*/
#ifdef SQLITE_64BIT_STATS
 typedef u64 tRowcnt;    /* 64-bit only if requested at compile-time */
#else
 typedef u32 tRowcnt;    /* 32-bit is the default */
#endif


























/*
** Macros to determine whether the machine is big or little endian,
** evaluated at runtime.
*/
#ifdef SQLITE_AMALGAMATION
const int sqlite3one = 1;
#else
................................................................................
  ExprList *pCheck;    /* All CHECK constraints */
#endif
  tRowcnt nRowEst;     /* Estimated rows in table - from sqlite_stat1 table */
  int tnum;            /* Root BTree node for this table (see note above) */
  i16 iPKey;           /* If not negative, use aCol[iPKey] as the primary key */
  i16 nCol;            /* Number of columns in this table */
  u16 nRef;            /* Number of pointers to this Table */

  u8 tabFlags;         /* Mask of TF_* values */
  u8 keyConf;          /* What to do in case of uniqueness conflict on iPKey */
#ifndef SQLITE_OMIT_ALTERTABLE
  int addColOffset;    /* Offset in CREATE TABLE stmt to add a new column */
#endif
#ifndef SQLITE_OMIT_VIRTUALTABLE
  int nModuleArg;      /* Number of arguments to the module */
................................................................................
  char *zColAff;           /* String defining the affinity of each column */
  Index *pNext;            /* The next index associated with the same table */
  Schema *pSchema;         /* Schema containing this index */
  u8 *aSortOrder;          /* for each column: True==DESC, False==ASC */
  char **azColl;           /* Array of collation sequence names for index */
  Expr *pPartIdxWhere;     /* WHERE clause for partial indices */
  int tnum;                /* DB Page containing root of this index */

  u16 nColumn;             /* Number of columns in table used by this index */
  u8 iScanRatio;           /* Scan rate relative to the main table */
  unsigned onError:4;      /* OE_Abort, OE_Ignore, OE_Replace, or OE_None */
  unsigned autoIndex:2;    /* 1==UNIQUE, 2==PRIMARY KEY, 0==CREATE INDEX */
  unsigned bUnordered:1;   /* Use this index for == or IN queries only */
  unsigned uniqNotNull:1;  /* True if UNIQUE and NOT NULL for all columns */
#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
  int nSample;             /* Number of elements in aSample[] */
  int nSampleCol;          /* Size of IndexSample.anEq[] and so on */
  tRowcnt *aAvgEq;         /* Average nEq values for keys not in aSample */
................................................................................
int sqlite3FixTriggerStep(DbFixer*, TriggerStep*);
int sqlite3AtoF(const char *z, double*, int, u8);
int sqlite3GetInt32(const char *, int*);
int sqlite3Atoi(const char*);
int sqlite3Utf16ByteLen(const void *pData, int nChar);
int sqlite3Utf8CharLen(const char *pData, int nByte);
u32 sqlite3Utf8Read(const u8**);







/*
** Routines to read and write variable-length integers.  These used to
** be defined locally, but now we use the varint routines in the util.c
** file.  Code should use the MACRO forms below, as the Varint32 versions
** are coded to assume the single byte case is already handled (which 
** the MACRO form does).

*/
#ifdef SQLITE_64BIT_STATS
 typedef u64 tRowcnt;    /* 64-bit only if requested at compile-time */
#else
 typedef u32 tRowcnt;    /* 32-bit is the default */
#endif

/*
** Estimated quantities used for query planning are stored as 16-bit
** logarithms.  For quantity X, the value stored is 10*log2(X).  This
** gives a possible range of values of approximately 1.0e986 to 1e-986.
** But the allowed values are "grainy".  Not every value is representable.
** For example, quantities 16 and 17 are both represented by a LogEst
** of 40.  However, since LogEst quantatites are suppose to be estimates,
** not exact values, this imprecision is not a problem.
**
** "LogEst" is short for "Logarithimic Estimate".
**
** Examples:
**      1 -> 0              20 -> 43          10000 -> 132
**      2 -> 10             25 -> 46          25000 -> 146
**      3 -> 16            100 -> 66        1000000 -> 199
**      4 -> 20           1000 -> 99        1048576 -> 200
**     10 -> 33           1024 -> 100    4294967296 -> 320
**
** The LogEst can be negative to indicate fractional values. 
** Examples:
**
**    0.5 -> -10           0.1 -> -33        0.0625 -> -40
*/
typedef INT16_TYPE LogEst;

/*
** Macros to determine whether the machine is big or little endian,
** evaluated at runtime.
*/
#ifdef SQLITE_AMALGAMATION
const int sqlite3one = 1;
#else
................................................................................
  ExprList *pCheck;    /* All CHECK constraints */
#endif
  tRowcnt nRowEst;     /* Estimated rows in table - from sqlite_stat1 table */
  int tnum;            /* Root BTree node for this table (see note above) */
  i16 iPKey;           /* If not negative, use aCol[iPKey] as the primary key */
  i16 nCol;            /* Number of columns in this table */
  u16 nRef;            /* Number of pointers to this Table */
  LogEst szTabRow;     /* Estimated size of each table row in bytes */
  u8 tabFlags;         /* Mask of TF_* values */
  u8 keyConf;          /* What to do in case of uniqueness conflict on iPKey */
#ifndef SQLITE_OMIT_ALTERTABLE
  int addColOffset;    /* Offset in CREATE TABLE stmt to add a new column */
#endif
#ifndef SQLITE_OMIT_VIRTUALTABLE
  int nModuleArg;      /* Number of arguments to the module */
................................................................................
  char *zColAff;           /* String defining the affinity of each column */
  Index *pNext;            /* The next index associated with the same table */
  Schema *pSchema;         /* Schema containing this index */
  u8 *aSortOrder;          /* for each column: True==DESC, False==ASC */
  char **azColl;           /* Array of collation sequence names for index */
  Expr *pPartIdxWhere;     /* WHERE clause for partial indices */
  int tnum;                /* DB Page containing root of this index */
  LogEst szIdxRow;         /* Estimated average row size in bytes */
  u16 nColumn;             /* Number of columns in table used by this index */
  u8 iScanRatio;           /* Scan rate relative to the main table */
  u8 onError;              /* OE_Abort, OE_Ignore, OE_Replace, or OE_None */
  unsigned autoIndex:2;    /* 1==UNIQUE, 2==PRIMARY KEY, 0==CREATE INDEX */
  unsigned bUnordered:1;   /* Use this index for == or IN queries only */
  unsigned uniqNotNull:1;  /* True if UNIQUE and NOT NULL for all columns */
#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
  int nSample;             /* Number of elements in aSample[] */
  int nSampleCol;          /* Size of IndexSample.anEq[] and so on */
  tRowcnt *aAvgEq;         /* Average nEq values for keys not in aSample */
................................................................................
int sqlite3FixTriggerStep(DbFixer*, TriggerStep*);
int sqlite3AtoF(const char *z, double*, int, u8);
int sqlite3GetInt32(const char *, int*);
int sqlite3Atoi(const char*);
int sqlite3Utf16ByteLen(const void *pData, int nChar);
int sqlite3Utf8CharLen(const char *pData, int nByte);
u32 sqlite3Utf8Read(const u8**);
LogEst sqlite3LogEst(u64);
LogEst sqlite3LogEstAdd(LogEst,LogEst);
#ifndef SQLITE_OMIT_VIRTUALTABLE
LogEst sqlite3LogEstFromDouble(double);
#endif
u64 sqlite3LogEstToInt(LogEst);

/*
** Routines to read and write variable-length integers.  These used to
** be defined locally, but now we use the varint routines in the util.c
** file.  Code should use the MACRO forms below, as the Varint32 versions
** are coded to assume the single byte case is already handled (which 
** the MACRO form does).

    int i, sz;
    sz = sqlite3Strlen30(z);
    for(i=sz-1; i>0 && z[i]!='/' && z[i]!='.'; i--){}
    if( z[i]=='.' && ALWAYS(sz>i+4) ) memmove(&z[i+1], &z[sz-3], 4);
  }
}
#endif

    int i, sz;
    sz = sqlite3Strlen30(z);
    for(i=sz-1; i>0 && z[i]!='/' && z[i]!='.'; i--){}
    if( z[i]=='.' && ALWAYS(sz>i+4) ) memmove(&z[i+1], &z[sz-3], 4);
  }
}
#endif

/* 
** Find (an approximate) sum of two LogEst values.  This computation is
** not a simple "+" operator because LogEst is stored as a logarithmic
** value.
** 
*/
LogEst sqlite3LogEstAdd(LogEst a, LogEst b){
  static const unsigned char x[] = {
     10, 10,                         /* 0,1 */
      9, 9,                          /* 2,3 */
      8, 8,                          /* 4,5 */
      7, 7, 7,                       /* 6,7,8 */
      6, 6, 6,                       /* 9,10,11 */
      5, 5, 5,                       /* 12-14 */
      4, 4, 4, 4,                    /* 15-18 */
      3, 3, 3, 3, 3, 3,              /* 19-24 */
      2, 2, 2, 2, 2, 2, 2,           /* 25-31 */
  };
  if( a>=b ){
    if( a>b+49 ) return a;
    if( a>b+31 ) return a+1;
    return a+x[a-b];
  }else{
    if( b>a+49 ) return b;
    if( b>a+31 ) return b+1;
    return b+x[b-a];
  }
}

/*
** Convert an integer into a LogEst.  In other words, compute a
** good approximatation for 10*log2(x).
*/
LogEst sqlite3LogEst(u64 x){
  static LogEst a[] = { 0, 2, 3, 5, 6, 7, 8, 9 };
  LogEst y = 40;
  if( x<8 ){
    if( x<2 ) return 0;
    while( x<8 ){  y -= 10; x <<= 1; }
  }else{
    while( x>255 ){ y += 40; x >>= 4; }
    while( x>15 ){  y += 10; x >>= 1; }
  }
  return a[x&7] + y - 10;
}

#ifndef SQLITE_OMIT_VIRTUALTABLE
/*
** Convert a double into a LogEst
** In other words, compute an approximation for 10*log2(x).
*/
LogEst sqlite3LogEstFromDouble(double x){
  u64 a;
  LogEst e;
  assert( sizeof(x)==8 && sizeof(a)==8 );
  if( x<=1 ) return 0;
  if( x<=2000000000 ) return sqlite3LogEst((u64)x);
  memcpy(&a, &x, 8);
  e = (a>>52) - 1022;
  return e*10;
}
#endif /* SQLITE_OMIT_VIRTUALTABLE */

