text: /*-------------------------------------------------------------------------
   *
   * 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.
   *----------
   */
  
  #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/acl.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, Oid collation,
  							  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, Oid collation,
  							 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, Oid collation,
  							   Datum *min, Datum *max);
  static void get_stats_slot_range(AttStatsSlot *sslot,
  								 Oid opfuncoid, FmgrInfo *opproc,
  								 Oid collation, int16 typLen, bool typByVal,
  								 Datum *min, Datum *max, bool *p_have_data);
  static bool get_actual_variable_range(PlannerInfo *root,
  									  VariableStatData *vardata,
  									  Oid sortop, Oid collation,
  									  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);
  	Oid			collation = PG_GET_COLLATION();
  	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, collation,
  							 ((Const *) other)->constvalue,
  							 ((Const *) other)->constisnull,
  							 varonleft, negate);
  	else
  		selec = var_eq_non_const(&vardata, operator, collation, 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, Oid collation,
  			 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))
  		{
  			LOCAL_FCINFO(fcinfo, 2);
  			FmgrInfo	eqproc;
  
  			fmgr_info(opfuncoid, &eqproc);
  
  			/*
  			 * Save a few cycles by setting up the fcinfo struct just once.
  			 * Using FunctionCallInvoke directly also avoids failure if the
  			 * eqproc returns NULL, though really equality functions should
  			 * never do that.
  			 */
  			InitFunctionCallInfoData(*fcinfo, &eqproc, 2, collation,
  									 NULL, NULL);
  			fcinfo->args[0].isnull = false;
  			fcinfo->args[1].isnull = false;
  			/* be careful to apply operator right way 'round */
  			if (varonleft)
  				fcinfo->args[1].value = constval;
  			else
  				fcinfo->args[0].value = constval;
  
  			for (i = 0; i < sslot.nvalues; i++)
  			{
  				Datum		fresult;
  
  				if (varonleft)
  					fcinfo->args[0].value = sslot.values[i];
  				else
  					fcinfo->args[1].value = sslot.values[i];
  				fcinfo->isnull = false;
  				fresult = FunctionCallInvoke(fcinfo);
  				if (!fcinfo->isnull && DatumGetBool(fresult))
  				{
  					match = true;
  					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, Oid collation,
  				 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,
  			  Oid collation,
  			  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, collation, 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,
  											operator, &opproc, isgt, iseq,
  											collation,
  											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, Oid collation,
  				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))
  	{
  		LOCAL_FCINFO(fcinfo, 2);
  
  		/*
  		 * We invoke the opproc "by hand" so that we won't fail on NULL
  		 * results.  Such cases won't arise for normal comparison functions,
  		 * but generic_restriction_selectivity could perhaps be used with
  		 * operators that can return NULL.  A small side benefit is to not
  		 * need to re-initialize the fcinfo struct from scratch each time.
  		 */
  		InitFunctionCallInfoData(*fcinfo, opproc, 2, collation,
  								 NULL, NULL);
  		fcinfo->args[0].isnull = false;
  		fcinfo->args[1].isnull = false;
  		/* be careful to apply operator right way 'round */
  		if (varonleft)
  			fcinfo->args[1].value = constval;
  		else
  			fcinfo->args[0].value = constval;
  
  		for (i = 0; i < sslot.nvalues; i++)
  		{
  			Datum		fresult;
  
  			if (varonleft)
  				fcinfo->args[0].value = sslot.values[i];
  			else
  				fcinfo->args[1].value = sslot.values[i];
  			fcinfo->isnull = false;
  			fresult = FunctionCallInvoke(fcinfo);
  			if (!fcinfo->isnull && DatumGetBool(fresult))
  				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, Oid collation,
  					  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)
  		{
  			LOCAL_FCINFO(fcinfo, 2);
  			int			nmatch = 0;
  			int			i;
  
  			/*
  			 * We invoke the opproc "by hand" so that we won't fail on NULL
  			 * results.  Such cases won't arise for normal comparison
  			 * functions, but generic_restriction_selectivity could perhaps be
  			 * used with operators that can return NULL.  A small side benefit
  			 * is to not need to re-initialize the fcinfo struct from scratch
  			 * each time.
  			 */
  			InitFunctionCallInfoData(*fcinfo, opproc, 2, collation,
  									 NULL, NULL);
  			fcinfo->args[0].isnull = false;
  			fcinfo->args[1].isnull = false;
  			/* be careful to apply operator right way 'round */
  			if (varonleft)
  				fcinfo->args[1].value = constval;
  			else
  				fcinfo->args[0].value = constval;
  
  			for (i = n_skip; i < sslot.nvalues - n_skip; i++)
  			{
  				Datum		fresult;
  
  				if (varonleft)
  					fcinfo->args[0].value = sslot.values[i];
  				else
  					fcinfo->args[1].value = sslot.values[i];
  				fcinfo->isnull = false;
  				fresult = FunctionCallInvoke(fcinfo);
  				if (!fcinfo->isnull && DatumGetBool(fresult))
  					nmatch++;
  			}
  			result = ((double) nmatch) / ((double) (sslot.nvalues - 2 * n_skip));
  		}
  		else
  			result = -1;
  		free_attstatsslot(&sslot);
  	}
  	else
  	{
  		*hist_size = 0;
  		result = -1;
  	}
  
  	return result;
  }
  
  /*
   *	generic_restriction_selectivity		- Selectivity for almost anything
   *
   * This function estimates selectivity for operators that we don't have any
   * special knowledge about, but are on data types that we collect standard
   * MCV and/or histogram statistics for.  (Additional assumptions are that
   * the operator is strict and immutable, or at least stable.)
   *
   * If we have "VAR OP CONST" or "CONST OP VAR", selectivity is estimated by
   * applying the operator to each element of the column's MCV and/or histogram
   * stats, and merging the results using the assumption that the histogram is
   * a reasonable random sample of the column's non-MCV population.  Note that
   * if the operator's semantics are related to the histogram ordering, this
   * might not be such a great assumption; other functions such as
   * scalarineqsel() are probably a better match in such cases.
   *
   * Otherwise, fall back to the default selectivity provided by the caller.
   */
  double
  generic_restriction_selectivity(PlannerInfo *root, Oid oproid, Oid collation,
  								List *args, int varRelid,
  								double default_selectivity)
  {
  	double		selec;
  	VariableStatData vardata;
  	Node	   *other;
  	bool		varonleft;
  
  	/*
  	 * If expression is not variable OP something or something OP variable,
  	 * then punt and return the default estimate.
  	 */
  	if (!get_restriction_variable(root, args, varRelid,
  								  &vardata, &other, &varonleft))
  		return default_selectivity;
  
  	/*
  	 * If the something is a NULL constant, assume operator is strict and
  	 * return zero, ie, operator will never return TRUE.
  	 */
  	if (IsA(other, Const) &&
  		((Const *) other)->constisnull)
  	{
  		ReleaseVariableStats(vardata);
  		return 0.0;
  	}
  
  	if (IsA(other, Const))
  	{
  		/* Variable is being compared to a known non-null constant */
  		Datum		constval = ((Const *) other)->constvalue;
  		FmgrInfo	opproc;
  		double		mcvsum;
  		double		mcvsel;
  		double		nullfrac;
  		int			hist_size;
  
  		fmgr_info(get_opcode(oproid), &opproc);
  
