(pageunion);
}
 
/*
 * We store boxes as boxes in GiST indexes, so we do not need
 * compress, decompress, or fetch functions.
 */
 
/*
 * The GiST Penalty method for boxes (also used for points)
 *
 * As in the R-tree paper, we use change in area as our penalty metric
 */
Datum
gist_box_penalty(PG_FUNCTION_ARGS)
{
    GISTENTRY  *origentry = (GISTENTRY *) PG_GETARG_POINTER(0);
    GISTENTRY  *newentry = (GISTENTRY *) PG_GETARG_POINTER(1);
    float      *result = (float *) PG_GETARG_POINTER(2);
    BOX        *origbox = DatumGetBoxP(origentry->key);
    BOX        *newbox = DatumGetBoxP(newentry->key);
 
    *result = (float) box_penalty(origbox, newbox);
    PG_RETURN_POINTER(result);
}
 
/*
 * Trivial split: half of entries will be placed on one page
 * and another half - to another
 */
static void
fallbackSplit(GistEntryVector *entryvec, GIST_SPLITVEC *v)
{
    OffsetNumber i,
                maxoff;
    BOX        *unionL = NULL,
               *unionR = NULL;
    int         nbytes;
 
    maxoff = entryvec->n - 1;
 
    nbytes = (maxoff + 2) * sizeof(OffsetNumber);
    v->spl_left = (OffsetNumber *) palloc(nbytes);
    v->spl_right = (OffsetNumber *) palloc(nbytes);
    v->spl_nleft = v->spl_nright = 0;
 
    for (i = FirstOffsetNumber; i <= maxoff; i = OffsetNumberNext(i))
    {
        BOX        *cur = DatumGetBoxP(entryvec->vector[i].key);
 
        if (i <= (maxoff - FirstOffsetNumber + 1) / 2)
        {
            v->spl_left[v->spl_nleft] = i;
            if (unionL == NULL)
            {
                unionL = (BOX *) palloc(sizeof(BOX));
                *unionL = *cur;
            }
            else
                adjustBox(unionL, cur);
 
            v->spl_nleft++;
        }
        else
        {
            v->spl_right[v->spl_nright] = i;
            if (unionR == NULL)
            {
                unionR = (BOX *) palloc(sizeof(BOX));
                *unionR = *cur;
            }
            else
                adjustBox(unionR, cur);
 
            v->spl_nright++;
        }
    }
 
    v->spl_ldatum = BoxPGetDatum(unionL);
    v->spl_rdatum = BoxPGetDatum(unionR);
}
 
/*
 * Represents information about an entry that can be placed to either group
 * without affecting overlap over selected axis ("common entry").
 */
typedef struct
{
    /* Index of entry in the initial array */
    int         index;
    /* Delta between penalties of entry insertion into different groups */
    float8      delta;
} CommonEntry;
 
/*
 * Context for g_box_consider_split. Contains information about currently
 * selected split and some general information.
 */
typedef struct
{
    int         entriesCount;   /* total number of entries being split */
    BOX         boundingBox;    /* minimum bounding box across all entries */
 
    /* Information about currently selected split follows */
 
    bool        first;          /* true if no split was selected yet */
 
    float8      leftUpper;      /* upper bound of left interval */
    float8      rightLower;     /* lower bound of right interval */
 
    float4      ratio;
    float4      overlap;
    int         dim;            /* axis of this split */
    float8      range;          /* width of general MBR projection to the
                                 * selected axis */
} ConsiderSplitContext;
 
/*
 * Interval represents projection of box to axis.
 */
typedef struct
{
    float8      lower,
                upper;
} SplitInterval;
 
/*
 * Interval comparison function by lower bound of the interval;
 */
static int
interval_cmp_lower(const void *i1, const void *i2)
{
    float8      lower1 = ((const SplitInterval *) i1)->lower,
                lower2 = ((const SplitInterval *) i2)->lower;
 
    return float8_cmp_internal(lower1, lower2);
}
 
/*
 * Interval comparison function by upper bound of the interval;
 */
static int
interval_cmp_upper(const void *i1, const void *i2)
{
    float8      upper1 = ((const SplitInterval *) i1)->upper,
                upper2 = ((const SplitInterval *) i2)->upper;
 
    return float8_cmp_internal(upper1, upper2);
}
 
/*
 * Replace negative (or NaN) value with zero.
 */
static inline float
non_negative(float val)
{
    if (val >= 0.0f)
        return val;
    else
        return 0.0f;
}
 
/*
 * Consider replacement of currently selected split with the better one.
 */
static inline void
g_box_consider_split(ConsiderSplitContext *context, int dimNum,
                     float8 rightLower, int minLeftCount,
                     float8 leftUpper, int maxLeftCount)
{
    int         leftCount,
                rightCount;
    float4      ratio,
                overlap;
    float8      range;
 
    /*
     * Calculate entries distribution ratio assuming most uniform distribution
     * of common entries.
     */
    if (minLeftCount >= (context->entriesCount + 1) / 2)
    {
        leftCount = minLeftCount;
    }
    else
    {
        if (maxLeftCount <= context->entriesCount / 2)
            leftCount = maxLeftCount;
        else
            leftCount = context->entriesCount / 2;
    }
    rightCount = context->entriesCount - leftCount;
 
    /*
     * Ratio of split - quotient between size of lesser group and total
     * entries count.
     */
    ratio = float4_div(Min(leftCount, rightCount), context->entriesCount);
 
    if (ratio > LIMIT_RATIO)
    {
        bool        selectthis = false;
 