/*
** Convert a LogEst into an integer.
*/
u64 sqlite3LogEstToInt(LogEst x){
  u64 n;
  if( x<10 ) return 1;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
  if( x>=3 ) return (n+8)<<(x-3);
  return (n+8)>>(3-x);
}

typedef struct WherePath WherePath;
typedef struct WhereTerm WhereTerm;
typedef struct WhereLoopBuilder WhereLoopBuilder;
typedef struct WhereScan WhereScan;
typedef struct WhereOrCost WhereOrCost;
typedef struct WhereOrSet WhereOrSet;

/*
** Cost X is tracked as 10*log2(X) stored in a 16-bit integer.  The
** maximum cost for ordinary tables is 64*(2**63) which becomes 6900.
** (Virtual tables can return a larger cost, but let's assume they do not.)
** So all costs can be stored in a 16-bit integer without risk
** of overflow.
**
** Costs are estimates, so no effort is made to compute 10*log2(X) exactly.
** Instead, a close estimate is used.  Any value of X=1 is stored as 0.
** X=2 is 10.  X=3 is 16.  X=1000 is 99. etc.  Negative values are allowed.
** A WhereCost of -10 means 0.5.  WhereCost of -20 means 0.25.  And so forth.
**
** The tool/wherecosttest.c source file implements a command-line program
** that will convert WhereCosts to integers, convert integers to WhereCosts
** and do addition and multiplication on WhereCost values.  The wherecosttest
** command-line program is a useful utility to have around when working with
** this module.
*/
typedef short int WhereCost;

/*
** This object contains information needed to implement a single nested
** loop in WHERE clause.
**
** Contrast this object with WhereLoop.  This object describes the
** implementation of the loop.  WhereLoop describes the algorithm.
** This object contains a pointer to the WhereLoop algorithm as one of
................................................................................
  Bitmask prereq;       /* Bitmask of other loops that must run first */
  Bitmask maskSelf;     /* Bitmask identifying table iTab */
#ifdef SQLITE_DEBUG
  char cId;             /* Symbolic ID of this loop for debugging use */
#endif
  u8 iTab;              /* Position in FROM clause of table for this loop */
  u8 iSortIdx;          /* Sorting index number.  0==None */
  WhereCost rSetup;     /* One-time setup cost (ex: create transient index) */
  WhereCost rRun;       /* Cost of running each loop */
  WhereCost nOut;       /* Estimated number of output rows */
  union {
    struct {               /* Information for internal btree tables */
      int nEq;               /* Number of equality constraints */
      Index *pIndex;         /* Index used, or NULL */
    } btree;
    struct {               /* Information for virtual tables */
      int idxNum;            /* Index number */
................................................................................

/* This object holds the prerequisites and the cost of running a
** subquery on one operand of an OR operator in the WHERE clause.
** See WhereOrSet for additional information 
*/
struct WhereOrCost {
  Bitmask prereq;     /* Prerequisites */
  WhereCost rRun;     /* Cost of running this subquery */
  WhereCost nOut;     /* Number of outputs for this subquery */
};

/* The WhereOrSet object holds a set of possible WhereOrCosts that
** correspond to the subquery(s) of OR-clause processing.  Only the
** best N_OR_COST elements are retained.
*/
#define N_OR_COST 3
................................................................................
** of length 2.  And so forth until the length of WherePaths equals the
** number of nodes in the FROM clause.  The best (lowest cost) WherePath
** at the end is the choosen query plan.
*/
struct WherePath {
  Bitmask maskLoop;     /* Bitmask of all WhereLoop objects in this path */
  Bitmask revLoop;      /* aLoop[]s that should be reversed for ORDER BY */
  WhereCost nRow;       /* Estimated number of rows generated by this path */
  WhereCost rCost;      /* Total cost of this path */
  u8 isOrdered;         /* True if this path satisfies ORDER BY */
  u8 isOrderedValid;    /* True if the isOrdered field is valid */
  WhereLoop **aLoop;    /* Array of WhereLoop objects implementing this path */
};

/*
** The query generator uses an array of instances of this structure to
................................................................................
  int iParent;            /* Disable pWC->a[iParent] when this term disabled */
  int leftCursor;         /* Cursor number of X in "X <op> <expr>" */
  union {
    int leftColumn;         /* Column number of X in "X <op> <expr>" */
    WhereOrInfo *pOrInfo;   /* Extra information if (eOperator & WO_OR)!=0 */
    WhereAndInfo *pAndInfo; /* Extra information if (eOperator& WO_AND)!=0 */
  } u;
  WhereCost truthProb;    /* Probability of truth for this expression */
  u16 eOperator;          /* A WO_xx value describing <op> */
  u8 wtFlags;             /* TERM_xxx bit flags.  See below */
  u8 nChild;              /* Number of children that must disable us */
  WhereClause *pWC;       /* The clause this term is part of */
  Bitmask prereqRight;    /* Bitmask of tables used by pExpr->pRight */
  Bitmask prereqAll;      /* Bitmask of tables referenced by pExpr */
};
................................................................................
struct WhereInfo {
  Parse *pParse;            /* Parsing and code generating context */
  SrcList *pTabList;        /* List of tables in the join */
  ExprList *pOrderBy;       /* The ORDER BY clause or NULL */
  ExprList *pResultSet;     /* Result set. DISTINCT operates on these */
  WhereLoop *pLoops;        /* List of all WhereLoop objects */
  Bitmask revMask;          /* Mask of ORDER BY terms that need reversing */
  WhereCost nRowOut;        /* Estimated number of output rows */
  u16 wctrlFlags;           /* Flags originally passed to sqlite3WhereBegin() */
  u8 bOBSat;                /* ORDER BY satisfied by indices */
  u8 okOnePass;             /* Ok to use one-pass algorithm for UPDATE/DELETE */
  u8 untestedTerms;         /* Not all WHERE terms resolved by outer loop */
  u8 eDistinct;             /* One of the WHERE_DISTINCT_* values below */
  u8 nLevel;                /* Number of nested loop */
  int iTop;                 /* The very beginning of the WHERE loop */
................................................................................
#define WHERE_INDEXED      0x00000200  /* WhereLoop.u.btree.pIndex is valid */
#define WHERE_VIRTUALTABLE 0x00000400  /* WhereLoop.u.vtab is valid */
#define WHERE_IN_ABLE      0x00000800  /* Able to support an IN operator */
#define WHERE_ONEROW       0x00001000  /* Selects no more than one row */
#define WHERE_MULTI_OR     0x00002000  /* OR using multiple indices */
#define WHERE_AUTO_INDEX   0x00004000  /* Uses an ephemeral index */


/* Convert a WhereCost value (10 times log2(X)) into its integer value X.
** A rough approximation is used.  The value returned is not exact.
*/
static u64 whereCostToInt(WhereCost x){
  u64 n;
  if( x<10 ) return 1;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
  if( x>=3 ) return (n+8)<<(x-3);
  return (n+8)>>(3-x);
}

/*
** Return the estimated number of output rows from a WHERE clause
*/
u64 sqlite3WhereOutputRowCount(WhereInfo *pWInfo){
  return whereCostToInt(pWInfo->nRowOut);
}

/*
** Return one of the WHERE_DISTINCT_xxxxx values to indicate how this
** WHERE clause returns outputs for DISTINCT processing.
*/
int sqlite3WhereIsDistinct(WhereInfo *pWInfo){
................................................................................
** The new entry might overwrite an existing entry, or it might be
** appended, or it might be discarded.  Do whatever is the right thing
** so that pSet keeps the N_OR_COST best entries seen so far.
*/
static int whereOrInsert(
  WhereOrSet *pSet,      /* The WhereOrSet to be updated */
  Bitmask prereq,        /* Prerequisites of the new entry */
  WhereCost rRun,        /* Run-cost of the new entry */
  WhereCost nOut         /* Number of outputs for the new entry */
){
  u16 i;
  WhereOrCost *p;
  for(i=pSet->n, p=pSet->a; i>0; i--, p++){
    if( rRun<=p->rRun && (prereq & p->prereq)==prereq ){
      goto whereOrInsert_done;
    }
................................................................................
    }
  }
  if( pWC->a!=pWC->aStatic ){
    sqlite3DbFree(db, pWC->a);
  }
}

/* Forward declaration */
static WhereCost whereCost(tRowcnt x);

/*
** Add a single new WhereTerm entry to the WhereClause object pWC.
** The new WhereTerm object is constructed from Expr p and with wtFlags.
** The index in pWC->a[] of the new WhereTerm is returned on success.
** 0 is returned if the new WhereTerm could not be added due to a memory
** allocation error.  The memory allocation failure will be recorded in
** the db->mallocFailed flag so that higher-level functions can detect it.
................................................................................
    if( pOld!=pWC->aStatic ){
      sqlite3DbFree(db, pOld);
    }
    pWC->nSlot = sqlite3DbMallocSize(db, pWC->a)/sizeof(pWC->a[0]);
  }
  pTerm = &pWC->a[idx = pWC->nTerm++];
  if( p && ExprHasProperty(p, EP_Unlikely) ){
    pTerm->truthProb = whereCost(p->iTable) - 99;
  }else{
    pTerm->truthProb = -1;
  }
  pTerm->pExpr = sqlite3ExprSkipCollate(p);
  pTerm->wtFlags = wtFlags;
  pTerm->pWC = pWC;
  pTerm->iParent = -1;
................................................................................
      return 1;
    }
  }

  return 0;
}

/* 
** Find (an approximate) sum of two WhereCosts.  This computation is
** not a simple "+" operator because WhereCost is stored as a logarithmic
** value.
** 
*/
static WhereCost whereCostAdd(WhereCost a, WhereCost b){
  static const unsigned char x[] = {
     10, 10,                         /* 0,1 */
      9, 9,                          /* 2,3 */
      8, 8,                          /* 4,5 */
      7, 7, 7,                       /* 6,7,8 */
      6, 6, 6,                       /* 9,10,11 */
      5, 5, 5,                       /* 12-14 */
      4, 4, 4, 4,                    /* 15-18 */
      3, 3, 3, 3, 3, 3,              /* 19-24 */
      2, 2, 2, 2, 2, 2, 2,           /* 25-31 */
  };
  if( a>=b ){
    if( a>b+49 ) return a;
    if( a>b+31 ) return a+1;
    return a+x[a-b];
  }else{
    if( b>a+49 ) return b;
    if( b>a+31 ) return b+1;
    return b+x[b-a];
  }
}