  		/*
  		 * Calculate the selectivity for the column's most common values.
  		 */
  		mcvsel = mcv_selectivity(&vardata, &opproc, collation,
  								 constval, varonleft,
  								 &mcvsum);
  
  		/*
  		 * If the histogram is large enough, see what fraction of it matches
  		 * the query, and assume that's representative of the non-MCV
  		 * population.  Otherwise use the default selectivity for the non-MCV
  		 * population.
  		 */
  		selec = histogram_selectivity(&vardata, &opproc, collation,
  									  constval, varonleft,
  									  10, 1, &hist_size);
  		if (selec < 0)
  		{
  			/* Nope, fall back on default */
  			selec = default_selectivity;
  		}
  		else if (hist_size < 100)
  		{
  			/*
  			 * For histogram sizes from 10 to 100, we combine the histogram
  			 * and default selectivities, putting increasingly more trust in
  			 * the histogram for larger sizes.
  			 */
  			double		hist_weight = hist_size / 100.0;
  
  			selec = selec * hist_weight +
  				default_selectivity * (1.0 - hist_weight);
  		}
  
  		/* In any case, don't believe extremely small or large estimates. */
  		if (selec < 0.0001)
  			selec = 0.0001;
  		else if (selec > 0.9999)
  			selec = 0.9999;
  
  		/* Don't forget to account for nulls. */
  		if (HeapTupleIsValid(vardata.statsTuple))
  			nullfrac = ((Form_pg_statistic) GETSTRUCT(vardata.statsTuple))->stanullfrac;
  		else
  			nullfrac = 0.0;
  
  		/*
  		 * 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 - nullfrac - mcvsum;
  		selec += mcvsel;
  	}
  	else
  	{
  		/* Comparison value is not constant, so we can't do anything */
  		selec = default_selectivity;
  	}
  
  	ReleaseVariableStats(vardata);
  
  	/* result should be in range, but make sure... */
  	CLAMP_PROBABILITY(selec);
  
  	return selec;
  }
  
  /*
   *	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.
   *
   * While opproc could be looked up from the operator OID, common callers
   * also need to call it separately, so we make the caller pass both.
   *
   * 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,
  						   Oid opoid, FmgrInfo *opproc, bool isgt, bool iseq,
  						   Oid collation,
  						   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.  Right now, we know there is only one, so just grab it and
  	 * see if it matches the query.
  	 *
  	 * Note that we can't use opoid as search argument; the staop appearing in
  	 * pg_statistic will be for the relevant '<' operator, but what we have
  	 * might be some other inequality operator such as '>='.  (Even if opoid
  	 * is a '<' operator, it could be cross-type.)  Hence we must use
  	 * comparison_ops_are_compatible() to see if the operators match.
  	 */
  	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 &&
  			sslot.stacoll == collation &&
  			comparison_ops_are_compatible(sslot.staop, opoid))
  		{
  			/*
  			 * 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).
  			 *
  			 * 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,
  													 collation,
  													 &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,
  														 collation,
  														 &sslot.values[0],
  														 NULL);
  				else if (probe == sslot.nvalues - 1 && sslot.nvalues > 2)
  					have_end = get_actual_variable_range(root,
  														 vardata,
  														 sslot.staop,
  														 collation,
  														 NULL,
  														 &sslot.values[probe]);
  
  				ltcmp = DatumGetBool(FunctionCall2Coll(opproc,
  													   collation,
  													   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, collation,
  									  &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;
  			}
  		}
  		else if (sslot.nvalues > 1)
  		{
  			/*
  			 * If we get here, we have a histogram but it's not sorted the way
  			 * we want.  Do a brute-force search to see how many of the
  			 * entries satisfy the comparison condition, and take that
  			 * fraction as our estimate.  (This is identical to the inner loop
  			 * of histogram_selectivity; maybe share code?)
  			 */
  			LOCAL_FCINFO(fcinfo, 2);
  			int			nmatch = 0;
  
  			InitFunctionCallInfoData(*fcinfo, opproc, 2, collation,
  									 NULL, NULL);
  			fcinfo->args[0].isnull = false;
  			fcinfo->args[1].isnull = false;
  			fcinfo->args[1].value = constval;
  			for (int i = 0; i < sslot.nvalues; i++)
  			{
  				Datum		fresult;
  
  				fcinfo->args[0].value = sslot.values[i];
  				fcinfo->isnull = false;
  				fresult = FunctionCallInvoke(fcinfo);
  				if (!fcinfo->isnull && DatumGetBool(fresult))
  					nmatch++;
  			}
  			hist_selec = ((double) nmatch) / ((double) sslot.nvalues);
  
  			/*
  			 * As above, clamp to a hundredth of the histogram resolution.
  			 * This case is surely even less trustworthy than the normal one,
  			 * so we shouldn't believe exact 0 or 1 selectivity.  (Maybe the
  			 * clamp should be more restrictive in this case?)
  			 */
  			{
  				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);
  	Oid			collation = PG_GET_COLLATION();
  	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, collation,
  						  &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, InvalidOid,
  							 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);
  	Oid			collation = PG_GET_COLLATION();
  	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, collation,
  								  &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, collation,
  									   &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, collation,
  									   &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, Oid collation,
  				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).
  		 */
  		LOCAL_FCINFO(fcinfo, 2);
  		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);
  
  		/*
  		 * Save a few cycles by setting up the fcinfo struct just once. Using
  		 * FunctionCallInvoke directly also avoids failure if the eqproc
  		 * returns NULL, though really equality functions should never do
  		 * that.
  		 */
  		InitFunctionCallInfoData(*fcinfo, &eqproc, 2, collation,
  								 NULL, NULL);
  		fcinfo->args[0].isnull = false;
  		fcinfo->args[1].isnull = false;
  
  		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;
  
  			fcinfo->args[0].value = sslot1->values[i];
  
  			for (j = 0; j < sslot2->nvalues; j++)
  			{
  				Datum		fresult;
  
  				if (hasmatch2[j])
  					continue;
  				fcinfo->args[1].value = sslot2->values[j];
  				fcinfo->isnull = false;
  				fresult = FunctionCallInvoke(fcinfo);
  				if (!fcinfo->isnull && DatumGetBool(fresult))
  				{
  					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, Oid collation,
  			   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.
  		 */
  		LOCAL_FCINFO(fcinfo, 2);
  		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);
  
  		/*
  		 * Save a few cycles by setting up the fcinfo struct just once. Using
  		 * FunctionCallInvoke directly also avoids failure if the eqproc
  		 * returns NULL, though really equality functions should never do
  		 * that.
  		 */
  		InitFunctionCallInfoData(*fcinfo, &eqproc, 2, collation,
  								 NULL, NULL);
  		fcinfo->args[0].isnull = false;
  		fcinfo->args[1].isnull = false;
  