        /*
         * The ratio is acceptable, so compare current split with previously
         * selected one. Between splits of one dimension we search for minimal
         * overlap (allowing negative values) and minimal ration (between same
         * overlaps. We switch dimension if find less overlap (non-negative)
         * or less range with same overlap.
         */
        if (dimNum == 0)
            range = float8_mi(context->boundingBox.high.x,
                              context->boundingBox.low.x);
        else
            range = float8_mi(context->boundingBox.high.y,
                              context->boundingBox.low.y);
 
        overlap = float8_div(float8_mi(leftUpper, rightLower), range);
 
        /* If there is no previous selection, select this */
        if (context->first)
            selectthis = true;
        else if (context->dim == dimNum)
        {
            /*
             * Within the same dimension, choose the new split if it has a
             * smaller overlap, or same overlap but better ratio.
             */
            if (overlap < context->overlap ||
                (overlap == context->overlap && ratio > context->ratio))
                selectthis = true;
        }
        else
        {
            /*
             * Across dimensions, choose the new split if it has a smaller
             * *non-negative* overlap, or same *non-negative* overlap but
             * bigger range. This condition differs from the one described in
             * the article. On the datasets where leaf MBRs don't overlap
             * themselves, non-overlapping splits (i.e. splits which have zero
             * *non-negative* overlap) are frequently possible. In this case
             * splits tends to be along one dimension, because most distant
             * non-overlapping splits (i.e. having lowest negative overlap)
             * appears to be in the same dimension as in the previous split.
             * Therefore MBRs appear to be very prolonged along another
             * dimension, which leads to bad search performance. Using range
             * as the second split criteria makes MBRs more quadratic. Using
             * *non-negative* overlap instead of overlap as the first split
             * criteria gives to range criteria a chance to matter, because
             * non-overlapping splits are equivalent in this criteria.
             */
            if (non_negative(overlap) < non_negative(context->overlap) ||
                (range > context->range &&
                 non_negative(overlap) <= non_negative(context->overlap)))
                selectthis = true;
        }
 
        if (selectthis)
        {
            /* save information about selected split */
            context->first = false;
            context->ratio = ratio;
            context->range = range;
            context->overlap = overlap;
            context->rightLower = rightLower;
            context->leftUpper = leftUpper;
            context->dim = dimNum;
        }
    }
}
 
/*
 * Compare common entries by their deltas.
 */
static int
common_entry_cmp(const void *i1, const void *i2)
{
    float8      delta1 = ((const CommonEntry *) i1)->delta,
                delta2 = ((const CommonEntry *) i2)->delta;
 
    return float8_cmp_internal(delta1, delta2);
}
 
/*
 * --------------------------------------------------------------------------
 * Double sorting split algorithm. This is used for both boxes and points.
 *
 * The algorithm finds split of boxes by considering splits along each axis.
 * Each entry is first projected as an interval on the X-axis, and different
 * ways to split the intervals into two groups are considered, trying to
 * minimize the overlap of the groups. Then the same is repeated for the
 * Y-axis, and the overall best split is chosen. The quality of a split is
 * determined by overlap along that axis and some other criteria (see
 * g_box_consider_split).
 *
 * After that, all the entries are divided into three groups:
 *
 * 1) Entries which should be placed to the left group
 * 2) Entries which should be placed to the right group
 * 3) "Common entries" which can be placed to any of groups without affecting
 *    of overlap along selected axis.
 *
 * The common entries are distributed by minimizing penalty.
 *
 * For details see:
 * "A new double sorting-based node splitting algorithm for R-tree", A. Korotkov
 * http://syrcose.ispras.ru/2011/files/SYRCoSE2011_Proceedings.pdf#page=36
 * --------------------------------------------------------------------------
 */
Datum
gist_box_picksplit(PG_FUNCTION_ARGS)
{
    GistEntryVector *entryvec = (GistEntryVector *) PG_GETARG_POINTER(0);
    GIST_SPLITVEC *v = (GIST_SPLITVEC *) PG_GETARG_POINTER(1);
    OffsetNumber i,
                maxoff;
    ConsiderSplitContext context;
    BOX        *box,
               *leftBox,
               *rightBox;
    int         dim,
                commonEntriesCount;
    SplitInterval *intervalsLower,
               *intervalsUpper;
    CommonEntry *commonEntries;
    int         nentries;
 
    memset(&context, 0, sizeof(ConsiderSplitContext));
 
    maxoff = entryvec->n - 1;
    nentries = context.entriesCount = maxoff - FirstOffsetNumber + 1;
 
    /* Allocate arrays for intervals along axes */
    intervalsLower = (SplitInterval *) palloc(nentries * sizeof(SplitInterval));
    intervalsUpper = (SplitInterval *) palloc(nentries * sizeof(SplitInterval));
 
    /*
     * Calculate the overall minimum bounding box over all the entries.
     */
    for (i = FirstOffsetNumber; i <= maxoff; i = OffsetNumberNext(i))
    {
        box = DatumGetBoxP(entryvec->vector[i].key);
        if (i == FirstOffsetNumber)
            context.boundingBox = *box;
        else
            adjustBox(&context.boundingBox, box);
    }
 
    /*
     * Iterate over axes for optimal split searching.
     */
    context.first = true;       /* nothing selected yet */
    for (dim = 0; dim < 2; dim++)
    {
        float8      leftUpper,
                    rightLower;
        int         i1,
                    i2;
 