/*
** Convert an integer into a WhereCost.  In other words, compute a
** good approximatation for 10*log2(x).
*/
static WhereCost whereCost(tRowcnt x){
  static WhereCost a[] = { 0, 2, 3, 5, 6, 7, 8, 9 };
  WhereCost y = 40;
  if( x<8 ){
    if( x<2 ) return 0;
    while( x<8 ){  y -= 10; x <<= 1; }
  }else{
    while( x>255 ){ y += 40; x >>= 4; }
    while( x>15 ){  y += 10; x >>= 1; }
  }
  return a[x&7] + y - 10;
}

#ifndef SQLITE_OMIT_VIRTUALTABLE
/*
** Convert a double (as received from xBestIndex of a virtual table)
** into a WhereCost.  In other words, compute an approximation for
** 10*log2(x).
*/
static WhereCost whereCostFromDouble(double x){
  u64 a;
  WhereCost e;
  assert( sizeof(x)==8 && sizeof(a)==8 );
  if( x<=1 ) return 0;
  if( x<=2000000000 ) return whereCost((tRowcnt)x);
  memcpy(&a, &x, 8);
  e = (a>>52) - 1022;
  return e*10;
}
#endif /* SQLITE_OMIT_VIRTUALTABLE */

/*
** Estimate the logarithm of the input value to base 2.
*/
static WhereCost estLog(WhereCost N){
  WhereCost x = whereCost(N);


  return x>33 ? x - 33 : 0;
}

/*
** Two routines for printing the content of an sqlite3_index_info
** structure.  Used for testing and debugging only.  If neither
** SQLITE_TEST or SQLITE_DEBUG are defined, then these routines
................................................................................
** then nEq is set to 1 (as the range restricted column, b, is the second 
** left-most column of the index). Or, if the query is:
**
**   ... FROM t1 WHERE a > ? AND a < ? ...
**
** then nEq is set to 0.
**
** When this function is called, *pnOut is set to the whereCost() of the
** number of rows that the index scan is expected to visit without 
** considering the range constraints. If nEq is 0, this is the number of 
** rows in the index. Assuming no error occurs, *pnOut is adjusted (reduced)
** to account for the range contraints pLower and pUpper.
** 
** In the absence of sqlite_stat4 ANALYZE data, or if such data cannot be
** used, each range inequality reduces the search space by a factor of 4. 
................................................................................
** rows visited by a factor of 16.
*/
static int whereRangeScanEst(
  Parse *pParse,       /* Parsing & code generating context */
  WhereLoopBuilder *pBuilder,
  WhereTerm *pLower,   /* Lower bound on the range. ex: "x>123" Might be NULL */
  WhereTerm *pUpper,   /* Upper bound on the range. ex: "x<455" Might be NULL */
  WhereCost *pnOut     /* IN/OUT: Number of rows visited */
){
  int rc = SQLITE_OK;
  int nOut = (int)*pnOut;
  WhereCost nNew;

#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
  Index *p = pBuilder->pNew->u.btree.pIndex;
  int nEq = pBuilder->pNew->u.btree.nEq;

  if( p->nSample>0
   && nEq==pBuilder->nRecValid
................................................................................
        nOut--;
      }
    }

    pBuilder->pRec = pRec;
    if( rc==SQLITE_OK ){
      if( iUpper>iLower ){
        nNew = whereCost(iUpper - iLower);
      }else{
        nNew = 10;        assert( 10==whereCost(2) );
      }
      if( nNew<nOut ){
        nOut = nNew;
      }
      *pnOut = (WhereCost)nOut;
      WHERETRACE(0x100, ("range scan regions: %u..%u  est=%d\n",
                         (u32)iLower, (u32)iUpper, nOut));
      return SQLITE_OK;
    }
  }
#else
  UNUSED_PARAMETER(pParse);
................................................................................
  UNUSED_PARAMETER(pBuilder);
#endif
  assert( pLower || pUpper );
  /* TUNING:  Each inequality constraint reduces the search space 4-fold.
  ** A BETWEEN operator, therefore, reduces the search space 16-fold */
  nNew = nOut;
  if( pLower && (pLower->wtFlags & TERM_VNULL)==0 ){
    nNew -= 20;        assert( 20==whereCost(4) );
    nOut--;
  }
  if( pUpper ){
    nNew -= 20;        assert( 20==whereCost(4) );
    nOut--;
  }
  if( nNew<10 ) nNew = 10;
  if( nNew<nOut ) nOut = nNew;
  *pnOut = (WhereCost)nOut;
  return rc;
}

#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
/*
** Estimate the number of rows that will be returned based on
** an equality constraint x=VALUE and where that VALUE occurs in
................................................................................
** If pProbe->tnum==0, that means pIndex is a fake index used for the
** INTEGER PRIMARY KEY.
*/
static int whereLoopAddBtreeIndex(
  WhereLoopBuilder *pBuilder,     /* The WhereLoop factory */
  struct SrcList_item *pSrc,      /* FROM clause term being analyzed */
  Index *pProbe,                  /* An index on pSrc */
  WhereCost nInMul                /* log(Number of iterations due to IN) */
){
  WhereInfo *pWInfo = pBuilder->pWInfo;  /* WHERE analyse context */
  Parse *pParse = pWInfo->pParse;        /* Parsing context */
  sqlite3 *db = pParse->db;       /* Database connection malloc context */
  WhereLoop *pNew;                /* Template WhereLoop under construction */
  WhereTerm *pTerm;               /* A WhereTerm under consideration */
  int opMask;                     /* Valid operators for constraints */
  WhereScan scan;                 /* Iterator for WHERE terms */
  Bitmask saved_prereq;           /* Original value of pNew->prereq */
  u16 saved_nLTerm;               /* Original value of pNew->nLTerm */
  int saved_nEq;                  /* Original value of pNew->u.btree.nEq */
  u32 saved_wsFlags;              /* Original value of pNew->wsFlags */
  WhereCost saved_nOut;           /* Original value of pNew->nOut */
  int iCol;                       /* Index of the column in the table */
  int rc = SQLITE_OK;             /* Return code */
  WhereCost nRowEst;              /* Estimated index selectivity */
  WhereCost rLogSize;             /* Logarithm of table size */
  WhereTerm *pTop = 0, *pBtm = 0; /* Top and bottom range constraints */

  pNew = pBuilder->pNew;
  if( db->mallocFailed ) return SQLITE_NOMEM;

  assert( (pNew->wsFlags & WHERE_VIRTUALTABLE)==0 );
  assert( (pNew->wsFlags & WHERE_TOP_LIMIT)==0 );
................................................................................
    opMask = WO_EQ|WO_IN|WO_ISNULL|WO_GT|WO_GE|WO_LT|WO_LE;
  }
  if( pProbe->bUnordered ) opMask &= ~(WO_GT|WO_GE|WO_LT|WO_LE);

  assert( pNew->u.btree.nEq<=pProbe->nColumn );
  if( pNew->u.btree.nEq < pProbe->nColumn ){
    iCol = pProbe->aiColumn[pNew->u.btree.nEq];
    nRowEst = whereCost(pProbe->aiRowEst[pNew->u.btree.nEq+1]);
    if( nRowEst==0 && pProbe->onError==OE_None ) nRowEst = 1;
  }else{
    iCol = -1;
    nRowEst = 0;
  }
  pTerm = whereScanInit(&scan, pBuilder->pWC, pSrc->iCursor, iCol,
                        opMask, pProbe);
  saved_nEq = pNew->u.btree.nEq;
  saved_nLTerm = pNew->nLTerm;
  saved_wsFlags = pNew->wsFlags;
  saved_prereq = pNew->prereq;
  saved_nOut = pNew->nOut;
  pNew->rSetup = 0;
  rLogSize = estLog(whereCost(pProbe->aiRowEst[0]));
  for(; rc==SQLITE_OK && pTerm!=0; pTerm = whereScanNext(&scan)){
    int nIn = 0;
#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
    int nRecValid = pBuilder->nRecValid;
#endif
    if( (pTerm->eOperator==WO_ISNULL || (pTerm->wtFlags&TERM_VNULL)!=0)
     && (iCol<0 || pSrc->pTab->aCol[iCol].notNull)
................................................................................
    pNew->prereq = (saved_prereq | pTerm->prereqRight) & ~pNew->maskSelf;
    pNew->rRun = rLogSize; /* Baseline cost is log2(N).  Adjustments below */
    if( pTerm->eOperator & WO_IN ){
      Expr *pExpr = pTerm->pExpr;
      pNew->wsFlags |= WHERE_COLUMN_IN;
      if( ExprHasProperty(pExpr, EP_xIsSelect) ){
        /* "x IN (SELECT ...)":  TUNING: the SELECT returns 25 rows */
        nIn = 46;  assert( 46==whereCost(25) );
      }else if( ALWAYS(pExpr->x.pList && pExpr->x.pList->nExpr) ){
        /* "x IN (value, value, ...)" */
        nIn = whereCost(pExpr->x.pList->nExpr);
      }
      pNew->rRun += nIn;
      pNew->u.btree.nEq++;
      pNew->nOut = nRowEst + nInMul + nIn;
    }else if( pTerm->eOperator & (WO_EQ) ){
      assert( (pNew->wsFlags & (WHERE_COLUMN_NULL|WHERE_COLUMN_IN))!=0
                  || nInMul==0 );
................................................................................
      }
      pNew->u.btree.nEq++;
      pNew->nOut = nRowEst + nInMul;
    }else if( pTerm->eOperator & (WO_ISNULL) ){
      pNew->wsFlags |= WHERE_COLUMN_NULL;
      pNew->u.btree.nEq++;
      /* TUNING: IS NULL selects 2 rows */
      nIn = 10;  assert( 10==whereCost(2) );
      pNew->nOut = nRowEst + nInMul + nIn;
    }else if( pTerm->eOperator & (WO_GT|WO_GE) ){
      testcase( pTerm->eOperator & WO_GT );
      testcase( pTerm->eOperator & WO_GE );
      pNew->wsFlags |= WHERE_COLUMN_RANGE|WHERE_BTM_LIMIT;
      pBtm = pTerm;
      pTop = 0;
................................................................................
        rc = whereEqualScanEst(pParse, pBuilder, pExpr->pRight, &nOut);
      }else if( (pTerm->eOperator & WO_IN)
             &&  !ExprHasProperty(pExpr, EP_xIsSelect)  ){
        rc = whereInScanEst(pParse, pBuilder, pExpr->x.pList, &nOut);
      }
      assert( nOut==0 || rc==SQLITE_OK );
      if( nOut ){
        nOut = whereCost(nOut);
        pNew->nOut = MIN(nOut, saved_nOut);
      }
    }
#endif
    if( (pNew->wsFlags & (WHERE_IDX_ONLY|WHERE_IPK))==0 ){
      /* Each row involves a step of the index, then a binary search of
      ** the main table */
      pNew->rRun =  whereCostAdd(pNew->rRun, rLogSize>27 ? rLogSize-17 : 10);
    }
    /* Step cost for each output row */
    pNew->rRun = whereCostAdd(pNew->rRun, pNew->nOut);
    whereLoopOutputAdjust(pBuilder->pWC, pNew, pSrc->iCursor);
    rc = whereLoopInsert(pBuilder, pNew);
    if( (pNew->wsFlags & WHERE_TOP_LIMIT)==0
     && pNew->u.btree.nEq<(pProbe->nColumn + (pProbe->zName!=0))
    ){
      whereLoopAddBtreeIndex(pBuilder, pSrc, pProbe, nInMul+nIn);
    }
................................................................................
  int aiColumnPk = -1;        /* The aColumn[] value for the sPk index */
  SrcList *pTabList;          /* The FROM clause */
  struct SrcList_item *pSrc;  /* The FROM clause btree term to add */
  WhereLoop *pNew;            /* Template WhereLoop object */
  int rc = SQLITE_OK;         /* Return code */
  int iSortIdx = 1;           /* Index number */
  int b;                      /* A boolean value */
  WhereCost rSize;            /* number of rows in the table */
  WhereCost rLogSize;         /* Logarithm of the number of rows in the table */
  WhereClause *pWC;           /* The parsed WHERE clause */
  