  		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;
  
  			fcinfo->args[0].value = sslot1->values[i];
  
  			for (j = 0; j < clamped_nvalues2; j++)
  			{
  				Datum		fresult;
  
  				if (hasmatch2[j])
  					continue;
  				fcinfo->args[1].value = sslot2->values[j];
  				fcinfo->isnull = false;
  				fresult = FunctionCallInvoke(fcinfo);
  				if (!fcinfo->isnull && DatumGetBool(fresult))
  				{
  					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,
  				collation,
  				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;
  	collation = ((OpExpr *) clause)->inputcollid;
  	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, collation,
  								&leftmin, &leftmax))
  			goto fail;			/* no range available from stats */
  		if (!get_variable_range(root, &rightvar, rstatop, collation,
  								&rightmin, &rightmax))
  			goto fail;			/* no range available from stats */
  	}
  	else
  	{
  		/* need to swap the max and min */
  		if (!get_variable_range(root, &leftvar, lstatop, collation,
  								&leftmax, &leftmin))
  			goto fail;			/* no range available from stats */
  		if (!get_variable_range(root, &rightvar, rstatop, collation,
  								&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, collation, &leftvar,
  						  rightmax, op_righttype);
  	if (selec != DEFAULT_INEQ_SEL)
  		*leftend = selec;
  
  	/* And similarly for the right variable. */
  	selec = scalarineqsel(root, revleop, isgt, true, collation, &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, collation, &leftvar,
  						  rightmin, op_righttype);
  	if (selec != DEFAULT_INEQ_SEL)
  		*leftstart = selec;
  
  	/* And similarly for the right variable. */
  	selec = scalarineqsel(root, revltop, isgt, false, collation, &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);
  }
  
  
  /*
   *	matchingsel -- generic matching-operator selectivity support
   *
   * Use these for any operators that (a) are on data types for which we collect
   * standard statistics, and (b) have behavior for which the default estimate
   * (twice DEFAULT_EQ_SEL) is sane.  Typically that is good for match-like
   * operators.
   */
  
  Datum
  matchingsel(PG_FUNCTION_ARGS)
  {
  	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);
  	Oid			collation = PG_GET_COLLATION();
  	double		selec;
  
  	/* Use generic restriction selectivity logic. */
  	selec = generic_restriction_selectivity(root, operator, collation,
  											args, varRelid,
  											DEFAULT_MATCHING_SEL);
  
  	PG_RETURN_FLOAT8((float8) selec);
  }
  
  Datum
  matchingjoinsel(PG_FUNCTION_ARGS)
  {
  	/* Just punt, for the moment. */
  	PG_RETURN_FLOAT8(DEFAULT_MATCHING_SEL);
  }
  
  
  /*
   * 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 = hash_agg_entry_size(agg_costs->numAggs,
  													path->pathtarget->width,
  													agg_costs->transitionSpace);
  
  	/*
  	 * 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.
   * The collation must be specified too.
   */
  static bool
  get_variable_range(PlannerInfo *root, VariableStatData *vardata,
  				   Oid sortop, Oid collation,
  				   Datum *min, Datum *max)
  {
  	Datum		tmin = 0;
  	Datum		tmax = 0;
  	bool		have_data = false;
  	int16		typLen;
  	bool		typByVal;
  	Oid			opfuncoid;
  	FmgrInfo	opproc;
  	AttStatsSlot sslot;
  
  	/*
  	 * 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, collation, 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;
  
  	opproc.fn_oid = InvalidOid; /* mark this as not looked up yet */
  
  	get_typlenbyval(vardata->atttype, &typLen, &typByVal);
  
  	/*
  	 * If there is a histogram with the ordering we want, grab the first and
  	 * last values.
  	 */
  	if (get_attstatsslot(&sslot, vardata->statsTuple,
  						 STATISTIC_KIND_HISTOGRAM, sortop,
  						 ATTSTATSSLOT_VALUES))
  	{
  		if (sslot.stacoll == collation && 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);
  	}
  
  	/*
  	 * Otherwise, if there is a histogram with some other ordering, scan it
  	 * and get the min and max values according to the ordering we want.  This
  	 * of course may not find values that are really extremal according to our
  	 * ordering, but it beats ignoring available data.
  	 */
  	if (!have_data &&
  		get_attstatsslot(&sslot, vardata->statsTuple,
  						 STATISTIC_KIND_HISTOGRAM, InvalidOid,
  						 ATTSTATSSLOT_VALUES))
  	{
  		get_stats_slot_range(&sslot, opfuncoid, &opproc,
  							 collation, typLen, typByVal,
  							 &tmin, &tmax, &have_data);
  		free_attstatsslot(&sslot);
  	}
  
  	/*
  	 * 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.
  	 */
  	if (get_attstatsslot(&sslot, vardata->statsTuple,
  						 STATISTIC_KIND_MCV, InvalidOid,
  						 ATTSTATSSLOT_VALUES))
  	{
  		get_stats_slot_range(&sslot, opfuncoid, &opproc,
  							 collation, typLen, typByVal,
  							 &tmin, &tmax, &have_data);
  		free_attstatsslot(&sslot);
  	}
  
  	*min = tmin;
  	*max = tmax;
  	return have_data;
  }
  
  /*
   * get_stats_slot_range: scan sslot for min/max values
   *
   * Subroutine for get_variable_range: update min/max/have_data according
   * to what we find in the statistics array.
   */
  static void
  get_stats_slot_range(AttStatsSlot *sslot, Oid opfuncoid, FmgrInfo *opproc,
  					 Oid collation, int16 typLen, bool typByVal,
  					 Datum *min, Datum *max, bool *p_have_data)
  {
  	Datum		tmin = *min;
  	Datum		tmax = *max;
  	bool		have_data = *p_have_data;
  	bool		found_tmin = false;
  	bool		found_tmax = false;
  
  	/* Look up the comparison function, if we didn't already do so */
  	if (opproc->fn_oid != opfuncoid)
  		fmgr_info(opfuncoid, opproc);
  
  	/* Scan all the slot's values */
  	for (int i = 0; i < sslot->nvalues; i++)
  	{
  		if (!have_data)
  		{
  			tmin = tmax = sslot->values[i];
  			found_tmin = found_tmax = true;
  			*p_have_data = have_data = true;
  			continue;
  		}
  		if (DatumGetBool(FunctionCall2Coll(opproc,
  										   collation,
  										   sslot->values[i], tmin)))
  		{
  			tmin = sslot->values[i];
  			found_tmin = true;
  		}
  		if (DatumGetBool(FunctionCall2Coll(opproc,
  										   collation,
  										   tmax, sslot->values[i])))
  		{
  			tmax = sslot->values[i];
  			found_tmax = true;
  		}
  	}
  
  	/*
  	 * Copy the slot's values, if we found new extreme values.
  	 */
  	if (found_tmin)
  		*min = datumCopy(tmin, typByVal, typLen);
  	if (found_tmax)
  		*max = datumCopy(tmax, typByVal, typLen);
  }
  
  
  /*
   * 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.
   * collation is the required collation.
   */
  static bool
  get_actual_variable_range(PlannerInfo *root, VariableStatData *vardata,
  						  Oid sortop, Oid collation,
  						  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, sortop, and
  		 * collation --- but we can use a descending-order index.
  		 */
  		if (collation != index->indexcollations[0])
  			continue;			/* test first 'cause it's cheapest */
  		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(&sslot, vardata.statsTuple,
  							 STATISTIC_KIND_CORRELATION, sortop,
  							 ATTSTATSSLOT_NUMBERS))
  		{
  			double		varCorrelation;
  
  			Assert(sslot.nnumbers == 1);
  			varCorrelation = sslot.numbers[0];
  
  			if (index->reverse_sort[0])
  				varCorrelation = -varCorrelation;
  
  			if (index->nkeycolumns > 1)
  				costs.indexCorrelation = varCorrelation * 0.75;
  			else
  				costs.indexCorrelation = varCorrelation;
  
  			free_attstatsslot(&sslot);
  		}
  	}
  
  	ReleaseVariableStats(vardata);
  