        /* Project each entry as an interval on the selected axis. */
        for (i = FirstOffsetNumber; i <= maxoff; i = OffsetNumberNext(i))
        {
            box = DatumGetBoxP(entryvec->vector[i].key);
            if (dim == 0)
            {
                intervalsLower[i - FirstOffsetNumber].lower = box->low.x;
                intervalsLower[i - FirstOffsetNumber].upper = box->high.x;
            }
            else
            {
                intervalsLower[i - FirstOffsetNumber].lower = box->low.y;
                intervalsLower[i - FirstOffsetNumber].upper = box->high.y;
            }
        }
 
        /*
         * Make two arrays of intervals: one sorted by lower bound and another
         * sorted by upper bound.
         */
        memcpy(intervalsUpper, intervalsLower,
               sizeof(SplitInterval) * nentries);
        qsort(intervalsLower, nentries, sizeof(SplitInterval),
              interval_cmp_lower);
        qsort(intervalsUpper, nentries, sizeof(SplitInterval),
              interval_cmp_upper);
 
        /*----
         * The goal is to form a left and right interval, so that every entry
         * interval is contained by either left or right interval (or both).
         *
         * For example, with the intervals (0,1), (1,3), (2,3), (2,4):
         *
         * 0 1 2 3 4
         * +-+
         *   +---+
         *     +-+
         *     +---+
         *
         * The left and right intervals are of the form (0,a) and (b,4).
         * We first consider splits where b is the lower bound of an entry.
         * We iterate through all entries, and for each b, calculate the
         * smallest possible a. Then we consider splits where a is the
         * upper bound of an entry, and for each a, calculate the greatest
         * possible b.
         *
         * In the above example, the first loop would consider splits:
         * b=0: (0,1)-(0,4)
         * b=1: (0,1)-(1,4)
         * b=2: (0,3)-(2,4)
         *
         * And the second loop:
         * a=1: (0,1)-(1,4)
         * a=3: (0,3)-(2,4)
         * a=4: (0,4)-(2,4)
         */
 
        /*
         * Iterate over lower bound of right group, finding smallest possible
         * upper bound of left group.
         */
        i1 = 0;
        i2 = 0;
        rightLower = intervalsLower[i1].lower;
        leftUpper = intervalsUpper[i2].lower;
        while (true)
        {
            /*
             * Find next lower bound of right group.
             */
            while (i1 < nentries &&
                   float8_eq(rightLower, intervalsLower[i1].lower))
            {
                if (float8_lt(leftUpper, intervalsLower[i1].upper))
                    leftUpper = intervalsLower[i1].upper;
                i1++;
            }
            if (i1 >= nentries)
                break;
            rightLower = intervalsLower[i1].lower;
 
            /*
             * Find count of intervals which anyway should be placed to the
             * left group.
             */
            while (i2 < nentries &&
                   float8_le(intervalsUpper[i2].upper, leftUpper))
                i2++;
 
            /*
             * Consider found split.
             */
            g_box_consider_split(&context, dim, rightLower, i1, leftUpper, i2);
        }
 
        /*
         * Iterate over upper bound of left group finding greatest possible
         * lower bound of right group.
         */
        i1 = nentries - 1;
        i2 = nentries - 1;
        rightLower = intervalsLower[i1].upper;
        leftUpper = intervalsUpper[i2].upper;
        while (true)
        {
            /*
             * Find next upper bound of left group.
             */
            while (i2 >= 0 && float8_eq(leftUpper, intervalsUpper[i2].upper))
            {
                if (float8_gt(rightLower, intervalsUpper[i2].lower))
                    rightLower = intervalsUpper[i2].lower;
                i2--;
            }
            if (i2 < 0)
                break;
            leftUpper = intervalsUpper[i2].upper;
 
            /*
             * Find count of intervals which anyway should be placed to the
             * right group.
             */
            while (i1 >= 0 && float8_ge(intervalsLower[i1].lower, rightLower))
                i1--;
 
            /*
             * Consider found split.
             */
            g_box_consider_split(&context, dim,
                                 rightLower, i1 + 1, leftUpper, i2 + 1);
        }
    }
 
    /*
     * If we failed to find any acceptable splits, use trivial split.
     */
    if (context.first)
    {
        fallbackSplit(entryvec, v);
        PG_RETURN_POINTER(v);
    }
 
    /*
     * Ok, we have now selected the split across one axis.
     *
     * While considering the splits, we already determined that there will be
     * enough entries in both groups to reach the desired ratio, but we did
     * not memorize which entries go to which group. So determine that now.
     */
 
    /* Allocate vectors for results */
    v->spl_left = (OffsetNumber *) palloc(nentries * sizeof(OffsetNumber));
    v->spl_right = (OffsetNumber *) palloc(nentries * sizeof(OffsetNumber));
    v->spl_nleft = 0;
    v->spl_nright = 0;
 
    /* Allocate bounding boxes of left and right groups */
    leftBox = palloc0(sizeof(BOX));
    rightBox = palloc0(sizeof(BOX));
 
    /*
     * Allocate an array for "common entries" - entries which can be placed to
     * either group without affecting overlap along selected axis.
     */
    commonEntriesCount = 0;
    commonEntries = (CommonEntry *) palloc(nentries * sizeof(CommonEntry));
 
    /* Helper macros to place an entry in the left or right group */
#define PLACE_LEFT(box, off)                    \
    do {                                        \
        if (v->spl_nleft > 0)                   \
            adjustBox(leftBox, box);            \
        else                                    \
            *leftBox = *(box);                  \
        v->spl_left[v->spl_nleft++] = off;      \
    } while(0)
 