  pNew = pBuilder->pNew;
  pWInfo = pBuilder->pWInfo;
  pTabList = pWInfo->pTabList;
  pSrc = pTabList->a + pNew->iTab;
  pWC = pBuilder->pWC;
................................................................................
    if( pSrc->notIndexed==0 ){
      /* The real indices of the table are only considered if the
      ** NOT INDEXED qualifier is omitted from the FROM clause */
      sPk.pNext = pFirst;
    }
    pProbe = &sPk;
  }
  rSize = whereCost(pSrc->pTab->nRowEst);
  rLogSize = estLog(rSize);

#ifndef SQLITE_OMIT_AUTOMATIC_INDEX
  /* Automatic indexes */
  if( !pBuilder->pOrSet
   && (pWInfo->pParse->db->flags & SQLITE_AutoIndex)!=0
   && pSrc->pIndex==0
................................................................................
        pNew->u.btree.nEq = 1;
        pNew->u.btree.pIndex = 0;
        pNew->nLTerm = 1;
        pNew->aLTerm[0] = pTerm;
        /* TUNING: One-time cost for computing the automatic index is
        ** approximately 7*N*log2(N) where N is the number of rows in
        ** the table being indexed. */
        pNew->rSetup = rLogSize + rSize + 28;  assert( 28==whereCost(7) );
        /* TUNING: Each index lookup yields 20 rows in the table.  This
        ** is more than the usual guess of 10 rows, since we have no way
        ** of knowning how selective the index will ultimately be.  It would
        ** not be unreasonable to make this value much larger. */
        pNew->nOut = 43;  assert( 43==whereCost(20) );
        pNew->rRun = whereCostAdd(rLogSize,pNew->nOut);
        pNew->wsFlags = WHERE_AUTO_INDEX;
        pNew->prereq = mExtra | pTerm->prereqRight;
        rc = whereLoopInsert(pBuilder, pNew);
      }
    }
  }
#endif /* SQLITE_OMIT_AUTOMATIC_INDEX */
................................................................................
    if( pProbe->tnum<=0 ){
      /* Integer primary key index */
      pNew->wsFlags = WHERE_IPK;

      /* Full table scan */
      pNew->iSortIdx = b ? iSortIdx : 0;
      /* TUNING: Cost of full table scan is 3*(N + log2(N)).
      **  +  The extra 4 factor is to encourage the use of indexed lookups
      **     over full scans. */
      pNew->rRun = whereCostAdd(rSize,rLogSize) + 16;
      whereLoopOutputAdjust(pWC, pNew, pSrc->iCursor);
      rc = whereLoopInsert(pBuilder, pNew);
      pNew->nOut = rSize;
      if( rc ) break;
    }else{
      Bitmask m = pSrc->colUsed & ~columnsInIndex(pProbe);
      pNew->wsFlags = (m==0) ? (WHERE_IDX_ONLY|WHERE_INDEXED) : WHERE_INDEXED;
................................................................................
         && sqlite3GlobalConfig.bUseCis
         && OptimizationEnabled(pWInfo->pParse->db, SQLITE_CoverIdxScan)
          )
      ){
        pNew->iSortIdx = b ? iSortIdx : 0;
        if( m==0 ){
          /* TUNING: Cost of a covering index scan is K*(N + log2(N)).
          **  +  The extra factor K of between 1.0 and 3.0 is added to

          **     encourage the use of indexed lookups.  The value of K
          **     depends on the iScanRatio value for the index.
          */
          pNew->rRun = whereCostAdd(rSize,rLogSize) + pProbe->iScanRatio/9 + 1;
        }else{
          assert( b!=0 ); 
          /* TUNING: Cost of scanning a non-covering index is (N+1)*log2(N)
          ** which we will simplify to just N*log2(N) */
          pNew->rRun = rSize + rLogSize;
        }
        whereLoopOutputAdjust(pWC, pNew, pSrc->iCursor);
................................................................................
      pNew->u.vtab.idxNum = pIdxInfo->idxNum;
      pNew->u.vtab.needFree = pIdxInfo->needToFreeIdxStr;
      pIdxInfo->needToFreeIdxStr = 0;
      pNew->u.vtab.idxStr = pIdxInfo->idxStr;
      pNew->u.vtab.isOrdered = (u8)((pIdxInfo->nOrderBy!=0)
                                     && pIdxInfo->orderByConsumed);
      pNew->rSetup = 0;
      pNew->rRun = whereCostFromDouble(pIdxInfo->estimatedCost);
      /* TUNING: Every virtual table query returns 25 rows */
      pNew->nOut = 46;  assert( 46==whereCost(25) );
      whereLoopInsert(pBuilder, pNew);
      if( pNew->u.vtab.needFree ){
        sqlite3_free(pNew->u.vtab.idxStr);
        pNew->u.vtab.needFree = 0;
      }
    }
  }  
................................................................................
          once = 0;
        }else{
          whereOrMove(&sPrev, &sSum);
          sSum.n = 0;
          for(i=0; i<sPrev.n; i++){
            for(j=0; j<sCur.n; j++){
              whereOrInsert(&sSum, sPrev.a[i].prereq | sCur.a[j].prereq,
                            whereCostAdd(sPrev.a[i].rRun, sCur.a[j].rRun),
                            whereCostAdd(sPrev.a[i].nOut, sCur.a[j].nOut));
            }
          }
        }
      }
      pNew->nLTerm = 1;
      pNew->aLTerm[0] = pTerm;
      pNew->wsFlags = WHERE_MULTI_OR;
................................................................................
** Assume that the total number of output rows that will need to be sorted
** will be nRowEst (in the 10*log2 representation).  Or, ignore sorting
** costs if nRowEst==0.
**
** Return SQLITE_OK on success or SQLITE_NOMEM of a memory allocation
** error occurs.
*/
static int wherePathSolver(WhereInfo *pWInfo, WhereCost nRowEst){
  int mxChoice;             /* Maximum number of simultaneous paths tracked */
  int nLoop;                /* Number of terms in the join */
  Parse *pParse;            /* Parsing context */
  sqlite3 *db;              /* The database connection */
  int iLoop;                /* Loop counter over the terms of the join */
  int ii, jj;               /* Loop counters */
  int mxI = 0;              /* Index of next entry to replace */
  WhereCost rCost;          /* Cost of a path */
  WhereCost nOut;           /* Number of outputs */
  WhereCost mxCost = 0;     /* Maximum cost of a set of paths */
  WhereCost mxOut = 0;      /* Maximum nOut value on the set of paths */
  WhereCost rSortCost;      /* Cost to do a sort */
  int nTo, nFrom;           /* Number of valid entries in aTo[] and aFrom[] */
  WherePath *aFrom;         /* All nFrom paths at the previous level */
  WherePath *aTo;           /* The nTo best paths at the current level */
  WherePath *pFrom;         /* An element of aFrom[] that we are working on */
  WherePath *pTo;           /* An element of aTo[] that we are working on */
  WhereLoop *pWLoop;        /* One of the WhereLoop objects */
  WhereLoop **pX;           /* Used to divy up the pSpace memory */
................................................................................
  }

  /* Seed the search with a single WherePath containing zero WhereLoops.
  **
  ** TUNING: Do not let the number of iterations go above 25.  If the cost
  ** of computing an automatic index is not paid back within the first 25
  ** rows, then do not use the automatic index. */
  aFrom[0].nRow = MIN(pParse->nQueryLoop, 46);  assert( 46==whereCost(25) );
  nFrom = 1;