  	*indexStartupCost = costs.indexStartupCost;
  	*indexTotalCost = costs.indexTotalCost;
  	*indexSelectivity = costs.indexSelectivity;
  	*indexCorrelation = costs.indexCorrelation;
  	*indexPages = costs.numIndexPages;
  }
  
  void
  hashcostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
  				 Cost *indexStartupCost, Cost *indexTotalCost,
  				 Selectivity *indexSelectivity, double *indexCorrelation,
  				 double *indexPages)
  {
  	GenericCosts costs;
  
  	MemSet(&costs, 0, sizeof(costs));
  
  	genericcostestimate(root, path, loop_count, &costs);
  
  	/*
  	 * A hash index has no descent costs as such, since the index AM can go
  	 * directly to the target bucket after computing the hash value.  There
  	 * are a couple of other hash-specific costs that we could conceivably add
  	 * here, though:
  	 *
  	 * Ideally we'd charge spc_random_page_cost for each page in the target
  	 * bucket, not just the numIndexPages pages that genericcostestimate
  	 * thought we'd visit.  However in most cases we don't know which bucket
  	 * that will be.  There's no point in considering the average bucket size
  	 * because the hash AM makes sure that's always one page.
  	 *
  	 * Likewise, we could consider charging some CPU for each index tuple in
  	 * the bucket, if we knew how many there were.  But the per-tuple cost is
  	 * just a hash value comparison, not a general datatype-dependent
  	 * comparison, so any such charge ought to be quite a bit less than
  	 * cpu_operator_cost; which makes it probably not worth worrying about.
  	 *
  	 * A bigger issue is that chance hash-value collisions will result in
  	 * wasted probes into the heap.  We don't currently attempt to model this
  	 * cost on the grounds that it's rare, but maybe it's not rare enough.
  	 * (Any fix for this ought to consider the generic lossy-operator problem,
  	 * though; it's not entirely hash-specific.)
  	 */
  
  	*indexStartupCost = costs.indexStartupCost;
  	*indexTotalCost = costs.indexTotalCost;
  	*indexSelectivity = costs.indexSelectivity;
  	*indexCorrelation = costs.indexCorrelation;
  	*indexPages = costs.numIndexPages;
  }
  
  void
  gistcostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
  				 Cost *indexStartupCost, Cost *indexTotalCost,
  				 Selectivity *indexSelectivity, double *indexCorrelation,
  				 double *indexPages)
  {
  	IndexOptInfo *index = path->indexinfo;
  	GenericCosts costs;
  	Cost		descentCost;
  
  	MemSet(&costs, 0, sizeof(costs));
  
  	genericcostestimate(root, path, loop_count, &costs);
  
  	/*
  	 * We model index descent costs similarly to those for btree, but to do
  	 * that we first need an idea of the tree height.  We somewhat arbitrarily
  	 * assume that the fanout is 100, meaning the tree height is at most
  	 * log100(index->pages).
  	 *
  	 * Although this computation isn't really expensive enough to require
  	 * caching, we might as well use index->tree_height to cache it.
  	 */
  	if (index->tree_height < 0) /* unknown? */
  	{
  		if (index->pages > 1)	/* avoid computing log(0) */
  			index->tree_height = (int) (log(index->pages) / log(100.0));
  		else
  			index->tree_height = 0;
  	}
  
  	/*
  	 * Add a CPU-cost component to represent the costs of initial descent. We
  	 * just use log(N) here not log2(N) since the branching factor isn't
  	 * necessarily two anyway.  As for btree, charge once per SA scan.
  	 */
  	if (index->tuples > 1)		/* avoid computing log(0) */
  	{
  		descentCost = ceil(log(index->tuples)) * cpu_operator_cost;
  		costs.indexStartupCost += descentCost;
  		costs.indexTotalCost += costs.num_sa_scans * descentCost;
  	}
  
  	/*
  	 * Likewise add a per-page charge, calculated the same as for btrees.
  	 */
  	descentCost = (index->tree_height + 1) * 50.0 * cpu_operator_cost;
  	costs.indexStartupCost += descentCost;
  	costs.indexTotalCost += costs.num_sa_scans * descentCost;
  
  	*indexStartupCost = costs.indexStartupCost;
  	*indexTotalCost = costs.indexTotalCost;
  	*indexSelectivity = costs.indexSelectivity;
  	*indexCorrelation = costs.indexCorrelation;
  	*indexPages = costs.numIndexPages;
  }
  
  void
  spgcostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
  				Cost *indexStartupCost, Cost *indexTotalCost,
  				Selectivity *indexSelectivity, double *indexCorrelation,
  				double *indexPages)
  {
  	IndexOptInfo *index = path->indexinfo;
  	GenericCosts costs;
  	Cost		descentCost;
  
  	MemSet(&costs, 0, sizeof(costs));
  
  	genericcostestimate(root, path, loop_count, &costs);
  
  	/*
  	 * We model index descent costs similarly to those for btree, but to do
  	 * that we first need an idea of the tree height.  We somewhat arbitrarily
  	 * assume that the fanout is 100, meaning the tree height is at most
  	 * log100(index->pages).
  	 *
  	 * Although this computation isn't really expensive enough to require
  	 * caching, we might as well use index->tree_height to cache it.
  	 */
  	if (index->tree_height < 0) /* unknown? */
  	{
  		if (index->pages > 1)	/* avoid computing log(0) */
  			index->tree_height = (int) (log(index->pages) / log(100.0));
  		else
  			index->tree_height = 0;
  	}
  
  	/*
  	 * Add a CPU-cost component to represent the costs of initial descent. We
  	 * just use log(N) here not log2(N) since the branching factor isn't
  	 * necessarily two anyway.  As for btree, charge once per SA scan.
  	 */
  	if (index->tuples > 1)		/* avoid computing log(0) */
  	{
  		descentCost = ceil(log(index->tuples)) * cpu_operator_cost;
  		costs.indexStartupCost += descentCost;
  		costs.indexTotalCost += costs.num_sa_scans * descentCost;
  	}
  
  	/*
  	 * Likewise add a per-page charge, calculated the same as for btrees.
  	 */
  	descentCost = (index->tree_height + 1) * 50.0 * cpu_operator_cost;
  	costs.indexStartupCost += descentCost;
  	costs.indexTotalCost += costs.num_sa_scans * descentCost;
  
  	*indexStartupCost = costs.indexStartupCost;
  	*indexTotalCost = costs.indexTotalCost;
  	*indexSelectivity = costs.indexSelectivity;
  	*indexCorrelation = costs.indexCorrelation;
  	*indexPages = costs.numIndexPages;
  }
  
  
  /*
   * Support routines for gincostestimate
   */
  
  typedef struct
  {
  	bool		attHasFullScan[INDEX_MAX_KEYS];
  	bool		attHasNormalScan[INDEX_MAX_KEYS];
  	double		partialEntries;
  	double		exactEntries;
  	double		searchEntries;
  	double		arrayScans;
  } GinQualCounts;
  
