src/util.c

/*
** 2001 September 15
**
** The author disclaims copyright to this source code.  In place of
** a legal notice, here is a blessing:
**
**    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.
**
*************************************************************************
** Utility functions used throughout sqlite.
**
** This file contains functions for allocating memory, comparing
** strings, and stuff like that.
**
*/
#include "sqliteInt.h"
#include <stdarg.h>
#ifndef SQLITE_OMIT_FLOATING_POINT
#include <math.h>
#endif

/*
** Calls to sqlite3FaultSim() are used to simulate a failure during testing,
** or to bypass normal error detection during testing in order to let 
** execute proceed futher downstream.
**
** In deployment, sqlite3FaultSim() *always* return SQLITE_OK (0).  The
** sqlite3FaultSim() function only returns non-zero during testing.
**
** During testing, if the test harness has set a fault-sim callback using
** a call to sqlite3_test_control(SQLITE_TESTCTRL_FAULT_INSTALL), then
** each call to sqlite3FaultSim() is relayed to that application-supplied
** callback and the integer return value form the application-supplied
** callback is returned by sqlite3FaultSim().
**
** The integer argument to sqlite3FaultSim() is a code to identify which
** sqlite3FaultSim() instance is being invoked. Each call to sqlite3FaultSim()
** should have a unique code.  To prevent legacy testing applications from
** breaking, the codes should not be changed or reused.
*/
#ifndef SQLITE_UNTESTABLE
int sqlite3FaultSim(int iTest){
  int (*xCallback)(int) = sqlite3GlobalConfig.xTestCallback;
  return xCallback ? xCallback(iTest) : SQLITE_OK;
}
#endif

#ifndef SQLITE_OMIT_FLOATING_POINT
/*
** Return true if the floating point value is Not a Number (NaN).
**
** Use the math library isnan() function if compiled with SQLITE_HAVE_ISNAN.
** Otherwise, we have our own implementation that works on most systems.
*/
int sqlite3IsNaN(double x){
  int rc;   /* The value return */
#if !SQLITE_HAVE_ISNAN && !HAVE_ISNAN
  u64 y;
  memcpy(&y,&x,sizeof(y));
  rc = IsNaN(y);
#else
  rc = isnan(x);
#endif /* HAVE_ISNAN */
  testcase( rc );
  return rc;
}
#endif /* SQLITE_OMIT_FLOATING_POINT */

/*
** Compute a string length that is limited to what can be stored in
** lower 30 bits of a 32-bit signed integer.
**
** The value returned will never be negative.  Nor will it ever be greater
** than the actual length of the string.  For very long strings (greater
** than 1GiB) the value returned might be less than the true string length.
*/
int sqlite3Strlen30(const char *z){
  if( z==0 ) return 0;
  return 0x3fffffff & (int)strlen(z);
}

/*
** Return the declared type of a column.  Or return zDflt if the column 
** has no declared type.
**
** The column type is an extra string stored after the zero-terminator on
** the column name if and only if the COLFLAG_HASTYPE flag is set.
*/
char *sqlite3ColumnType(Column *pCol, char *zDflt){
  if( pCol->colFlags & COLFLAG_HASTYPE ){
    return pCol->zCnName + strlen(pCol->zCnName) + 1;
  }else if( pCol->eCType ){
    assert( pCol->eCType<=SQLITE_N_STDTYPE );
    return (char*)sqlite3StdType[pCol->eCType-1];
  }else{
    return zDflt;
  }
}

/*
** Helper function for sqlite3Error() - called rarely.  Broken out into
** a separate routine to avoid unnecessary register saves on entry to
** sqlite3Error().
*/
static SQLITE_NOINLINE void  sqlite3ErrorFinish(sqlite3 *db, int err_code){
  if( db->pErr ) sqlite3ValueSetNull(db->pErr);
  sqlite3SystemError(db, err_code);
}

/*
** Set the current error code to err_code and clear any prior error message.
** Also set iSysErrno (by calling sqlite3System) if the err_code indicates
** that would be appropriate.
*/
void sqlite3Error(sqlite3 *db, int err_code){
  assert( db!=0 );
  db->errCode = err_code;
  if( err_code || db->pErr ){
    sqlite3ErrorFinish(db, err_code);
  }else{
    db->errByteOffset = -1;
  }
}

/*
** The equivalent of sqlite3Error(db, SQLITE_OK).  Clear the error state
** and error message.
*/
void sqlite3ErrorClear(sqlite3 *db){
  assert( db!=0 );
  db->errCode = SQLITE_OK;
  db->errByteOffset = -1;
  if( db->pErr ) sqlite3ValueSetNull(db->pErr);
}