  /* Precompute the cost of sorting the final result set, if the caller
  ** to sqlite3WhereBegin() was concerned about sorting */
  rSortCost = 0;
  if( pWInfo->pOrderBy==0 || nRowEst==0 ){
    aFrom[0].isOrderedValid = 1;
................................................................................
        Bitmask revMask = 0;
        u8 isOrderedValid = pFrom->isOrderedValid;
        u8 isOrdered = pFrom->isOrdered;
        if( (pWLoop->prereq & ~pFrom->maskLoop)!=0 ) continue;
        if( (pWLoop->maskSelf & pFrom->maskLoop)!=0 ) continue;
        /* At this point, pWLoop is a candidate to be the next loop. 
        ** Compute its cost */
        rCost = whereCostAdd(pWLoop->rSetup,pWLoop->rRun + pFrom->nRow);
        rCost = whereCostAdd(rCost, pFrom->rCost);
        nOut = pFrom->nRow + pWLoop->nOut;
        maskNew = pFrom->maskLoop | pWLoop->maskSelf;
        if( !isOrderedValid ){
          switch( wherePathSatisfiesOrderBy(pWInfo,
                       pWInfo->pOrderBy, pFrom, pWInfo->wctrlFlags,
                       iLoop, pWLoop, &revMask) ){
            case 1:  /* Yes.  pFrom+pWLoop does satisfy the ORDER BY clause */
              isOrdered = 1;
              isOrderedValid = 1;
              break;
            case 0:  /* No.  pFrom+pWLoop will require a separate sort */
              isOrdered = 0;
              isOrderedValid = 1;
              rCost = whereCostAdd(rCost, rSortCost);
              break;
            default: /* Cannot tell yet.  Try again on the next iteration */
              break;
          }
        }else{
          revMask = pFrom->revLoop;
        }
................................................................................
  pTerm = findTerm(pWC, iCur, -1, 0, WO_EQ, 0);
  if( pTerm ){
    pLoop->wsFlags = WHERE_COLUMN_EQ|WHERE_IPK|WHERE_ONEROW;
    pLoop->aLTerm[0] = pTerm;
    pLoop->nLTerm = 1;
    pLoop->u.btree.nEq = 1;
    /* TUNING: Cost of a rowid lookup is 10 */
    pLoop->rRun = 33;  /* 33==whereCost(10) */
  }else{
    for(pIdx=pTab->pIndex; pIdx; pIdx=pIdx->pNext){
      assert( pLoop->aLTermSpace==pLoop->aLTerm );
      assert( ArraySize(pLoop->aLTermSpace)==4 );
      if( pIdx->onError==OE_None 
       || pIdx->pPartIdxWhere!=0 
       || pIdx->nColumn>ArraySize(pLoop->aLTermSpace) 
................................................................................
      if( (pItem->colUsed & ~columnsInIndex(pIdx))==0 ){
        pLoop->wsFlags |= WHERE_IDX_ONLY;
      }
      pLoop->nLTerm = j;
      pLoop->u.btree.nEq = j;
      pLoop->u.btree.pIndex = pIdx;
      /* TUNING: Cost of a unique index lookup is 15 */
      pLoop->rRun = 39;  /* 39==whereCost(15) */
      break;
    }
  }
  if( pLoop->wsFlags ){
    pLoop->nOut = (WhereCost)1;
    pWInfo->a[0].pWLoop = pLoop;
    pLoop->maskSelf = getMask(&pWInfo->sMaskSet, iCur);
    pWInfo->a[0].iTabCur = iCur;
    pWInfo->nRowOut = 1;
    if( pWInfo->pOrderBy ) pWInfo->bOBSat =  1;
    if( pWInfo->wctrlFlags & WHERE_WANT_DISTINCT ){
      pWInfo->eDistinct = WHERE_DISTINCT_UNIQUE;

typedef struct WherePath WherePath;
typedef struct WhereTerm WhereTerm;
typedef struct WhereLoopBuilder WhereLoopBuilder;
typedef struct WhereScan WhereScan;
typedef struct WhereOrCost WhereOrCost;
typedef struct WhereOrSet WhereOrSet;





















/*
** This object contains information needed to implement a single nested
** loop in WHERE clause.
**
** Contrast this object with WhereLoop.  This object describes the
** implementation of the loop.  WhereLoop describes the algorithm.
** This object contains a pointer to the WhereLoop algorithm as one of
................................................................................
  Bitmask prereq;       /* Bitmask of other loops that must run first */
  Bitmask maskSelf;     /* Bitmask identifying table iTab */
#ifdef SQLITE_DEBUG
  char cId;             /* Symbolic ID of this loop for debugging use */
#endif
  u8 iTab;              /* Position in FROM clause of table for this loop */
  u8 iSortIdx;          /* Sorting index number.  0==None */
  LogEst rSetup;        /* One-time setup cost (ex: create transient index) */
  LogEst rRun;          /* Cost of running each loop */
  LogEst nOut;          /* Estimated number of output rows */
  union {
    struct {               /* Information for internal btree tables */
      int nEq;               /* Number of equality constraints */
      Index *pIndex;         /* Index used, or NULL */
    } btree;
    struct {               /* Information for virtual tables */
      int idxNum;            /* Index number */
................................................................................

/* This object holds the prerequisites and the cost of running a
** subquery on one operand of an OR operator in the WHERE clause.
** See WhereOrSet for additional information 
*/
struct WhereOrCost {
  Bitmask prereq;     /* Prerequisites */
  LogEst rRun;        /* Cost of running this subquery */
  LogEst nOut;        /* Number of outputs for this subquery */
};

/* The WhereOrSet object holds a set of possible WhereOrCosts that
** correspond to the subquery(s) of OR-clause processing.  Only the
** best N_OR_COST elements are retained.
*/
#define N_OR_COST 3
................................................................................
** of length 2.  And so forth until the length of WherePaths equals the
** number of nodes in the FROM clause.  The best (lowest cost) WherePath
** at the end is the choosen query plan.
*/
struct WherePath {
  Bitmask maskLoop;     /* Bitmask of all WhereLoop objects in this path */
  Bitmask revLoop;      /* aLoop[]s that should be reversed for ORDER BY */
  LogEst nRow;          /* Estimated number of rows generated by this path */
  LogEst rCost;         /* Total cost of this path */
  u8 isOrdered;         /* True if this path satisfies ORDER BY */
  u8 isOrderedValid;    /* True if the isOrdered field is valid */
  WhereLoop **aLoop;    /* Array of WhereLoop objects implementing this path */
};

/*
** The query generator uses an array of instances of this structure to
................................................................................
  int iParent;            /* Disable pWC->a[iParent] when this term disabled */
  int leftCursor;         /* Cursor number of X in "X <op> <expr>" */
  union {
    int leftColumn;         /* Column number of X in "X <op> <expr>" */
    WhereOrInfo *pOrInfo;   /* Extra information if (eOperator & WO_OR)!=0 */
    WhereAndInfo *pAndInfo; /* Extra information if (eOperator& WO_AND)!=0 */
  } u;
  LogEst truthProb;       /* Probability of truth for this expression */
  u16 eOperator;          /* A WO_xx value describing <op> */
  u8 wtFlags;             /* TERM_xxx bit flags.  See below */
  u8 nChild;              /* Number of children that must disable us */
  WhereClause *pWC;       /* The clause this term is part of */
  Bitmask prereqRight;    /* Bitmask of tables used by pExpr->pRight */
  Bitmask prereqAll;      /* Bitmask of tables referenced by pExpr */
};
................................................................................
struct WhereInfo {
  Parse *pParse;            /* Parsing and code generating context */
  SrcList *pTabList;        /* List of tables in the join */
  ExprList *pOrderBy;       /* The ORDER BY clause or NULL */
  ExprList *pResultSet;     /* Result set. DISTINCT operates on these */
  WhereLoop *pLoops;        /* List of all WhereLoop objects */
  Bitmask revMask;          /* Mask of ORDER BY terms that need reversing */
  LogEst nRowOut;           /* Estimated number of output rows */
  u16 wctrlFlags;           /* Flags originally passed to sqlite3WhereBegin() */
  u8 bOBSat;                /* ORDER BY satisfied by indices */
  u8 okOnePass;             /* Ok to use one-pass algorithm for UPDATE/DELETE */
  u8 untestedTerms;         /* Not all WHERE terms resolved by outer loop */
  u8 eDistinct;             /* One of the WHERE_DISTINCT_* values below */
  u8 nLevel;                /* Number of nested loop */
  int iTop;                 /* The very beginning of the WHERE loop */
................................................................................
#define WHERE_INDEXED      0x00000200  /* WhereLoop.u.btree.pIndex is valid */
#define WHERE_VIRTUALTABLE 0x00000400  /* WhereLoop.u.vtab is valid */
#define WHERE_IN_ABLE      0x00000800  /* Able to support an IN operator */
#define WHERE_ONEROW       0x00001000  /* Selects no more than one row */
#define WHERE_MULTI_OR     0x00002000  /* OR using multiple indices */
#define WHERE_AUTO_INDEX   0x00004000  /* Uses an ephemeral index */
















/*
** Return the estimated number of output rows from a WHERE clause
*/
u64 sqlite3WhereOutputRowCount(WhereInfo *pWInfo){
  return sqlite3LogEstToInt(pWInfo->nRowOut);
}

/*
** Return one of the WHERE_DISTINCT_xxxxx values to indicate how this
** WHERE clause returns outputs for DISTINCT processing.
*/
int sqlite3WhereIsDistinct(WhereInfo *pWInfo){
................................................................................
** The new entry might overwrite an existing entry, or it might be
** appended, or it might be discarded.  Do whatever is the right thing
** so that pSet keeps the N_OR_COST best entries seen so far.
*/
static int whereOrInsert(
  WhereOrSet *pSet,      /* The WhereOrSet to be updated */
  Bitmask prereq,        /* Prerequisites of the new entry */
  LogEst rRun,           /* Run-cost of the new entry */
  LogEst nOut            /* Number of outputs for the new entry */
){
  u16 i;
  WhereOrCost *p;
  for(i=pSet->n, p=pSet->a; i>0; i--, p++){
    if( rRun<=p->rRun && (prereq & p->prereq)==prereq ){
      goto whereOrInsert_done;
    }
................................................................................
    }
  }
  if( pWC->a!=pWC->aStatic ){
    sqlite3DbFree(db, pWC->a);
  }
}