  /*
   * Estimate the number of index terms that need to be searched for while
   * testing the given GIN query, and increment the counts in *counts
   * appropriately.  If the query is unsatisfiable, return false.
   */
  static bool
  gincost_pattern(IndexOptInfo *index, int indexcol,
  				Oid clause_op, Datum query,
  				GinQualCounts *counts)
  {
  	FmgrInfo	flinfo;
  	Oid			extractProcOid;
  	Oid			collation;
  	int			strategy_op;
  	Oid			lefttype,
  				righttype;
  	int32		nentries = 0;
  	bool	   *partial_matches = NULL;
  	Pointer    *extra_data = NULL;
  	bool	   *nullFlags = NULL;
  	int32		searchMode = GIN_SEARCH_MODE_DEFAULT;
  	int32		i;
  
  	Assert(indexcol < index->nkeycolumns);
  
  	/*
  	 * Get the operator's strategy number and declared input data types within
  	 * the index opfamily.  (We don't need the latter, but we use
  	 * get_op_opfamily_properties because it will throw error if it fails to
  	 * find a matching pg_amop entry.)
  	 */
  	get_op_opfamily_properties(clause_op, index->opfamily[indexcol], false,
  							   &strategy_op, &lefttype, &righttype);
  
  	/*
  	 * GIN always uses the "default" support functions, which are those with
  	 * lefttype == righttype == the opclass' opcintype (see
  	 * IndexSupportInitialize in relcache.c).
  	 */
  	extractProcOid = get_opfamily_proc(index->opfamily[indexcol],
  									   index->opcintype[indexcol],
  									   index->opcintype[indexcol],
  									   GIN_EXTRACTQUERY_PROC);
  
  	if (!OidIsValid(extractProcOid))
  	{
  		/* should not happen; throw same error as index_getprocinfo */
  		elog(ERROR, "missing support function %d for attribute %d of index \"%s\"",
  			 GIN_EXTRACTQUERY_PROC, indexcol + 1,
  			 get_rel_name(index->indexoid));
  	}
  
  	/*
  	 * Choose collation to pass to extractProc (should match initGinState).
  	 */
  	if (OidIsValid(index->indexcollations[indexcol]))
  		collation = index->indexcollations[indexcol];
  	else
  		collation = DEFAULT_COLLATION_OID;
  
  	fmgr_info(extractProcOid, &flinfo);
  
  	set_fn_opclass_options(&flinfo, index->opclassoptions[indexcol]);
  
  	FunctionCall7Coll(&flinfo,
  					  collation,
  					  query,
  					  PointerGetDatum(&nentries),
  					  UInt16GetDatum(strategy_op),
  					  PointerGetDatum(&partial_matches),
  					  PointerGetDatum(&extra_data),
  					  PointerGetDatum(&nullFlags),
  					  PointerGetDatum(&searchMode));
  
  	if (nentries <= 0 && searchMode == GIN_SEARCH_MODE_DEFAULT)
  	{
  		/* No match is possible */
  		return false;
  	}
  
  	for (i = 0; i < nentries; i++)
  	{
  		/*
  		 * For partial match we haven't any information to estimate number of
  		 * matched entries in index, so, we just estimate it as 100
  		 */
  		if (partial_matches && partial_matches[i])
  			counts->partialEntries += 100;
  		else
  			counts->exactEntries++;
  
  		counts->searchEntries++;
  	}
  
  	if (searchMode == GIN_SEARCH_MODE_DEFAULT)
  	{
  		counts->attHasNormalScan[indexcol] = true;
  	}
  	else if (searchMode == GIN_SEARCH_MODE_INCLUDE_EMPTY)
  	{
  		/* Treat "include empty" like an exact-match item */
  		counts->attHasNormalScan[indexcol] = true;
  		counts->exactEntries++;
  		counts->searchEntries++;
  	}
  	else
  	{
  		/* It's GIN_SEARCH_MODE_ALL */
  		counts->attHasFullScan[indexcol] = true;
  	}
  
  	return true;
  }
  
  /*
   * Estimate the number of index terms that need to be searched for while
   * testing the given GIN index clause, and increment the counts in *counts
   * appropriately.  If the query is unsatisfiable, return false.
   */
  static bool
  gincost_opexpr(PlannerInfo *root,
  			   IndexOptInfo *index,
  			   int indexcol,
  			   OpExpr *clause,
  			   GinQualCounts *counts)
  {
  	Oid			clause_op = clause->opno;
  	Node	   *operand = (Node *) lsecond(clause->args);
  
  	/* aggressively reduce to a constant, and look through relabeling */
  	operand = estimate_expression_value(root, operand);
  
  	if (IsA(operand, RelabelType))
  		operand = (Node *) ((RelabelType *) operand)->arg;
  
  	/*
  	 * It's impossible to call extractQuery method for unknown operand. So
  	 * unless operand is a Const we can't do much; just assume there will be
  	 * one ordinary search entry from the operand at runtime.
  	 */
  	if (!IsA(operand, Const))
  	{
  		counts->exactEntries++;
  		counts->searchEntries++;
  		return true;
  	}
  
  	/* If Const is null, there can be no matches */
  	if (((Const *) operand)->constisnull)
  		return false;
  
  	/* Otherwise, apply extractQuery and get the actual term counts */
  	return gincost_pattern(index, indexcol, clause_op,
  						   ((Const *) operand)->constvalue,
  						   counts);
  }
  
  /*
   * Estimate the number of index terms that need to be searched for while
   * testing the given GIN index clause, and increment the counts in *counts
   * appropriately.  If the query is unsatisfiable, return false.
   *
   * A ScalarArrayOpExpr will give rise to N separate indexscans at runtime,
   * each of which involves one value from the RHS array, plus all the
   * non-array quals (if any).  To model this, we average the counts across
   * the RHS elements, and add the averages to the counts in *counts (which
   * correspond to per-indexscan costs).  We also multiply counts->arrayScans
   * by N, causing gincostestimate to scale up its estimates accordingly.
   */
  static bool
  gincost_scalararrayopexpr(PlannerInfo *root,
  						  IndexOptInfo *index,
  						  int indexcol,
  						  ScalarArrayOpExpr *clause,
  						  double numIndexEntries,
  						  GinQualCounts *counts)
  {
  	Oid			clause_op = clause->opno;
  	Node	   *rightop = (Node *) lsecond(clause->args);
  	ArrayType  *arrayval;
  	int16		elmlen;
  	bool		elmbyval;
  	char		elmalign;
  	int			numElems;
  	Datum	   *elemValues;
  	bool	   *elemNulls;
  	GinQualCounts arraycounts;
  	int			numPossible = 0;
  	int			i;
  
  	Assert(clause->useOr);
  
  	/* aggressively reduce to a constant, and look through relabeling */
  	rightop = estimate_expression_value(root, rightop);
  
  	if (IsA(rightop, RelabelType))
  		rightop = (Node *) ((RelabelType *) rightop)->arg;
  
  	/*
  	 * It's impossible to call extractQuery method for unknown operand. So
  	 * unless operand is a Const we can't do much; just assume there will be
  	 * one ordinary search entry from each array entry at runtime, and fall
  	 * back on a probably-bad estimate of the number of array entries.
  	 */
  	if (!IsA(rightop, Const))
  	{
  		counts->exactEntries++;
  		counts->searchEntries++;
  		counts->arrayScans *= estimate_array_length(rightop);
  		return true;
  	}
  