#define PLACE_RIGHT(box, off)                   \
    do {                                        \
        if (v->spl_nright > 0)                  \
            adjustBox(rightBox, box);           \
        else                                    \
            *rightBox = *(box);                 \
        v->spl_right[v->spl_nright++] = off;    \
    } while(0)
 
    /*
     * Distribute entries which can be distributed unambiguously, and collect
     * common entries.
     */
    for (i = FirstOffsetNumber; i <= maxoff; i = OffsetNumberNext(i))
    {
        float8      lower,
                    upper;
 
        /*
         * Get upper and lower bounds along selected axis.
         */
        box = DatumGetBoxP(entryvec->vector[i].key);
        if (context.dim == 0)
        {
            lower = box->low.x;
            upper = box->high.x;
        }
        else
        {
            lower = box->low.y;
            upper = box->high.y;
        }
 
        if (float8_le(upper, context.leftUpper))
        {
            /* Fits to the left group */
            if (float8_ge(lower, context.rightLower))
            {
                /* Fits also to the right group, so "common entry" */
                commonEntries[commonEntriesCount++].index = i;
            }
            else
            {
                /* Doesn't fit to the right group, so join to the left group */
                PLACE_LEFT(box, i);
            }
        }
        else
        {
            /*
             * Each entry should fit on either left or right group. Since this
             * entry didn't fit on the left group, it better fit in the right
             * group.
             */
            Assert(float8_ge(lower, context.rightLower));
 
            /* Doesn't fit to the left group, so join to the right group */
            PLACE_RIGHT(box, i);
        }
    }
 
    /*
     * Distribute "common entries", if any.
     */
    if (commonEntriesCount > 0)
    {
        /*
         * Calculate minimum number of entries that must be placed in both
         * groups, to reach LIMIT_RATIO.
         */
        int         m = ceil(LIMIT_RATIO * nentries);
 
        /*
         * Calculate delta between penalties of join "common entries" to
         * different groups.
         */
        for (i = 0; i < commonEntriesCount; i++)
        {
            box = DatumGetBoxP(entryvec->vector[commonEntries[i].index].key);
            commonEntries[i].delta = fabs(float8_mi(box_penalty(leftBox, box),
                                                    box_penalty(rightBox, box)));
        }
 
        /*
         * Sort "common entries" by calculated deltas in order to distribute
         * the most ambiguous entries first.
         */
        qsort(commonEntries, commonEntriesCount, sizeof(CommonEntry), common_entry_cmp);
 
        /*
         * Distribute "common entries" between groups.
         */
        for (i = 0; i < commonEntriesCount; i++)
        {
            box = DatumGetBoxP(entryvec->vector[commonEntries[i].index].key);
 
            /*
             * Check if we have to place this entry in either group to achieve
             * LIMIT_RATIO.
             */
            if (v->spl_nleft + (commonEntriesCount - i) <= m)
                PLACE_LEFT(box, commonEntries[i].index);
            else if (v->spl_nright + (commonEntriesCount - i) <= m)
                PLACE_RIGHT(box, commonEntries[i].index);
            else
            {
                /* Otherwise select the group by minimal penalty */
                if (box_penalty(leftBox, box) < box_penalty(rightBox, box))
                    PLACE_LEFT(box, commonEntries[i].index);
                else
                    PLACE_RIGHT(box, commonEntries[i].index);
            }
        }
    }
 
    v->spl_ldatum = PointerGetDatum(leftBox);
    v->spl_rdatum = PointerGetDatum(rightBox);
    PG_RETURN_POINTER(v);
}
 
/*
 * Equality method
 *
 * This is used for boxes, points, circles, and polygons, all of which store
 * boxes as GiST index entries.
 *
 * Returns true only when boxes are exactly the same.  We can't use fuzzy
 * comparisons here without breaking index consistency; therefore, this isn't
 * equivalent to box_same().
 */
Datum
gist_box_same(PG_FUNCTION_ARGS)
{
    BOX        *b1 = PG_GETARG_BOX_P(0);
    BOX        *b2 = PG_GETARG_BOX_P(1);
    bool       *result = (bool *) PG_GETARG_POINTER(2);
 
    if (b1 && b2)
        *result = (float8_eq(b1->low.x, b2->low.x) &&
                   float8_eq(b1->low.y, b2->low.y) &&
                   float8_eq(b1->high.x, b2->high.x) &&
                   float8_eq(b1->high.y, b2->high.y));
    else
        *result = (b1 == NULL && b2 == NULL);
    PG_RETURN_POINTER(result);
}
 
/*
 * Leaf-level consistency for boxes: just apply the query operator
 */
static bool
gist_box_leaf_consistent(BOX *key, BOX *query, StrategyNumber strategy)
{
    bool        retval;
 
    switch (strategy)
    {
        case RTLeftStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_left,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverLeftStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overleft,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverlapStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overlap,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverRightStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overright,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTRightStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_right,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTSameStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_same,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTContainsStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_contain,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTContainedByStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_contained,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverBelowStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overbelow,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTBelowStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_below,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTAboveStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_above,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverAboveStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overabove,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            retval = false;     /* keep compiler quiet */
            break;
    }
    return retval;
}
 
/*****************************************
 * Common rtree functions (for boxes, polygons, and circles)
 *****************************************/
 