/*
** Add a single new WhereTerm entry to the WhereClause object pWC.
** The new WhereTerm object is constructed from Expr p and with wtFlags.
** The index in pWC->a[] of the new WhereTerm is returned on success.
** 0 is returned if the new WhereTerm could not be added due to a memory
** allocation error.  The memory allocation failure will be recorded in
** the db->mallocFailed flag so that higher-level functions can detect it.
................................................................................
    if( pOld!=pWC->aStatic ){
      sqlite3DbFree(db, pOld);
    }
    pWC->nSlot = sqlite3DbMallocSize(db, pWC->a)/sizeof(pWC->a[0]);
  }
  pTerm = &pWC->a[idx = pWC->nTerm++];
  if( p && ExprHasProperty(p, EP_Unlikely) ){
    pTerm->truthProb = sqlite3LogEst(p->iTable) - 99;
  }else{
    pTerm->truthProb = -1;
  }
  pTerm->pExpr = sqlite3ExprSkipCollate(p);
  pTerm->wtFlags = wtFlags;
  pTerm->pWC = pWC;
  pTerm->iParent = -1;
................................................................................
      return 1;
    }
  }

  return 0;
}






























/*



































** Estimate the logarithm of the input value to base 2.
*/


static LogEst estLog(LogEst N){
  LogEst x = sqlite3LogEst(N);
  return x>33 ? x - 33 : 0;
}

/*
** Two routines for printing the content of an sqlite3_index_info
** structure.  Used for testing and debugging only.  If neither
** SQLITE_TEST or SQLITE_DEBUG are defined, then these routines
................................................................................
** then nEq is set to 1 (as the range restricted column, b, is the second 
** left-most column of the index). Or, if the query is:
**
**   ... FROM t1 WHERE a > ? AND a < ? ...
**
** then nEq is set to 0.
**
** When this function is called, *pnOut is set to the sqlite3LogEst() of the
** number of rows that the index scan is expected to visit without 
** considering the range constraints. If nEq is 0, this is the number of 
** rows in the index. Assuming no error occurs, *pnOut is adjusted (reduced)
** to account for the range contraints pLower and pUpper.
** 
** In the absence of sqlite_stat4 ANALYZE data, or if such data cannot be
** used, each range inequality reduces the search space by a factor of 4. 
................................................................................
** rows visited by a factor of 16.
*/
static int whereRangeScanEst(
  Parse *pParse,       /* Parsing & code generating context */
  WhereLoopBuilder *pBuilder,
  WhereTerm *pLower,   /* Lower bound on the range. ex: "x>123" Might be NULL */
  WhereTerm *pUpper,   /* Upper bound on the range. ex: "x<455" Might be NULL */
  LogEst *pnOut        /* IN/OUT: Number of rows visited */
){
  int rc = SQLITE_OK;
  int nOut = (int)*pnOut;
  LogEst nNew;

#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
  Index *p = pBuilder->pNew->u.btree.pIndex;
  int nEq = pBuilder->pNew->u.btree.nEq;

  if( p->nSample>0
   && nEq==pBuilder->nRecValid
................................................................................
        nOut--;
      }
    }

    pBuilder->pRec = pRec;
    if( rc==SQLITE_OK ){
      if( iUpper>iLower ){
        nNew = sqlite3LogEst(iUpper - iLower);
      }else{
        nNew = 10;        assert( 10==sqlite3LogEst(2) );
      }
      if( nNew<nOut ){
        nOut = nNew;
      }
      *pnOut = (LogEst)nOut;
      WHERETRACE(0x100, ("range scan regions: %u..%u  est=%d\n",
                         (u32)iLower, (u32)iUpper, nOut));
      return SQLITE_OK;
    }
  }
#else
  UNUSED_PARAMETER(pParse);
................................................................................
  UNUSED_PARAMETER(pBuilder);
#endif
  assert( pLower || pUpper );
  /* TUNING:  Each inequality constraint reduces the search space 4-fold.
  ** A BETWEEN operator, therefore, reduces the search space 16-fold */
  nNew = nOut;
  if( pLower && (pLower->wtFlags & TERM_VNULL)==0 ){
    nNew -= 20;        assert( 20==sqlite3LogEst(4) );
    nOut--;
  }
  if( pUpper ){
    nNew -= 20;        assert( 20==sqlite3LogEst(4) );
    nOut--;
  }
  if( nNew<10 ) nNew = 10;
  if( nNew<nOut ) nOut = nNew;
  *pnOut = (LogEst)nOut;
  return rc;
}

#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
/*
** Estimate the number of rows that will be returned based on
** an equality constraint x=VALUE and where that VALUE occurs in
................................................................................
** If pProbe->tnum==0, that means pIndex is a fake index used for the
** INTEGER PRIMARY KEY.
*/
static int whereLoopAddBtreeIndex(
  WhereLoopBuilder *pBuilder,     /* The WhereLoop factory */
  struct SrcList_item *pSrc,      /* FROM clause term being analyzed */
  Index *pProbe,                  /* An index on pSrc */
  LogEst nInMul                   /* log(Number of iterations due to IN) */
){
  WhereInfo *pWInfo = pBuilder->pWInfo;  /* WHERE analyse context */
  Parse *pParse = pWInfo->pParse;        /* Parsing context */
  sqlite3 *db = pParse->db;       /* Database connection malloc context */
  WhereLoop *pNew;                /* Template WhereLoop under construction */
  WhereTerm *pTerm;               /* A WhereTerm under consideration */
  int opMask;                     /* Valid operators for constraints */
  WhereScan scan;                 /* Iterator for WHERE terms */
  Bitmask saved_prereq;           /* Original value of pNew->prereq */
  u16 saved_nLTerm;               /* Original value of pNew->nLTerm */
  int saved_nEq;                  /* Original value of pNew->u.btree.nEq */
  u32 saved_wsFlags;              /* Original value of pNew->wsFlags */
  LogEst saved_nOut;              /* Original value of pNew->nOut */
  int iCol;                       /* Index of the column in the table */
  int rc = SQLITE_OK;             /* Return code */
  LogEst nRowEst;                 /* Estimated index selectivity */
  LogEst rLogSize;                /* Logarithm of table size */
  WhereTerm *pTop = 0, *pBtm = 0; /* Top and bottom range constraints */

  pNew = pBuilder->pNew;
  if( db->mallocFailed ) return SQLITE_NOMEM;

  assert( (pNew->wsFlags & WHERE_VIRTUALTABLE)==0 );
  assert( (pNew->wsFlags & WHERE_TOP_LIMIT)==0 );
................................................................................
    opMask = WO_EQ|WO_IN|WO_ISNULL|WO_GT|WO_GE|WO_LT|WO_LE;
  }
  if( pProbe->bUnordered ) opMask &= ~(WO_GT|WO_GE|WO_LT|WO_LE);

  assert( pNew->u.btree.nEq<=pProbe->nColumn );
  if( pNew->u.btree.nEq < pProbe->nColumn ){
    iCol = pProbe->aiColumn[pNew->u.btree.nEq];
    nRowEst = sqlite3LogEst(pProbe->aiRowEst[pNew->u.btree.nEq+1]);
    if( nRowEst==0 && pProbe->onError==OE_None ) nRowEst = 1;
  }else{
    iCol = -1;
    nRowEst = 0;
  }
  pTerm = whereScanInit(&scan, pBuilder->pWC, pSrc->iCursor, iCol,
                        opMask, pProbe);
  saved_nEq = pNew->u.btree.nEq;
  saved_nLTerm = pNew->nLTerm;
  saved_wsFlags = pNew->wsFlags;
  saved_prereq = pNew->prereq;
  saved_nOut = pNew->nOut;
  pNew->rSetup = 0;
  rLogSize = estLog(sqlite3LogEst(pProbe->aiRowEst[0]));
  for(; rc==SQLITE_OK && pTerm!=0; pTerm = whereScanNext(&scan)){
    int nIn = 0;
#ifdef SQLITE_ENABLE_STAT3_OR_STAT4
    int nRecValid = pBuilder->nRecValid;
#endif
    if( (pTerm->eOperator==WO_ISNULL || (pTerm->wtFlags&TERM_VNULL)!=0)
     && (iCol<0 || pSrc->pTab->aCol[iCol].notNull)
................................................................................
    pNew->prereq = (saved_prereq | pTerm->prereqRight) & ~pNew->maskSelf;
    pNew->rRun = rLogSize; /* Baseline cost is log2(N).  Adjustments below */
    if( pTerm->eOperator & WO_IN ){
      Expr *pExpr = pTerm->pExpr;
      pNew->wsFlags |= WHERE_COLUMN_IN;
      if( ExprHasProperty(pExpr, EP_xIsSelect) ){
        /* "x IN (SELECT ...)":  TUNING: the SELECT returns 25 rows */
        nIn = 46;  assert( 46==sqlite3LogEst(25) );
      }else if( ALWAYS(pExpr->x.pList && pExpr->x.pList->nExpr) ){
        /* "x IN (value, value, ...)" */
        nIn = sqlite3LogEst(pExpr->x.pList->nExpr);
      }
      pNew->rRun += nIn;
      pNew->u.btree.nEq++;
      pNew->nOut = nRowEst + nInMul + nIn;
    }else if( pTerm->eOperator & (WO_EQ) ){
      assert( (pNew->wsFlags & (WHERE_COLUMN_NULL|WHERE_COLUMN_IN))!=0
                  || nInMul==0 );
................................................................................
      }
      pNew->u.btree.nEq++;
      pNew->nOut = nRowEst + nInMul;
    }else if( pTerm->eOperator & (WO_ISNULL) ){
      pNew->wsFlags |= WHERE_COLUMN_NULL;
      pNew->u.btree.nEq++;
      /* TUNING: IS NULL selects 2 rows */
      nIn = 10;  assert( 10==sqlite3LogEst(2) );
      pNew->nOut = nRowEst + nInMul + nIn;
    }else if( pTerm->eOperator & (WO_GT|WO_GE) ){
      testcase( pTerm->eOperator & WO_GT );
      testcase( pTerm->eOperator & WO_GE );
      pNew->wsFlags |= WHERE_COLUMN_RANGE|WHERE_BTM_LIMIT;
      pBtm = pTerm;
      pTop = 0;
................................................................................
        rc = whereEqualScanEst(pParse, pBuilder, pExpr->pRight, &nOut);
      }else if( (pTerm->eOperator & WO_IN)
             &&  !ExprHasProperty(pExpr, EP_xIsSelect)  ){
        rc = whereInScanEst(pParse, pBuilder, pExpr->x.pList, &nOut);
      }
      assert( nOut==0 || rc==SQLITE_OK );
      if( nOut ){
        nOut = sqlite3LogEst(nOut);
        pNew->nOut = MIN(nOut, saved_nOut);
      }
    }
#endif
    if( (pNew->wsFlags & (WHERE_IDX_ONLY|WHERE_IPK))==0 ){
      /* Each row involves a step of the index, then a binary search of
      ** the main table */
      pNew->rRun =  sqlite3LogEstAdd(pNew->rRun,rLogSize>27 ? rLogSize-17 : 10);
    }
    /* Step cost for each output row */
    pNew->rRun = sqlite3LogEstAdd(pNew->rRun, pNew->nOut);
    whereLoopOutputAdjust(pBuilder->pWC, pNew, pSrc->iCursor);
    rc = whereLoopInsert(pBuilder, pNew);
    if( (pNew->wsFlags & WHERE_TOP_LIMIT)==0
     && pNew->u.btree.nEq<(pProbe->nColumn + (pProbe->zName!=0))
    ){
      whereLoopAddBtreeIndex(pBuilder, pSrc, pProbe, nInMul+nIn);
    }
................................................................................
  int aiColumnPk = -1;        /* The aColumn[] value for the sPk index */
  SrcList *pTabList;          /* The FROM clause */
  struct SrcList_item *pSrc;  /* The FROM clause btree term to add */
  WhereLoop *pNew;            /* Template WhereLoop object */
  int rc = SQLITE_OK;         /* Return code */
  int iSortIdx = 1;           /* Index number */
  int b;                      /* A boolean value */
  LogEst rSize;               /* number of rows in the table */
  LogEst rLogSize;            /* Logarithm of the number of rows in the table */
  WhereClause *pWC;           /* The parsed WHERE clause */
  