  	/* If Const is null, there can be no matches */
  	if (((Const *) rightop)->constisnull)
  		return false;
  
  	/* Otherwise, extract the array elements and iterate over them */
  	arrayval = DatumGetArrayTypeP(((Const *) rightop)->constvalue);
  	get_typlenbyvalalign(ARR_ELEMTYPE(arrayval),
  						 &elmlen, &elmbyval, &elmalign);
  	deconstruct_array(arrayval,
  					  ARR_ELEMTYPE(arrayval),
  					  elmlen, elmbyval, elmalign,
  					  &elemValues, &elemNulls, &numElems);
  
  	memset(&arraycounts, 0, sizeof(arraycounts));
  
  	for (i = 0; i < numElems; i++)
  	{
  		GinQualCounts elemcounts;
  
  		/* NULL can't match anything, so ignore, as the executor will */
  		if (elemNulls[i])
  			continue;
  
  		/* Otherwise, apply extractQuery and get the actual term counts */
  		memset(&elemcounts, 0, sizeof(elemcounts));
  
  		if (gincost_pattern(index, indexcol, clause_op, elemValues[i],
  							&elemcounts))
  		{
  			/* We ignore array elements that are unsatisfiable patterns */
  			numPossible++;
  
  			if (elemcounts.attHasFullScan[indexcol] &&
  				!elemcounts.attHasNormalScan[indexcol])
  			{
  				/*
  				 * Full index scan will be required.  We treat this as if
  				 * every key in the index had been listed in the query; is
  				 * that reasonable?
  				 */
  				elemcounts.partialEntries = 0;
  				elemcounts.exactEntries = numIndexEntries;
  				elemcounts.searchEntries = numIndexEntries;
  			}
  			arraycounts.partialEntries += elemcounts.partialEntries;
  			arraycounts.exactEntries += elemcounts.exactEntries;
  			arraycounts.searchEntries += elemcounts.searchEntries;
  		}
  	}
  
  	if (numPossible == 0)
  	{
  		/* No satisfiable patterns in the array */
  		return false;
  	}
  
  	/*
  	 * Now add the averages to the global counts.  This will give us an
  	 * estimate of the average number of terms searched for in each indexscan,
  	 * including contributions from both array and non-array quals.
  	 */
  	counts->partialEntries += arraycounts.partialEntries / numPossible;
  	counts->exactEntries += arraycounts.exactEntries / numPossible;
  	counts->searchEntries += arraycounts.searchEntries / numPossible;
  
  	counts->arrayScans *= numPossible;
  
  	return true;
  }
  
  /*
   * GIN has search behavior completely different from other index types
   */
  void
  gincostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
  				Cost *indexStartupCost, Cost *indexTotalCost,
  				Selectivity *indexSelectivity, double *indexCorrelation,
  				double *indexPages)
  {
  	IndexOptInfo *index = path->indexinfo;
  	List	   *indexQuals = get_quals_from_indexclauses(path->indexclauses);
  	List	   *selectivityQuals;
  	double		numPages = index->pages,
  				numTuples = index->tuples;
  	double		numEntryPages,
  				numDataPages,
  				numPendingPages,
  				numEntries;
  	GinQualCounts counts;
  	bool		matchPossible;
  	bool		fullIndexScan;
  	double		partialScale;
  	double		entryPagesFetched,
  				dataPagesFetched,
  				dataPagesFetchedBySel;
  	double		qual_op_cost,
  				qual_arg_cost,
  				spc_random_page_cost,
  				outer_scans;
  	Relation	indexRel;
  	GinStatsData ginStats;
  	ListCell   *lc;
  	int			i;
  
  	/*
  	 * Obtain statistical information from the meta page, if possible.  Else
  	 * set ginStats to zeroes, and we'll cope below.
  	 */
  	if (!index->hypothetical)
  	{
  		/* Lock should have already been obtained in plancat.c */
  		indexRel = index_open(index->indexoid, NoLock);
  		ginGetStats(indexRel, &ginStats);
  		index_close(indexRel, NoLock);
  	}
  	else
  	{
  		memset(&ginStats, 0, sizeof(ginStats));
  	}
  
  	/*
  	 * Assuming we got valid (nonzero) stats at all, nPendingPages can be
  	 * trusted, but the other fields are data as of the last VACUUM.  We can
  	 * scale them up to account for growth since then, but that method only
  	 * goes so far; in the worst case, the stats might be for a completely
  	 * empty index, and scaling them will produce pretty bogus numbers.
  	 * Somewhat arbitrarily, set the cutoff for doing scaling at 4X growth; if
  	 * it's grown more than that, fall back to estimating things only from the
  	 * assumed-accurate index size.  But we'll trust nPendingPages in any case
  	 * so long as it's not clearly insane, ie, more than the index size.
  	 */
  	if (ginStats.nPendingPages < numPages)
  		numPendingPages = ginStats.nPendingPages;
  	else
  		numPendingPages = 0;
  
  	if (numPages > 0 && ginStats.nTotalPages <= numPages &&
  		ginStats.nTotalPages > numPages / 4 &&
  		ginStats.nEntryPages > 0 && ginStats.nEntries > 0)
  	{
  		/*
  		 * OK, the stats seem close enough to sane to be trusted.  But we
  		 * still need to scale them by the ratio numPages / nTotalPages to
  		 * account for growth since the last VACUUM.
  		 */
  		double		scale = numPages / ginStats.nTotalPages;
  
  		numEntryPages = ceil(ginStats.nEntryPages * scale);
  		numDataPages = ceil(ginStats.nDataPages * scale);
  		numEntries = ceil(ginStats.nEntries * scale);
  		/* ensure we didn't round up too much */
  		numEntryPages = Min(numEntryPages, numPages - numPendingPages);
  		numDataPages = Min(numDataPages,
  						   numPages - numPendingPages - numEntryPages);
  	}
  	else
  	{
  		/*
  		 * We might get here because it's a hypothetical index, or an index
  		 * created pre-9.1 and never vacuumed since upgrading (in which case
  		 * its stats would read as zeroes), or just because it's grown too
  		 * much since the last VACUUM for us to put our faith in scaling.
  		 *
  		 * Invent some plausible internal statistics based on the index page
  		 * count (and clamp that to at least 10 pages, just in case).  We
  		 * estimate that 90% of the index is entry pages, and the rest is data
  		 * pages.  Estimate 100 entries per entry page; this is rather bogus
  		 * since it'll depend on the size of the keys, but it's more robust
  		 * than trying to predict the number of entries per heap tuple.
  		 */
  		numPages = Max(numPages, 10);
  		numEntryPages = floor((numPages - numPendingPages) * 0.90);
  		numDataPages = numPages - numPendingPages - numEntryPages;
  		numEntries = floor(numEntryPages * 100);
  	}
  
  	/* In an empty index, numEntries could be zero.  Avoid divide-by-zero */
  	if (numEntries < 1)
  		numEntries = 1;
  
  	/*
  	 * 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, indexQuals);
  
  	/* Estimate the fraction of main-table tuples that will be visited */
  	*indexSelectivity = clauselist_selectivity(root, selectivityQuals,
  											   index->rel->relid,
  											   JOIN_INNER,
  											   NULL);
  
  	/* fetch estimated page cost for tablespace containing index */
  	get_tablespace_page_costs(index->reltablespace,
  							  &spc_random_page_cost,
  							  NULL);
  