/*
 * Internal-page consistency for all these types
 *
 * We can use the same function since all types use bounding boxes as the
 * internal-page representation.
 */
static bool
rtree_internal_consistent(BOX *key, BOX *query, StrategyNumber strategy)
{
    bool        retval;
 
    switch (strategy)
    {
        case RTLeftStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_overright,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTOverLeftStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_right,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTOverlapStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overlap,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverRightStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_left,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTRightStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_overleft,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTSameStrategyNumber:
        case RTContainsStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_contain,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTContainedByStrategyNumber:
            retval = DatumGetBool(DirectFunctionCall2(box_overlap,
                                                      PointerGetDatum(key),
                                                      PointerGetDatum(query)));
            break;
        case RTOverBelowStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_above,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTBelowStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_overabove,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTAboveStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_overbelow,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        case RTOverAboveStrategyNumber:
            retval = !DatumGetBool(DirectFunctionCall2(box_below,
                                                       PointerGetDatum(key),
                                                       PointerGetDatum(query)));
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            retval = false;     /* keep compiler quiet */
            break;
    }
    return retval;
}
 
/**************************************************
 * Polygon ops
 **************************************************/
 
/*
 * GiST compress for polygons: represent a polygon by its bounding box
 */
Datum
gist_poly_compress(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    GISTENTRY  *retval;
 
    if (entry->leafkey)
    {
        POLYGON    *in = DatumGetPolygonP(entry->key);
        BOX        *r;
 
        r = (BOX *) palloc(sizeof(BOX));
        memcpy(r, &(in->boundbox), sizeof(BOX));
 
        retval = (GISTENTRY *) palloc(sizeof(GISTENTRY));
        gistentryinit(*retval, PointerGetDatum(r),
                      entry->rel, entry->page,
                      entry->offset, false);
    }
    else
        retval = entry;
    PG_RETURN_POINTER(retval);
}
 
/*
 * The GiST Consistent method for polygons
 */
Datum
gist_poly_consistent(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    POLYGON    *query = PG_GETARG_POLYGON_P(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid      subtype = PG_GETARG_OID(3); */
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
    bool        result;
 
    /* All cases served by this function are inexact */
    *recheck = true;
 
    if (DatumGetBoxP(entry->key) == NULL || query == NULL)
        PG_RETURN_BOOL(false);
 
    /*
     * Since the operators require recheck anyway, we can just use
     * rtree_internal_consistent even at leaf nodes.  (This works in part
     * because the index entries are bounding boxes not polygons.)
     */
    result = rtree_internal_consistent(DatumGetBoxP(entry->key),
                                       &(query->boundbox), strategy);
 
    /* Avoid memory leak if supplied poly is toasted */
    PG_FREE_IF_COPY(query, 1);
 
    PG_RETURN_BOOL(result);
}
 
/**************************************************
 * Circle ops
 **************************************************/
 
/*
 * GiST compress for circles: represent a circle by its bounding box
 */
Datum
gist_circle_compress(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    GISTENTRY  *retval;
 
    if (entry->leafkey)
    {
        CIRCLE     *in = DatumGetCircleP(entry->key);
        BOX        *r;
 
        r = (BOX *) palloc(sizeof(BOX));
        r->high.x = float8_pl(in->center.x, in->radius);
        r->low.x = float8_mi(in->center.x, in->radius);
        r->high.y = float8_pl(in->center.y, in->radius);
        r->low.y = float8_mi(in->center.y, in->radius);
 
        retval = (GISTENTRY *) palloc(sizeof(GISTENTRY));
        gistentryinit(*retval, PointerGetDatum(r),
                      entry->rel, entry->page,
                      entry->offset, false);
    }
    else
        retval = entry;
    PG_RETURN_POINTER(retval);
}
 
/*
 * The GiST Consistent method for circles
 */
Datum
gist_circle_consistent(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    CIRCLE     *query = PG_GETARG_CIRCLE_P(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid      subtype = PG_GETARG_OID(3); */
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
    BOX         bbox;
    bool        result;
 
    /* All cases served by this function are inexact */
    *recheck = true;
 
    if (DatumGetBoxP(entry->key) == NULL || query == NULL)
        PG_RETURN_BOOL(false);
 
    /*
     * Since the operators require recheck anyway, we can just use
     * rtree_internal_consistent even at leaf nodes.  (This works in part
     * because the index entries are bounding boxes not circles.)
     */
    bbox.high.x = float8_pl(query->center.x, query->radius);
    bbox.low.x = float8_mi(query->center.x, query->radius);
    bbox.high.y = float8_pl(query->center.y, query->radius);
    bbox.low.y = float8_mi(query->center.y, query->radius);
 
    result = rtree_internal_consistent(DatumGetBoxP(entry->key),
                                       &bbox, strategy);
 
    PG_RETURN_BOOL(result);
}
 
/**************************************************
 * Point ops
 **************************************************/
 
Datum
gist_point_compress(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
 
    if (entry->leafkey)         /* Point, actually */
    {
        BOX        *box = palloc(sizeof(BOX));
        Point      *point = DatumGetPointP(entry->key);
        GISTENTRY  *retval = palloc(sizeof(GISTENTRY));
 
        box->high = box->low = *point;
 
        gistentryinit(*retval, BoxPGetDatum(box),
                      entry->rel, entry->page, entry->offset, false);
 
        PG_RETURN_POINTER(retval);
    }
 
    PG_RETURN_POINTER(entry);
}
 
/*
 * GiST Fetch method for point
 *
 * Get point coordinates from its bounding box coordinates and form new
 * gistentry.
 */
Datum
gist_point_fetch(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    BOX        *in = DatumGetBoxP(entry->key);
    Point      *r;
    GISTENTRY  *retval;
 
    retval = palloc(sizeof(GISTENTRY));
 
    r = (Point *) palloc(sizeof(Point));
    r->x = in->high.x;
    r->y = in->high.y;
    gistentryinit(*retval, PointerGetDatum(r),
                  entry->rel, entry->page,
                  entry->offset, false);
 