/*
** Load the sqlite3.iSysErrno field if that is an appropriate thing
** to do based on the SQLite error code in rc.
*/
void sqlite3SystemError(sqlite3 *db, int rc){
  if( rc==SQLITE_IOERR_NOMEM ) return;
  rc &= 0xff;
  if( rc==SQLITE_CANTOPEN || rc==SQLITE_IOERR ){
    db->iSysErrno = sqlite3OsGetLastError(db->pVfs);
  }
}

/*
** Set the most recent error code and error string for the sqlite
** handle "db". The error code is set to "err_code".
**
** If it is not NULL, string zFormat specifies the format of the
** error string.  zFormat and any string tokens that follow it are
** assumed to be encoded in UTF-8.
**
** To clear the most recent error for sqlite handle "db", sqlite3Error
** should be called with err_code set to SQLITE_OK and zFormat set
** to NULL.
*/
void sqlite3ErrorWithMsg(sqlite3 *db, int err_code, const char *zFormat, ...){
  assert( db!=0 );
  db->errCode = err_code;
  sqlite3SystemError(db, err_code);
  if( zFormat==0 ){
    sqlite3Error(db, err_code);
  }else if( db->pErr || (db->pErr = sqlite3ValueNew(db))!=0 ){
    char *z;
    va_list ap;
    va_start(ap, zFormat);
    z = sqlite3VMPrintf(db, zFormat, ap);
    va_end(ap);
    sqlite3ValueSetStr(db->pErr, -1, z, SQLITE_UTF8, SQLITE_DYNAMIC);
  }
}

/*
** Add an error message to pParse->zErrMsg and increment pParse->nErr.
**
** This function should be used to report any error that occurs while
** compiling an SQL statement (i.e. within sqlite3_prepare()). The
** last thing the sqlite3_prepare() function does is copy the error
** stored by this function into the database handle using sqlite3Error().
** Functions sqlite3Error() or sqlite3ErrorWithMsg() should be used
** during statement execution (sqlite3_step() etc.).
*/
void sqlite3ErrorMsg(Parse *pParse, const char *zFormat, ...){
  char *zMsg;
  va_list ap;
  sqlite3 *db = pParse->db;
  db->errByteOffset = -2;
  va_start(ap, zFormat);
  zMsg = sqlite3VMPrintf(db, zFormat, ap);
  va_end(ap);
  if( db->errByteOffset<-1 ) db->errByteOffset = -1;
  if( db->suppressErr ){
    sqlite3DbFree(db, zMsg);
  }else{
    pParse->nErr++;
    sqlite3DbFree(db, pParse->zErrMsg);
    pParse->zErrMsg = zMsg;
    pParse->rc = SQLITE_ERROR;
    pParse->pWith = 0;
  }
}

/*
** If database connection db is currently parsing SQL, then transfer
** error code errCode to that parser if the parser has not already
** encountered some other kind of error.
*/
int sqlite3ErrorToParser(sqlite3 *db, int errCode){
  Parse *pParse;
  if( db==0 || (pParse = db->pParse)==0 ) return errCode;
  pParse->rc = errCode;
  pParse->nErr++;
  return errCode;
}

/*
** Convert an SQL-style quoted string into a normal string by removing
** the quote characters.  The conversion is done in-place.  If the
** input does not begin with a quote character, then this routine
** is a no-op.
**
** The input string must be zero-terminated.  A new zero-terminator
** is added to the dequoted string.
**
** The return value is -1 if no dequoting occurs or the length of the
** dequoted string, exclusive of the zero terminator, if dequoting does
** occur.
**
** 2002-02-14: This routine is extended to remove MS-Access style
** brackets from around identifiers.  For example:  "[a-b-c]" becomes
** "a-b-c".
*/
void sqlite3Dequote(char *z){
  char quote;
  int i, j;
  if( z==0 ) return;
  quote = z[0];
  if( !sqlite3Isquote(quote) ) return;
  if( quote=='[' ) quote = ']';
  for(i=1, j=0;; i++){
    assert( z[i] );
    if( z[i]==quote ){
      if( z[i+1]==quote ){
        z[j++] = quote;
        i++;
      }else{
        break;
      }
    }else{
      z[j++] = z[i];
    }
  }
  z[j] = 0;
}
void sqlite3DequoteExpr(Expr *p){
  assert( !ExprHasProperty(p, EP_IntValue) );
  assert( sqlite3Isquote(p->u.zToken[0]) );
  p->flags |= p->u.zToken[0]=='"' ? EP_Quoted|EP_DblQuoted : EP_Quoted;
  sqlite3Dequote(p->u.zToken);
}