  	/*
  	 * Generic assumption about index correlation: there isn't any.
  	 */
  	*indexCorrelation = 0.0;
  
  	/*
  	 * Examine quals to estimate number of search entries & partial matches
  	 */
  	memset(&counts, 0, sizeof(counts));
  	counts.arrayScans = 1;
  	matchPossible = true;
  
  	foreach(lc, path->indexclauses)
  	{
  		IndexClause *iclause = lfirst_node(IndexClause, lc);
  		ListCell   *lc2;
  
  		foreach(lc2, iclause->indexquals)
  		{
  			RestrictInfo *rinfo = lfirst_node(RestrictInfo, lc2);
  			Expr	   *clause = rinfo->clause;
  
  			if (IsA(clause, OpExpr))
  			{
  				matchPossible = gincost_opexpr(root,
  											   index,
  											   iclause->indexcol,
  											   (OpExpr *) clause,
  											   &counts);
  				if (!matchPossible)
  					break;
  			}
  			else if (IsA(clause, ScalarArrayOpExpr))
  			{
  				matchPossible = gincost_scalararrayopexpr(root,
  														  index,
  														  iclause->indexcol,
  														  (ScalarArrayOpExpr *) clause,
  														  numEntries,
  														  &counts);
  				if (!matchPossible)
  					break;
  			}
  			else
  			{
  				/* shouldn't be anything else for a GIN index */
  				elog(ERROR, "unsupported GIN indexqual type: %d",
  					 (int) nodeTag(clause));
  			}
  		}
  	}
  
  	/* Fall out if there were any provably-unsatisfiable quals */
  	if (!matchPossible)
  	{
  		*indexStartupCost = 0;
  		*indexTotalCost = 0;
  		*indexSelectivity = 0;
  		return;
  	}
  
  	/*
  	 * If attribute has a full scan and at the same time doesn't have normal
  	 * scan, then we'll have to scan all non-null entries of that attribute.
  	 * Currently, we don't have per-attribute statistics for GIN.  Thus, we
  	 * must assume the whole GIN index has to be scanned in this case.
  	 */
  	fullIndexScan = false;
  	for (i = 0; i < index->nkeycolumns; i++)
  	{
  		if (counts.attHasFullScan[i] && !counts.attHasNormalScan[i])
  		{
  			fullIndexScan = true;
  			break;
  		}
  	}
  
  	if (fullIndexScan || indexQuals == NIL)
  	{
  		/*
  		 * Full index scan will be required.  We treat this as if every key in
  		 * the index had been listed in the query; is that reasonable?
  		 */
  		counts.partialEntries = 0;
  		counts.exactEntries = numEntries;
  		counts.searchEntries = numEntries;
  	}
  
  	/* Will we have more than one iteration of a nestloop scan? */
  	outer_scans = loop_count;
  
  	/*
  	 * Compute cost to begin scan, first of all, pay attention to pending
  	 * list.
  	 */
  	entryPagesFetched = numPendingPages;
  
  	/*
  	 * Estimate number of entry pages read.  We need to do
  	 * counts.searchEntries searches.  Use a power function as it should be,
  	 * but tuples on leaf pages usually is much greater. Here we include all
  	 * searches in entry tree, including search of first entry in partial
  	 * match algorithm
  	 */
  	entryPagesFetched += ceil(counts.searchEntries * rint(pow(numEntryPages, 0.15)));
  
  	/*
  	 * Add an estimate of entry pages read by partial match algorithm. It's a
  	 * scan over leaf pages in entry tree.  We haven't any useful stats here,
  	 * so estimate it as proportion.  Because counts.partialEntries is really
  	 * pretty bogus (see code above), it's possible that it is more than
  	 * numEntries; clamp the proportion to ensure sanity.
  	 */
  	partialScale = counts.partialEntries / numEntries;
  	partialScale = Min(partialScale, 1.0);
  
  	entryPagesFetched += ceil(numEntryPages * partialScale);
  
  	/*
  	 * Partial match algorithm reads all data pages before doing actual scan,
  	 * so it's a startup cost.  Again, we haven't any useful stats here, so
  	 * estimate it as proportion.
  	 */
  	dataPagesFetched = ceil(numDataPages * partialScale);
  
  	/*
  	 * Calculate cache effects if more than one scan due to nestloops or array
  	 * quals.  The result is pro-rated per nestloop scan, but the array qual
  	 * factor shouldn't be pro-rated (compare genericcostestimate).
  	 */
  	if (outer_scans > 1 || counts.arrayScans > 1)
  	{
  		entryPagesFetched *= outer_scans * counts.arrayScans;
  		entryPagesFetched = index_pages_fetched(entryPagesFetched,
  												(BlockNumber) numEntryPages,
  												numEntryPages, root);
  		entryPagesFetched /= outer_scans;
  		dataPagesFetched *= outer_scans * counts.arrayScans;
  		dataPagesFetched = index_pages_fetched(dataPagesFetched,
  											   (BlockNumber) numDataPages,
  											   numDataPages, root);
  		dataPagesFetched /= outer_scans;
  	}
  
  	/*
  	 * Here we use random page cost because logically-close pages could be far
  	 * apart on disk.
  	 */
  	*indexStartupCost = (entryPagesFetched + dataPagesFetched) * spc_random_page_cost;
  
  	/*
  	 * Now compute the number of data pages fetched during the scan.
  	 *
  	 * We assume every entry to have the same number of items, and that there
  	 * is no overlap between them. (XXX: tsvector and array opclasses collect
  	 * statistics on the frequency of individual keys; it would be nice to use
  	 * those here.)
  	 */
  	dataPagesFetched = ceil(numDataPages * counts.exactEntries / numEntries);
  
  	/*
  	 * If there is a lot of overlap among the entries, in particular if one of
  	 * the entries is very frequent, the above calculation can grossly
  	 * under-estimate.  As a simple cross-check, calculate a lower bound based
  	 * on the overall selectivity of the quals.  At a minimum, we must read
  	 * one item pointer for each matching entry.
  	 *
  	 * The width of each item pointer varies, based on the level of
  	 * compression.  We don't have statistics on that, but an average of
  	 * around 3 bytes per item is fairly typical.
  	 */
  	dataPagesFetchedBySel = ceil(*indexSelectivity *
  								 (numTuples / (BLCKSZ / 3)));
  	if (dataPagesFetchedBySel > dataPagesFetched)
  		dataPagesFetched = dataPagesFetchedBySel;
  
  	/* Account for cache effects, the same as above */
  	if (outer_scans > 1 || counts.arrayScans > 1)
  	{
  		dataPagesFetched *= outer_scans * counts.arrayScans;
  		dataPagesFetched = index_pages_fetched(dataPagesFetched,
  											   (BlockNumber) numDataPages,
  											   numDataPages, root);
  		dataPagesFetched /= outer_scans;
  	}
  
  	/* And apply random_page_cost as the cost per page */
  	*indexTotalCost = *indexStartupCost +
  		dataPagesFetched * spc_random_page_cost;
  
  	/*
  	 * Add on index qual eval costs, much as in genericcostestimate.  But we
  	 * can disregard indexorderbys, since GIN doesn't support those.
  	 */
  	qual_arg_cost = index_other_operands_eval_cost(root, indexQuals);
  	qual_op_cost = cpu_operator_cost * list_length(indexQuals);
  