    PG_RETURN_POINTER(retval);
}
 
 
#define point_point_distance(p1,p2) \
    DatumGetFloat8(DirectFunctionCall2(point_distance, \
                                       PointPGetDatum(p1), PointPGetDatum(p2)))
 
static float8
computeDistance(bool isLeaf, BOX *box, Point *point)
{
    float8      result = 0.0;
 
    if (isLeaf)
    {
        /* simple point to point distance */
        result = point_point_distance(point, &box->low);
    }
    else if (point->x <= box->high.x && point->x >= box->low.x &&
             point->y <= box->high.y && point->y >= box->low.y)
    {
        /* point inside the box */
        result = 0.0;
    }
    else if (point->x <= box->high.x && point->x >= box->low.x)
    {
        /* point is over or below box */
        Assert(box->low.y <= box->high.y);
        if (point->y > box->high.y)
            result = float8_mi(point->y, box->high.y);
        else if (point->y < box->low.y)
            result = float8_mi(box->low.y, point->y);
        else
            elog(ERROR, "inconsistent point values");
    }
    else if (point->y <= box->high.y && point->y >= box->low.y)
    {
        /* point is to left or right of box */
        Assert(box->low.x <= box->high.x);
        if (point->x > box->high.x)
            result = float8_mi(point->x, box->high.x);
        else if (point->x < box->low.x)
            result = float8_mi(box->low.x, point->x);
        else
            elog(ERROR, "inconsistent point values");
    }
    else
    {
        /* closest point will be a vertex */
        Point       p;
        float8      subresult;
 
        result = point_point_distance(point, &box->low);
 
        subresult = point_point_distance(point, &box->high);
        if (result > subresult)
            result = subresult;
 
        p.x = box->low.x;
        p.y = box->high.y;
        subresult = point_point_distance(point, &p);
        if (result > subresult)
            result = subresult;
 
        p.x = box->high.x;
        p.y = box->low.y;
        subresult = point_point_distance(point, &p);
        if (result > subresult)
            result = subresult;
    }
 
    return result;
}
 
static bool
gist_point_consistent_internal(StrategyNumber strategy,
                               bool isLeaf, BOX *key, Point *query)
{
    bool        result = false;
 
    switch (strategy)
    {
        case RTLeftStrategyNumber:
            result = FPlt(key->low.x, query->x);
            break;
        case RTRightStrategyNumber:
            result = FPgt(key->high.x, query->x);
            break;
        case RTAboveStrategyNumber:
            result = FPgt(key->high.y, query->y);
            break;
        case RTBelowStrategyNumber:
            result = FPlt(key->low.y, query->y);
            break;
        case RTSameStrategyNumber:
            if (isLeaf)
            {
                /* key.high must equal key.low, so we can disregard it */
                result = (FPeq(key->low.x, query->x) &&
                          FPeq(key->low.y, query->y));
            }
            else
            {
                result = (FPle(query->x, key->high.x) &&
                          FPge(query->x, key->low.x) &&
                          FPle(query->y, key->high.y) &&
                          FPge(query->y, key->low.y));
            }
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            result = false;     /* keep compiler quiet */
            break;
    }
 
    return result;
}
 
#define GeoStrategyNumberOffset     20
#define PointStrategyNumberGroup    0
#define BoxStrategyNumberGroup      1
#define PolygonStrategyNumberGroup  2
#define CircleStrategyNumberGroup   3
 
Datum
gist_point_consistent(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
    bool        result;
    StrategyNumber strategyGroup;
 
    /*
     * We have to remap these strategy numbers to get this klugy
     * classification logic to work.
     */
    if (strategy == RTOldBelowStrategyNumber)
        strategy = RTBelowStrategyNumber;
    else if (strategy == RTOldAboveStrategyNumber)
        strategy = RTAboveStrategyNumber;
 
    strategyGroup = strategy / GeoStrategyNumberOffset;
    switch (strategyGroup)
    {
        case PointStrategyNumberGroup:
            result = gist_point_consistent_internal(strategy % GeoStrategyNumberOffset,
                                                    GIST_LEAF(entry),
                                                    DatumGetBoxP(entry->key),
                                                    PG_GETARG_POINT_P(1));
            *recheck = false;
            break;
        case BoxStrategyNumberGroup:
            {
                /*
                 * The only operator in this group is point <@ box (on_pb), so
                 * we needn't examine strategy again.
                 *
                 * For historical reasons, on_pb uses exact rather than fuzzy
                 * comparisons.  We could use box_overlap when at an internal
                 * page, but that would lead to possibly visiting child pages
                 * uselessly, because box_overlap uses fuzzy comparisons.
                 * Instead we write a non-fuzzy overlap test.  The same code
                 * will also serve for leaf-page tests, since leaf keys have
                 * high == low.
                 */
                BOX        *query,
                           *key;
 
                query = PG_GETARG_BOX_P(1);
                key = DatumGetBoxP(entry->key);
 
                result = (key->high.x >= query->low.x &&
                          key->low.x <= query->high.x &&
                          key->high.y >= query->low.y &&
                          key->low.y <= query->high.y);
                *recheck = false;
            }
            break;
        case PolygonStrategyNumberGroup:
            {
                POLYGON    *query = PG_GETARG_POLYGON_P(1);
 
                result = DatumGetBool(DirectFunctionCall5(gist_poly_consistent,
                                                          PointerGetDatum(entry),
                                                          PolygonPGetDatum(query),
                                                          Int16GetDatum(RTOverlapStrategyNumber),
                                                          0, PointerGetDatum(recheck)));
 
                if (GIST_LEAF(entry) && result)
                {
                    /*
                     * We are on leaf page and quick check shows overlapping
                     * of polygon's bounding box and point
                     */
                    BOX        *box = DatumGetBoxP(entry->key);
 