  pNew = pBuilder->pNew;
  pWInfo = pBuilder->pWInfo;
  pTabList = pWInfo->pTabList;
  pSrc = pTabList->a + pNew->iTab;
  pWC = pBuilder->pWC;
................................................................................
    if( pSrc->notIndexed==0 ){
      /* The real indices of the table are only considered if the
      ** NOT INDEXED qualifier is omitted from the FROM clause */
      sPk.pNext = pFirst;
    }
    pProbe = &sPk;
  }
  rSize = sqlite3LogEst(pSrc->pTab->nRowEst);
  rLogSize = estLog(rSize);

#ifndef SQLITE_OMIT_AUTOMATIC_INDEX
  /* Automatic indexes */
  if( !pBuilder->pOrSet
   && (pWInfo->pParse->db->flags & SQLITE_AutoIndex)!=0
   && pSrc->pIndex==0
................................................................................
        pNew->u.btree.nEq = 1;
        pNew->u.btree.pIndex = 0;
        pNew->nLTerm = 1;
        pNew->aLTerm[0] = pTerm;
        /* TUNING: One-time cost for computing the automatic index is
        ** approximately 7*N*log2(N) where N is the number of rows in
        ** the table being indexed. */
        pNew->rSetup = rLogSize + rSize + 28;  assert( 28==sqlite3LogEst(7) );
        /* TUNING: Each index lookup yields 20 rows in the table.  This
        ** is more than the usual guess of 10 rows, since we have no way
        ** of knowning how selective the index will ultimately be.  It would
        ** not be unreasonable to make this value much larger. */
        pNew->nOut = 43;  assert( 43==sqlite3LogEst(20) );
        pNew->rRun = sqlite3LogEstAdd(rLogSize,pNew->nOut);
        pNew->wsFlags = WHERE_AUTO_INDEX;
        pNew->prereq = mExtra | pTerm->prereqRight;
        rc = whereLoopInsert(pBuilder, pNew);
      }
    }
  }
#endif /* SQLITE_OMIT_AUTOMATIC_INDEX */
................................................................................
    if( pProbe->tnum<=0 ){
      /* Integer primary key index */
      pNew->wsFlags = WHERE_IPK;

      /* Full table scan */
      pNew->iSortIdx = b ? iSortIdx : 0;
      /* TUNING: Cost of full table scan is 3*(N + log2(N)).
      **  +  The extra 3 factor is to encourage the use of indexed lookups
      **     over full scans. */
      pNew->rRun = sqlite3LogEstAdd(rSize,rLogSize) + 16;
      whereLoopOutputAdjust(pWC, pNew, pSrc->iCursor);
      rc = whereLoopInsert(pBuilder, pNew);
      pNew->nOut = rSize;
      if( rc ) break;
    }else{
      Bitmask m = pSrc->colUsed & ~columnsInIndex(pProbe);
      pNew->wsFlags = (m==0) ? (WHERE_IDX_ONLY|WHERE_INDEXED) : WHERE_INDEXED;
................................................................................
         && sqlite3GlobalConfig.bUseCis
         && OptimizationEnabled(pWInfo->pParse->db, SQLITE_CoverIdxScan)
          )
      ){
        pNew->iSortIdx = b ? iSortIdx : 0;
        if( m==0 ){
          /* TUNING: Cost of a covering index scan is K*(N + log2(N)).
          **  +  The extra factor K of between 1.1 (iScanRatio between 0
          **     and 9) and 2.8 (iScanRatio between 126 and 127) is added
          **     to encourage the use of indexed lookups.

          */
          pNew->rRun = sqlite3LogEstAdd(rSize,rLogSize) + pProbe->iScanRatio/9 + 1;
        }else{
          assert( b!=0 ); 
          /* TUNING: Cost of scanning a non-covering index is (N+1)*log2(N)
          ** which we will simplify to just N*log2(N) */
          pNew->rRun = rSize + rLogSize;
        }
        whereLoopOutputAdjust(pWC, pNew, pSrc->iCursor);
................................................................................
      pNew->u.vtab.idxNum = pIdxInfo->idxNum;
      pNew->u.vtab.needFree = pIdxInfo->needToFreeIdxStr;
      pIdxInfo->needToFreeIdxStr = 0;
      pNew->u.vtab.idxStr = pIdxInfo->idxStr;
      pNew->u.vtab.isOrdered = (u8)((pIdxInfo->nOrderBy!=0)
                                     && pIdxInfo->orderByConsumed);
      pNew->rSetup = 0;
      pNew->rRun = sqlite3LogEstFromDouble(pIdxInfo->estimatedCost);
      /* TUNING: Every virtual table query returns 25 rows */
      pNew->nOut = 46;  assert( 46==sqlite3LogEst(25) );
      whereLoopInsert(pBuilder, pNew);
      if( pNew->u.vtab.needFree ){
        sqlite3_free(pNew->u.vtab.idxStr);
        pNew->u.vtab.needFree = 0;
      }
    }
  }  
................................................................................
          once = 0;
        }else{
          whereOrMove(&sPrev, &sSum);
          sSum.n = 0;
          for(i=0; i<sPrev.n; i++){
            for(j=0; j<sCur.n; j++){
              whereOrInsert(&sSum, sPrev.a[i].prereq | sCur.a[j].prereq,
                            sqlite3LogEstAdd(sPrev.a[i].rRun, sCur.a[j].rRun),
                            sqlite3LogEstAdd(sPrev.a[i].nOut, sCur.a[j].nOut));
            }
          }
        }
      }
      pNew->nLTerm = 1;
      pNew->aLTerm[0] = pTerm;
      pNew->wsFlags = WHERE_MULTI_OR;
................................................................................
** Assume that the total number of output rows that will need to be sorted
** will be nRowEst (in the 10*log2 representation).  Or, ignore sorting
** costs if nRowEst==0.
**
** Return SQLITE_OK on success or SQLITE_NOMEM of a memory allocation
** error occurs.
*/
static int wherePathSolver(WhereInfo *pWInfo, LogEst nRowEst){
  int mxChoice;             /* Maximum number of simultaneous paths tracked */
  int nLoop;                /* Number of terms in the join */
  Parse *pParse;            /* Parsing context */
  sqlite3 *db;              /* The database connection */
  int iLoop;                /* Loop counter over the terms of the join */
  int ii, jj;               /* Loop counters */
  int mxI = 0;              /* Index of next entry to replace */
  LogEst rCost;             /* Cost of a path */
  LogEst nOut;              /* Number of outputs */
  LogEst mxCost = 0;        /* Maximum cost of a set of paths */
  LogEst mxOut = 0;         /* Maximum nOut value on the set of paths */
  LogEst rSortCost;         /* Cost to do a sort */
  int nTo, nFrom;           /* Number of valid entries in aTo[] and aFrom[] */
  WherePath *aFrom;         /* All nFrom paths at the previous level */
  WherePath *aTo;           /* The nTo best paths at the current level */
  WherePath *pFrom;         /* An element of aFrom[] that we are working on */
  WherePath *pTo;           /* An element of aTo[] that we are working on */
  WhereLoop *pWLoop;        /* One of the WhereLoop objects */
  WhereLoop **pX;           /* Used to divy up the pSpace memory */
................................................................................
  }

  /* Seed the search with a single WherePath containing zero WhereLoops.
  **
  ** TUNING: Do not let the number of iterations go above 25.  If the cost
  ** of computing an automatic index is not paid back within the first 25
  ** rows, then do not use the automatic index. */
  aFrom[0].nRow = MIN(pParse->nQueryLoop, 46);  assert( 46==sqlite3LogEst(25) );
  nFrom = 1;