/*
** If the input token p is quoted, try to adjust the token to remove
** the quotes.  This is not always possible:
**
**     "abc"     ->   abc
**     "ab""cd"  ->   (not possible because of the interior "")
**
** Remove the quotes if possible.  This is a optimization.  The overall
** system should still return the correct answer even if this routine
** is always a no-op.
*/
void sqlite3DequoteToken(Token *p){
  unsigned int i;
  if( p->n<2 ) return;
  if( !sqlite3Isquote(p->z[0]) ) return;
  for(i=1; i<p->n-1; i++){
    if( sqlite3Isquote(p->z[i]) ) return;
  }
  p->n -= 2;
  p->z++;
}

/*
** Generate a Token object from a string
*/
void sqlite3TokenInit(Token *p, char *z){
  p->z = z;
  p->n = sqlite3Strlen30(z);
}

/* Convenient short-hand */
#define UpperToLower sqlite3UpperToLower

/*
** Some systems have stricmp().  Others have strcasecmp().  Because
** there is no consistency, we will define our own.
**
** IMPLEMENTATION-OF: R-30243-02494 The sqlite3_stricmp() and
** sqlite3_strnicmp() APIs allow applications and extensions to compare
** the contents of two buffers containing UTF-8 strings in a
** case-independent fashion, using the same definition of "case
** independence" that SQLite uses internally when comparing identifiers.
*/
int sqlite3_stricmp(const char *zLeft, const char *zRight){
  if( zLeft==0 ){
    return zRight ? -1 : 0;
  }else if( zRight==0 ){
    return 1;
  }
  return sqlite3StrICmp(zLeft, zRight);
}
int sqlite3StrICmp(const char *zLeft, const char *zRight){
  unsigned char *a, *b;
  int c, x;
  a = (unsigned char *)zLeft;
  b = (unsigned char *)zRight;
  for(;;){
    c = *a;
    x = *b;
    if( c==x ){
      if( c==0 ) break;
    }else{
      c = (int)UpperToLower[c] - (int)UpperToLower[x];
      if( c ) break;
    }
    a++;
    b++;
  }
  return c;
}
int sqlite3_strnicmp(const char *zLeft, const char *zRight, int N){
  register unsigned char *a, *b;
  if( zLeft==0 ){
    return zRight ? -1 : 0;
  }else if( zRight==0 ){
    return 1;
  }
  a = (unsigned char *)zLeft;
  b = (unsigned char *)zRight;
  while( N-- > 0 && *a!=0 && UpperToLower[*a]==UpperToLower[*b]){ a++; b++; }
  return N<0 ? 0 : UpperToLower[*a] - UpperToLower[*b];
}

/*
** Compute an 8-bit hash on a string that is insensitive to case differences
*/
u8 sqlite3StrIHash(const char *z){
  u8 h = 0;
  if( z==0 ) return 0;
  while( z[0] ){
    h += UpperToLower[(unsigned char)z[0]];
    z++;
  }
  return h;
}

/*
** Compute 10 to the E-th power.  Examples:  E==1 results in 10.
** E==2 results in 100.  E==50 results in 1.0e50.
**
** This routine only works for values of E between 1 and 341.
*/
static LONGDOUBLE_TYPE sqlite3Pow10(int E){
#if defined(_MSC_VER)
  static const LONGDOUBLE_TYPE x[] = {
    1.0e+001L,
    1.0e+002L,
    1.0e+004L,
    1.0e+008L,
    1.0e+016L,
    1.0e+032L,
    1.0e+064L,
    1.0e+128L,
    1.0e+256L
  };
  LONGDOUBLE_TYPE r = 1.0;
  int i;
  assert( E>=0 && E<=307 );
  for(i=0; E!=0; i++, E >>=1){
    if( E & 1 ) r *= x[i];
  }
  return r;
#else
  LONGDOUBLE_TYPE x = 10.0;
  LONGDOUBLE_TYPE r = 1.0;
  while(1){
    if( E & 1 ) r *= x;
    E >>= 1;
    if( E==0 ) break;
    x *= x;
  }
  return r; 
#endif
}