                    Assert(box->high.x == box->low.x
                           && box->high.y == box->low.y);
                    result = DatumGetBool(DirectFunctionCall2(poly_contain_pt,
                                                              PolygonPGetDatum(query),
                                                              PointPGetDatum(&box->high)));
                    *recheck = false;
                }
            }
            break;
        case CircleStrategyNumberGroup:
            {
                CIRCLE     *query = PG_GETARG_CIRCLE_P(1);
 
                result = DatumGetBool(DirectFunctionCall5(gist_circle_consistent,
                                                          PointerGetDatum(entry),
                                                          CirclePGetDatum(query),
                                                          Int16GetDatum(RTOverlapStrategyNumber),
                                                          0, PointerGetDatum(recheck)));
 
                if (GIST_LEAF(entry) && result)
                {
                    /*
                     * We are on leaf page and quick check shows overlapping
                     * of polygon's bounding box and point
                     */
                    BOX        *box = DatumGetBoxP(entry->key);
 
                    Assert(box->high.x == box->low.x
                           && box->high.y == box->low.y);
                    result = DatumGetBool(DirectFunctionCall2(circle_contain_pt,
                                                              CirclePGetDatum(query),
                                                              PointPGetDatum(&box->high)));
                    *recheck = false;
                }
            }
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            result = false;     /* keep compiler quiet */
            break;
    }
 
    PG_RETURN_BOOL(result);
}
 
Datum
gist_point_distance(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
    float8      distance;
    StrategyNumber strategyGroup = strategy / GeoStrategyNumberOffset;
 
    switch (strategyGroup)
    {
        case PointStrategyNumberGroup:
            distance = computeDistance(GIST_LEAF(entry),
                                       DatumGetBoxP(entry->key),
                                       PG_GETARG_POINT_P(1));
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            distance = 0.0;     /* keep compiler quiet */
            break;
    }
 
    PG_RETURN_FLOAT8(distance);
}
 
static float8
gist_bbox_distance(GISTENTRY *entry, Datum query, StrategyNumber strategy)
{
    float8      distance;
    StrategyNumber strategyGroup = strategy / GeoStrategyNumberOffset;
 
    switch (strategyGroup)
    {
        case PointStrategyNumberGroup:
            distance = computeDistance(false,
                                       DatumGetBoxP(entry->key),
                                       DatumGetPointP(query));
            break;
        default:
            elog(ERROR, "unrecognized strategy number: %d", strategy);
            distance = 0.0;     /* keep compiler quiet */
    }
 
    return distance;
}
 
Datum
gist_box_distance(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    Datum       query = PG_GETARG_DATUM(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid subtype = PG_GETARG_OID(3); */
    /* bool    *recheck = (bool *) PG_GETARG_POINTER(4); */
    float8      distance;
 
    distance = gist_bbox_distance(entry, query, strategy);
 
    PG_RETURN_FLOAT8(distance);
}
 
/*
 * The inexact GiST distance methods for geometric types that store bounding
 * boxes.
 *
 * Compute lossy distance from point to index entries.  The result is inexact
 * because index entries are bounding boxes, not the exact shapes of the
 * indexed geometric types.  We use distance from point to MBR of index entry.
 * This is a lower bound estimate of distance from point to indexed geometric
 * type.
 */
Datum
gist_circle_distance(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    Datum       query = PG_GETARG_DATUM(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid subtype = PG_GETARG_OID(3); */
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
    float8      distance;
 
    distance = gist_bbox_distance(entry, query, strategy);
    *recheck = true;
 
    PG_RETURN_FLOAT8(distance);
}
 
Datum
gist_poly_distance(PG_FUNCTION_ARGS)
{
    GISTENTRY  *entry = (GISTENTRY *) PG_GETARG_POINTER(0);
    Datum       query = PG_GETARG_DATUM(1);
    StrategyNumber strategy = (StrategyNumber) PG_GETARG_UINT16(2);
 
    /* Oid subtype = PG_GETARG_OID(3); */
    bool       *recheck = (bool *) PG_GETARG_POINTER(4);
    float8      distance;
 
    distance = gist_bbox_distance(entry, query, strategy);
    *recheck = true;
 
    PG_RETURN_FLOAT8(distance);
}
 
/*
 * Z-order routines for fast index build
 */
 
/*
 * Compute Z-value of a point
 *
 * Z-order (also known as Morton Code) maps a two-dimensional point to a
 * single integer, in a way that preserves locality. Points that are close in
 * the two-dimensional space are mapped to integer that are not far from each
 * other. We do that by interleaving the bits in the X and Y components.
 *
 * Morton Code is normally defined only for integers, but the X and Y values
 * of a point are floating point. We expect floats to be in IEEE format.
 */
static uint64
point_zorder_internal(float4 x, float4 y)
{
    uint32      ix = ieee_float32_to_uint32(x);
    uint32      iy = ieee_float32_to_uint32(y);
 