  /* Precompute the cost of sorting the final result set, if the caller
  ** to sqlite3WhereBegin() was concerned about sorting */
  rSortCost = 0;
  if( pWInfo->pOrderBy==0 || nRowEst==0 ){
    aFrom[0].isOrderedValid = 1;
................................................................................
        Bitmask revMask = 0;
        u8 isOrderedValid = pFrom->isOrderedValid;
        u8 isOrdered = pFrom->isOrdered;
        if( (pWLoop->prereq & ~pFrom->maskLoop)!=0 ) continue;
        if( (pWLoop->maskSelf & pFrom->maskLoop)!=0 ) continue;
        /* At this point, pWLoop is a candidate to be the next loop. 
        ** Compute its cost */
        rCost = sqlite3LogEstAdd(pWLoop->rSetup,pWLoop->rRun + pFrom->nRow);
        rCost = sqlite3LogEstAdd(rCost, pFrom->rCost);
        nOut = pFrom->nRow + pWLoop->nOut;
        maskNew = pFrom->maskLoop | pWLoop->maskSelf;
        if( !isOrderedValid ){
          switch( wherePathSatisfiesOrderBy(pWInfo,
                       pWInfo->pOrderBy, pFrom, pWInfo->wctrlFlags,
                       iLoop, pWLoop, &revMask) ){
            case 1:  /* Yes.  pFrom+pWLoop does satisfy the ORDER BY clause */
              isOrdered = 1;
              isOrderedValid = 1;
              break;
            case 0:  /* No.  pFrom+pWLoop will require a separate sort */
              isOrdered = 0;
              isOrderedValid = 1;
              rCost = sqlite3LogEstAdd(rCost, rSortCost);
              break;
            default: /* Cannot tell yet.  Try again on the next iteration */
              break;
          }
        }else{
          revMask = pFrom->revLoop;
        }
................................................................................
  pTerm = findTerm(pWC, iCur, -1, 0, WO_EQ, 0);
  if( pTerm ){
    pLoop->wsFlags = WHERE_COLUMN_EQ|WHERE_IPK|WHERE_ONEROW;
    pLoop->aLTerm[0] = pTerm;
    pLoop->nLTerm = 1;
    pLoop->u.btree.nEq = 1;
    /* TUNING: Cost of a rowid lookup is 10 */
    pLoop->rRun = 33;  /* 33==sqlite3LogEst(10) */
  }else{
    for(pIdx=pTab->pIndex; pIdx; pIdx=pIdx->pNext){
      assert( pLoop->aLTermSpace==pLoop->aLTerm );
      assert( ArraySize(pLoop->aLTermSpace)==4 );
      if( pIdx->onError==OE_None 
       || pIdx->pPartIdxWhere!=0 
       || pIdx->nColumn>ArraySize(pLoop->aLTermSpace) 
................................................................................
      if( (pItem->colUsed & ~columnsInIndex(pIdx))==0 ){
        pLoop->wsFlags |= WHERE_IDX_ONLY;
      }
      pLoop->nLTerm = j;
      pLoop->u.btree.nEq = j;
      pLoop->u.btree.pIndex = pIdx;
      /* TUNING: Cost of a unique index lookup is 15 */
      pLoop->rRun = 39;  /* 39==sqlite3LogEst(15) */
      break;
    }
  }
  if( pLoop->wsFlags ){
    pLoop->nOut = (LogEst)1;
    pWInfo->a[0].pWLoop = pLoop;
    pLoop->maskSelf = getMask(&pWInfo->sMaskSet, iCur);
    pWInfo->a[0].iTabCur = iCur;
    pWInfo->nRowOut = 1;
    if( pWInfo->pOrderBy ) pWInfo->bOBSat =  1;
    if( pWInfo->wctrlFlags & WHERE_WANT_DISTINCT ){
      pWInfo->eDistinct = WHERE_DISTINCT_UNIQUE;

**
**    May you do good and not evil.
**    May you find forgiveness for yourself and forgive others.
**    May you share freely, never taking more than you give.
**
*************************************************************************
** This file contains a simple command-line utility for converting from
** integers and WhereCost values and back again and for doing simple
** arithmetic operations (multiple and add) on WhereCost values.
**
** Usage:
**
**      ./wherecosttest ARGS
**
** Arguments:
**
**    'x'    Multiple the top two elements of the stack
**    '+'    Add the top two elements of the stack
**    NUM    Convert NUM from integer to WhereCost and push onto the stack
**   ^NUM    Interpret NUM as a WhereCost and push onto stack.
**
** Examples:
**
** To convert 123 from WhereCost to integer:
** 
**         ./wherecosttest ^123
**
** To convert 123456 from integer to WhereCost:
**
**         ./wherecosttest 123456
**
*/
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>




typedef unsigned short int WhereCost;  /* 10 times log2() */

WhereCost whereCostMultiply(WhereCost a, WhereCost b){ return a+b; }
WhereCost whereCostAdd(WhereCost a, WhereCost b){


  static const unsigned char x[] = {
     10, 10,                         /* 0,1 */
      9, 9,                          /* 2,3 */
      8, 8,                          /* 4,5 */
      7, 7, 7,                       /* 6,7,8 */
      6, 6, 6,                       /* 9,10,11 */
      5, 5, 5,                       /* 12-14 */
      4, 4, 4, 4,                    /* 15-18 */
      3, 3, 3, 3, 3, 3,              /* 19-24 */
      2, 2, 2, 2, 2, 2, 2,           /* 25-31 */
  };
  if( a<b ){ WhereCost t = a; a = b; b = t; }
  if( a>b+49 ) return a;
  if( a>b+31 ) return a+1;
  return a+x[a-b];
}
WhereCost whereCostFromInteger(int x){
  static WhereCost a[] = { 0, 2, 3, 5, 6, 7, 8, 9 };
  WhereCost y = 40;
  if( x<8 ){
    if( x<2 ) return 0;
    while( x<8 ){  y -= 10; x <<= 1; }
  }else{
    while( x>255 ){ y += 40; x >>= 4; }
    while( x>15 ){  y += 10; x >>= 1; }
  }
  return a[x&7] + y - 10;
}
static unsigned long int whereCostToInt(WhereCost x){
  unsigned long int n;
  if( x<10 ) return 1;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
  if( x>=3 ) return (n+8)<<(x-3);
  return (n+8)>>(3-x);
}




















int main(int argc, char **argv){
  int i;
  int n = 0;
  WhereCost a[100];
  for(i=1; i<argc; i++){
    const char *z = argv[i];
    if( z[0]=='+' ){
      if( n>=2 ){
        a[n-2] = whereCostAdd(a[n-2],a[n-1]);
        n--;
      }
    }else if( z[0]=='x' ){
      if( n>=2 ){
        a[n-2] = whereCostMultiply(a[n-2],a[n-1]);
        n--;
      }
    }else if( z[0]=='^' ){
      a[n++] = atoi(z+1);


    }else{
      a[n++] = whereCostFromInteger(atoi(z));
    }
  }
  for(i=n-1; i>=0; i--){



    printf("%d (%lu)\n", a[i], whereCostToInt(a[i]));

  }
  return 0;
}

**
**    May you do good and not evil.
**    May you find forgiveness for yourself and forgive others.
**    May you share freely, never taking more than you give.
**
*************************************************************************
** This file contains a simple command-line utility for converting from
** integers and LogEst values and back again and for doing simple
** arithmetic operations (multiple and add) on LogEst values.
**
** Usage:
**
**      ./LogEst ARGS
**
** Arguments:
**
**    'x'    Multiple the top two elements of the stack
**    '+'    Add the top two elements of the stack
**    NUM    Convert NUM from integer to LogEst and push onto the stack
**   ^NUM    Interpret NUM as a LogEst and push onto stack.
**
** Examples:
**
** To convert 123 from LogEst to integer:
** 
**         ./LogEst ^123
**
** To convert 123456 from integer to LogEst:
**
**         ./LogEst 123456
**
*/
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
#include <assert.h>
#include <string.h>
#include "sqlite3.h"

typedef short int LogEst;  /* 10 times log2() */



LogEst logEstMultiply(LogEst a, LogEst b){ return a+b; }
LogEst logEstAdd(LogEst a, LogEst b){
  static const unsigned char x[] = {
     10, 10,                         /* 0,1 */
      9, 9,                          /* 2,3 */
      8, 8,                          /* 4,5 */
      7, 7, 7,                       /* 6,7,8 */
      6, 6, 6,                       /* 9,10,11 */
      5, 5, 5,                       /* 12-14 */
      4, 4, 4, 4,                    /* 15-18 */
      3, 3, 3, 3, 3, 3,              /* 19-24 */
      2, 2, 2, 2, 2, 2, 2,           /* 25-31 */
  };
  if( a<b ){ LogEst t = a; a = b; b = t; }
  if( a>b+49 ) return a;
  if( a>b+31 ) return a+1;
  return a+x[a-b];
}
LogEst logEstFromInteger(sqlite3_uint64 x){
  static LogEst a[] = { 0, 2, 3, 5, 6, 7, 8, 9 };
  LogEst y = 40;
  if( x<8 ){
    if( x<2 ) return 0;
    while( x<8 ){  y -= 10; x <<= 1; }
  }else{
    while( x>255 ){ y += 40; x >>= 4; }
    while( x>15 ){  y += 10; x >>= 1; }
  }
  return a[x&7] + y - 10;
}
static unsigned long int logEstToInt(LogEst x){
  unsigned long int n;
  if( x<10 ) return 1;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
  if( x>=3 ) return (n+8)<<(x-3);
  return (n+8)>>(3-x);
}
static LogEst logEstFromDouble(double x){
  sqlite3_uint64 a;
  LogEst e;
  assert( sizeof(x)==8 && sizeof(a)==8 );
  if( x<=0 ) return -32768;
  if( x<1 ) return -logEstFromDouble(1/x);
  if( x<=2000000000 ) return logEstFromInteger((sqlite3_uint64)x);
  memcpy(&a, &x, 8);
  e = (a>>52) - 1022;
  return e*10;
}

int isFloat(const char *z){
  while( z[0] ){
    if( z[0]=='.' || z[0]=='E' || z[0]=='e' ) return 1;
    z++;
  }
  return 0;
}

int main(int argc, char **argv){
  int i;
  int n = 0;
  LogEst a[100];
  for(i=1; i<argc; i++){
    const char *z = argv[i];
    if( z[0]=='+' ){
      if( n>=2 ){
        a[n-2] = logEstAdd(a[n-2],a[n-1]);
        n--;
      }
    }else if( z[0]=='x' ){
      if( n>=2 ){
        a[n-2] = logEstMultiply(a[n-2],a[n-1]);
        n--;
      }
    }else if( z[0]=='^' ){
      a[n++] = atoi(z+1);
    }else if( isFloat(z) ){
      a[n++] = logEstFromDouble(atof(z));
    }else{
      a[n++] = logEstFromInteger(atoi(z));
    }
  }
  for(i=n-1; i>=0; i--){
    if( a[i]<0 ){
      printf("%d (%f)\n", a[i], 1.0/(double)logEstToInt(-a[i]));
    }else{
      printf("%d (%lu)\n", a[i], logEstToInt(a[i]));
    }
  }
  return 0;
}