/*
** The string z[] is an text representation of a real number.
** Convert this string to a double and write it into *pResult.
**
** The string z[] is length bytes in length (bytes, not characters) and
** uses the encoding enc.  The string is not necessarily zero-terminated.
**
** Return TRUE if the result is a valid real number (or integer) and FALSE
** if the string is empty or contains extraneous text.  More specifically
** return
**      1          =>  The input string is a pure integer
**      2 or more  =>  The input has a decimal point or eNNN clause
**      0 or less  =>  The input string is not a valid number
**     -1          =>  Not a valid number, but has a valid prefix which 
**                     includes a decimal point and/or an eNNN clause
**
** Valid numbers are in one of these formats:
**
**    [+-]digits[E[+-]digits]
**    [+-]digits.[digits][E[+-]digits]
**    [+-].digits[E[+-]digits]
**
** Leading and trailing whitespace is ignored for the purpose of determining
** validity.
**
** If some prefix of the input string is a valid number, this routine
** returns FALSE but it still converts the prefix and writes the result
** into *pResult.
*/
#if defined(_MSC_VER)
#pragma warning(disable : 4756)
#endif
int sqlite3AtoF(const char *z, double *pResult, int length, u8 enc){
#ifndef SQLITE_OMIT_FLOATING_POINT
  int incr;
  const char *zEnd;
  /* sign * significand * (10 ^ (esign * exponent)) */
  int sign = 1;    /* sign of significand */
  i64 s = 0;       /* significand */
  int d = 0;       /* adjust exponent for shifting decimal point */
  int esign = 1;   /* sign of exponent */
  int e = 0;       /* exponent */
  int eValid = 1;  /* True exponent is either not used or is well-formed */
  double result;
  int nDigit = 0;  /* Number of digits processed */
  int eType = 1;   /* 1: pure integer,  2+: fractional  -1 or less: bad UTF16 */

  assert( enc==SQLITE_UTF8 || enc==SQLITE_UTF16LE || enc==SQLITE_UTF16BE );
  *pResult = 0.0;   /* Default return value, in case of an error */
  if( length==0 ) return 0;

  if( enc==SQLITE_UTF8 ){
    incr = 1;
    zEnd = z + length;
  }else{
    int i;
    incr = 2;
    length &= ~1;
    assert( SQLITE_UTF16LE==2 && SQLITE_UTF16BE==3 );
    testcase( enc==SQLITE_UTF16LE );
    testcase( enc==SQLITE_UTF16BE );
    for(i=3-enc; i<length && z[i]==0; i+=2){}
    if( i<length ) eType = -100;
    zEnd = &z[i^1];
    z += (enc&1);
  }

  /* skip leading spaces */
  while( z<zEnd && sqlite3Isspace(*z) ) z+=incr;
  if( z>=zEnd ) return 0;

  /* get sign of significand */
  if( *z=='-' ){
    sign = -1;
    z+=incr;
  }else if( *z=='+' ){
    z+=incr;
  }

  /* copy max significant digits to significand */
  while( z<zEnd && sqlite3Isdigit(*z) ){
    s = s*10 + (*z - '0');
    z+=incr; nDigit++;
    if( s>=((LARGEST_INT64-9)/10) ){
      /* skip non-significant significand digits
      ** (increase exponent by d to shift decimal left) */
      while( z<zEnd && sqlite3Isdigit(*z) ){ z+=incr; d++; }
    }
  }
  if( z>=zEnd ) goto do_atof_calc;

  /* if decimal point is present */
  if( *z=='.' ){
    z+=incr;
    eType++;
    /* copy digits from after decimal to significand
    ** (decrease exponent by d to shift decimal right) */
    while( z<zEnd && sqlite3Isdigit(*z) ){
      if( s<((LARGEST_INT64-9)/10) ){
        s = s*10 + (*z - '0');
        d--;
        nDigit++;
      }
      z+=incr;
    }
  }
  if( z>=zEnd ) goto do_atof_calc;

  /* if exponent is present */
  if( *z=='e' || *z=='E' ){
    z+=incr;
    eValid = 0;
    eType++;

    /* This branch is needed to avoid a (harmless) buffer overread.  The 
    ** special comment alerts the mutation tester that the correct answer
    ** is obtained even if the branch is omitted */
    if( z>=zEnd ) goto do_atof_calc;              /*PREVENTS-HARMLESS-OVERREAD*/

    /* get sign of exponent */
    if( *z=='-' ){
      esign = -1;
      z+=incr;
    }else if( *z=='+' ){
      z+=incr;
    }
    /* copy digits to exponent */
    while( z<zEnd && sqlite3Isdigit(*z) ){
      e = e<10000 ? (e*10 + (*z - '0')) : 10000;
      z+=incr;
      eValid = 1;
    }
  }