  	*indexStartupCost += qual_arg_cost;
  	*indexTotalCost += qual_arg_cost;
  	*indexTotalCost += (numTuples * *indexSelectivity) * (cpu_index_tuple_cost + qual_op_cost);
  	*indexPages = dataPagesFetched;
  }
  
  /*
   * BRIN has search behavior completely different from other index types
   */
  void
  brincostestimate(PlannerInfo *root, IndexPath *path, double loop_count,
  				 Cost *indexStartupCost, Cost *indexTotalCost,
  				 Selectivity *indexSelectivity, double *indexCorrelation,
  				 double *indexPages)
  {
  	IndexOptInfo *index = path->indexinfo;
  	List	   *indexQuals = get_quals_from_indexclauses(path->indexclauses);
  	double		numPages = index->pages;
  	RelOptInfo *baserel = index->rel;
  	RangeTblEntry *rte = planner_rt_fetch(baserel->relid, root);
  	Cost		spc_seq_page_cost;
  	Cost		spc_random_page_cost;
  	double		qual_arg_cost;
  	double		qualSelectivity;
  	BrinStatsData statsData;
  	double		indexRanges;
  	double		minimalRanges;
  	double		estimatedRanges;
  	double		selec;
  	Relation	indexRel;
  	ListCell   *l;
  	VariableStatData vardata;
  
  	Assert(rte->rtekind == RTE_RELATION);
  
  	/* fetch estimated page cost for the tablespace containing the index */
  	get_tablespace_page_costs(index->reltablespace,
  							  &spc_random_page_cost,
  							  &spc_seq_page_cost);
  
  	/*
  	 * Obtain some data from the index itself, if possible.  Otherwise invent
  	 * some plausible internal statistics based on the relation page count.
  	 */
  	if (!index->hypothetical)
  	{
  		/*
  		 * A lock should have already been obtained on the index in plancat.c.
  		 */
  		indexRel = index_open(index->indexoid, NoLock);
  		brinGetStats(indexRel, &statsData);
  		index_close(indexRel, NoLock);
  
  		/* work out the actual number of ranges in the index */
  		indexRanges = Max(ceil((double) baserel->pages /
  							   statsData.pagesPerRange), 1.0);
  	}
  	else
  	{
  		/*
  		 * Assume default number of pages per range, and estimate the number
  		 * of ranges based on that.
  		 */
  		indexRanges = Max(ceil((double) baserel->pages /
  							   BRIN_DEFAULT_PAGES_PER_RANGE), 1.0);
  
  		statsData.pagesPerRange = BRIN_DEFAULT_PAGES_PER_RANGE;
  		statsData.revmapNumPages = (indexRanges / REVMAP_PAGE_MAXITEMS) + 1;
  	}
  
  	/*
  	 * Compute index correlation
  	 *
  	 * Because we can use all index quals equally when scanning, we can use
  	 * the largest correlation (in absolute value) among columns used by the
  	 * query.  Start at zero, the worst possible case.  If we cannot find any
  	 * correlation statistics, we will keep it as 0.
  	 */
  	*indexCorrelation = 0;
  
  	foreach(l, path->indexclauses)
  	{
  		IndexClause *iclause = lfirst_node(IndexClause, l);
  		AttrNumber	attnum = index->indexkeys[iclause->indexcol];
  
  		/* attempt to lookup stats in relation for this index column */
  		if (attnum != 0)
  		{
  			/* Simple variable -- look to stats for the underlying table */
  			if (get_relation_stats_hook &&
  				(*get_relation_stats_hook) (root, rte, attnum, &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(rte->relid),
  									Int16GetDatum(attnum),
  									BoolGetDatum(false));
  				vardata.freefunc = ReleaseSysCache;
  			}
  		}
  		else
  		{
  			/*
  			 * Looks like we've found an expression column in the index. Let's
  			 * see if there's any stats for it.
  			 */
  
  			/* get the attnum from the 0-based index. */
  			attnum = iclause->indexcol + 1;
  
  			if (get_index_stats_hook &&
  				(*get_index_stats_hook) (root, index->indexoid, attnum, &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(index->indexoid),
  													 Int16GetDatum(attnum),
  													 BoolGetDatum(false));
  				vardata.freefunc = ReleaseSysCache;
  			}
  		}
  
  		if (HeapTupleIsValid(vardata.statsTuple))
  		{
  			AttStatsSlot sslot;
  
  			if (get_attstatsslot(&sslot, vardata.statsTuple,
  								 STATISTIC_KIND_CORRELATION, InvalidOid,
  								 ATTSTATSSLOT_NUMBERS))
  			{
  				double		varCorrelation = 0.0;
  
  				if (sslot.nnumbers > 0)
  					varCorrelation = Abs(sslot.numbers[0]);
  
  				if (varCorrelation > *indexCorrelation)
  					*indexCorrelation = varCorrelation;
  
  				free_attstatsslot(&sslot);
  			}
  		}
  
  		ReleaseVariableStats(vardata);
  	}
  
  	qualSelectivity = clauselist_selectivity(root, indexQuals,
  											 baserel->relid,
  											 JOIN_INNER, NULL);
  
  	/*
  	 * Now calculate the minimum possible ranges we could match with if all of
  	 * the rows were in the perfect order in the table's heap.
  	 */
  	minimalRanges = ceil(indexRanges * qualSelectivity);
  
  	/*
  	 * Now estimate the number of ranges that we'll touch by using the
  	 * indexCorrelation from the stats. Careful not to divide by zero (note
  	 * we're using the absolute value of the correlation).
  	 */
  	if (*indexCorrelation < 1.0e-10)
  		estimatedRanges = indexRanges;
  	else
  		estimatedRanges = Min(minimalRanges / *indexCorrelation, indexRanges);
  
  	/* we expect to visit this portion of the table */
  	selec = estimatedRanges / indexRanges;
  
  	CLAMP_PROBABILITY(selec);
  
  	*indexSelectivity = selec;
  
  	/*
  	 * Compute the index qual costs, much as in genericcostestimate, to add to
  	 * the index costs.  We can disregard indexorderbys, since BRIN doesn't
  	 * support those.
  	 */
  	qual_arg_cost = index_other_operands_eval_cost(root, indexQuals);
  
  	/*
  	 * Compute the startup cost as the cost to read the whole revmap
  	 * sequentially, including the cost to execute the index quals.
  	 */
  	*indexStartupCost =
  		spc_seq_page_cost * statsData.revmapNumPages * loop_count;
  	*indexStartupCost += qual_arg_cost;
  
  	/*
  	 * To read a BRIN index there might be a bit of back and forth over
  	 * regular pages, as revmap might point to them out of sequential order;
  	 * calculate the total cost as reading the whole index in random order.
  	 */
  	*indexTotalCost = *indexStartupCost +
  		spc_random_page_cost * (numPages - statsData.revmapNumPages) * loop_count;
  
  	/*
  	 * Charge a small amount per range tuple which we expect to match to. This
  	 * is meant to reflect the costs of manipulating the bitmap. The BRIN scan
  	 * will set a bit for each page in the range when we find a matching
  	 * range, so we must multiply the charge by the number of pages in the
  	 * range.
  	 */
  	*indexTotalCost += 0.1 * cpu_operator_cost * estimatedRanges *
  		statsData.pagesPerRange;
  
  	*indexPages = index->pages;
  }
  