    /* Interleave the bits */
    return part_bits32_by2(ix) | (part_bits32_by2(iy) << 1);
}
 
/* Interleave 32 bits with zeroes */
static uint64
part_bits32_by2(uint32 x)
{
    uint64      n = x;
 
    n = (n | (n << 16)) & UINT64CONST(0x0000FFFF0000FFFF);
    n = (n | (n << 8)) & UINT64CONST(0x00FF00FF00FF00FF);
    n = (n | (n << 4)) & UINT64CONST(0x0F0F0F0F0F0F0F0F);
    n = (n | (n << 2)) & UINT64CONST(0x3333333333333333);
    n = (n | (n << 1)) & UINT64CONST(0x5555555555555555);
 
    return n;
}
 
/*
 * Convert a 32-bit IEEE float to uint32 in a way that preserves the ordering
 */
static uint32
ieee_float32_to_uint32(float f)
{
    /*----
     *
     * IEEE 754 floating point format
     * ------------------------------
     *
     * IEEE 754 floating point numbers have this format:
     *
     *   exponent (8 bits)
     *   |
     * s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
     * |          |
     * sign       mantissa (23 bits)
     *
     * Infinity has all bits in the exponent set and the mantissa is all
     * zeros. Negative infinity is the same but with the sign bit set.
     *
     * NaNs are represented with all bits in the exponent set, and the least
     * significant bit in the mantissa also set. The rest of the mantissa bits
     * can be used to distinguish different kinds of NaNs.
     *
     * The IEEE format has the nice property that when you take the bit
     * representation and interpret it as an integer, the order is preserved,
     * except for the sign. That holds for the +-Infinity values too.
     *
     * Mapping to uint32
     * -----------------
     *
     * In order to have a smooth transition from negative to positive numbers,
     * we map floats to unsigned integers like this:
     *
     * x < 0 to range 0-7FFFFFFF
     * x = 0 to value 8000000 (both positive and negative zero)
     * x > 0 to range 8000001-FFFFFFFF
     *
     * We don't care to distinguish different kind of NaNs, so they are all
     * mapped to the same arbitrary value, FFFFFFFF. Because of the IEEE bit
     * representation of NaNs, there aren't any non-NaN values that would be
     * mapped to FFFFFFFF. In fact, there is a range of unused values on both
     * ends of the uint32 space.
     */
    if (isnan(f))
        return 0xFFFFFFFF;
    else
    {
        union
        {
            float       f;
            uint32      i;
        }           u;
 
        u.f = f;
 
        /* Check the sign bit */
        if ((u.i & 0x80000000) != 0)
        {
            /*
             * Map the negative value to range 0-7FFFFFFF. This flips the sign
             * bit to 0 in the same instruction.
             */
            Assert(f <= 0);     /* can be -0 */
            u.i ^= 0xFFFFFFFF;
        }
        else
        {
            /* Map the positive value (or 0) to range 80000000-FFFFFFFF */
            u.i |= 0x80000000;
        }
 
        return u.i;
    }
}
 
/*
 * Compare the Z-order of points
 */
static int
gist_bbox_zorder_cmp(Datum a, Datum b, SortSupport ssup)
{
    Point      *p1 = &(DatumGetBoxP(a)->low);
    Point      *p2 = &(DatumGetBoxP(b)->low);
    uint64      z1;
    uint64      z2;
 
    /*
     * Do a quick check for equality first. It's not clear if this is worth it
     * in general, but certainly is when used as tie-breaker with abbreviated
     * keys,
     */
    if (p1->x == p2->x && p1->y == p2->y)
        return 0;
 
    z1 = point_zorder_internal(p1->x, p1->y);
    z2 = point_zorder_internal(p2->x, p2->y);
    if (z1 > z2)
        return 1;
    else if (z1 < z2)
        return -1;
    else
        return 0;
}
 
/*
 * Abbreviated version of Z-order comparison
 *
 * The abbreviated format is a Z-order value computed from the two 32-bit
 * floats. If SIZEOF_DATUM == 8, the 64-bit Z-order value fits fully in the
 * abbreviated Datum, otherwise use its most significant bits.
 */
static Datum
gist_bbox_zorder_abbrev_convert(Datum original, SortSupport ssup)
{
    Point      *p = &(DatumGetBoxP(original)->low);
    uint64      z;
 
    z = point_zorder_internal(p->x, p->y);
 
#if SIZEOF_DATUM == 8
    return (Datum) z;
#else
    return (Datum) (z >> 32);
#endif
}
 
/*
 * We never consider aborting the abbreviation.
 *
 * On 64-bit systems, the abbreviation is not lossy so it is always
 * worthwhile. (Perhaps it's not on 32-bit systems, but we don't bother
 * with logic to decide.)
 */
static bool
gist_bbox_zorder_abbrev_abort(int memtupcount, SortSupport ssup)
{
    return false;
}
 
/*
 * Sort support routine for fast GiST index build by sorting.
 */
Datum
gist_point_sortsupport(PG_FUNCTION_ARGS)
{
    SortSupport ssup = (SortSupport) PG_GETARG_POINTER(0);
 
    if (ssup->abbreviate)
    {
        ssup->comparator = ssup_datum_unsigned_cmp;
        ssup->abbrev_converter = gist_bbox_zorder_abbrev_convert;
        ssup->abbrev_abort = gist_bbox_zorder_abbrev_abort;
        ssup->abbrev_full_comparator = gist_bbox_zorder_cmp;
    }
    else
    {
        ssup->comparator = gist_bbox_zorder_cmp;
    }
    PG_RETURN_VOID();
}