  /* skip trailing spaces */
  while( z<zEnd && sqlite3Isspace(*z) ) z+=incr;

do_atof_calc:
  /* adjust exponent by d, and update sign */
  e = (e*esign) + d;
  if( e<0 ) {
    esign = -1;
    e *= -1;
  } else {
    esign = 1;
  }

  if( s==0 ) {
    /* In the IEEE 754 standard, zero is signed. */
    result = sign<0 ? -(double)0 : (double)0;
  } else {
    /* Attempt to reduce exponent.
    **
    ** Branches that are not required for the correct answer but which only
    ** help to obtain the correct answer faster are marked with special
    ** comments, as a hint to the mutation tester.
    */
    while( e>0 ){                                       /*OPTIMIZATION-IF-TRUE*/
      if( esign>0 ){
        if( s>=(LARGEST_INT64/10) ) break;             /*OPTIMIZATION-IF-FALSE*/
        s *= 10;
      }else{
        if( s%10!=0 ) break;                           /*OPTIMIZATION-IF-FALSE*/
        s /= 10;
      }
      e--;
    }

    /* adjust the sign of significand */
    s = sign<0 ? -s : s;

    if( e==0 ){                                         /*OPTIMIZATION-IF-TRUE*/
      result = (double)s;
    }else{
      /* attempt to handle extremely small/large numbers better */
      if( e>307 ){                                      /*OPTIMIZATION-IF-TRUE*/
        if( e<342 ){                                    /*OPTIMIZATION-IF-TRUE*/
          LONGDOUBLE_TYPE scale = sqlite3Pow10(e-308);
          if( esign<0 ){
            result = s / scale;
            result /= 1.0e+308;
          }else{
            result = s * scale;
            result *= 1.0e+308;
          }
        }else{ assert( e>=342 );
          if( esign<0 ){
            result = 0.0*s;
          }else{
#ifdef INFINITY
            result = INFINITY*s;
#else
            result = 1e308*1e308*s;  /* Infinity */
#endif
          }
        }
      }else{
        LONGDOUBLE_TYPE scale = sqlite3Pow10(e);
        if( esign<0 ){
          result = s / scale;
        }else{
          result = s * scale;
        }
      }
    }
  }

  /* store the result */
  *pResult = result;

  /* return true if number and no extra non-whitespace chracters after */
  if( z==zEnd && nDigit>0 && eValid && eType>0 ){
    return eType;
  }else if( eType>=2 && (eType==3 || eValid) && nDigit>0 ){
    return -1;
  }else{
    return 0;
  }
#else
  return !sqlite3Atoi64(z, pResult, length, enc);
#endif /* SQLITE_OMIT_FLOATING_POINT */
}
#if defined(_MSC_VER)
#pragma warning(default : 4756)
#endif

/*
** Render an signed 64-bit integer as text.  Store the result in zOut[].
**
** The caller must ensure that zOut[] is at least 21 bytes in size.
*/
void sqlite3Int64ToText(i64 v, char *zOut){
  int i;
  u64 x;
  char zTemp[22];
  if( v<0 ){
    x = (v==SMALLEST_INT64) ? ((u64)1)<<63 : (u64)-v;
  }else{
    x = v;
  }
  i = sizeof(zTemp)-2;
  zTemp[sizeof(zTemp)-1] = 0;
  do{
    zTemp[i--] = (x%10) + '0';
    x = x/10;
  }while( x );
  if( v<0 ) zTemp[i--] = '-';
  memcpy(zOut, &zTemp[i+1], sizeof(zTemp)-1-i);
}

/*
** Compare the 19-character string zNum against the text representation
** value 2^63:  9223372036854775808.  Return negative, zero, or positive
** if zNum is less than, equal to, or greater than the string.
** Note that zNum must contain exactly 19 characters.
**
** Unlike memcmp() this routine is guaranteed to return the difference
** in the values of the last digit if the only difference is in the
** last digit.  So, for example,
**
**      compare2pow63("9223372036854775800", 1)
**
** will return -8.
*/
static int compare2pow63(const char *zNum, int incr){
  int c = 0;
  int i;
                    /* 012345678901234567 */
  const char *pow63 = "922337203685477580";
  for(i=0; c==0 && i<18; i++){
    c = (zNum[i*incr]-pow63[i])*10;
  }
  if( c==0 ){
    c = zNum[18*incr] - '8';
    testcase( c==(-1) );
    testcase( c==0 );
    testcase( c==(+1) );
  }
  return c;
}

/*
** Convert zNum to a 64-bit signed integer.  zNum must be decimal. This
** routine does *not* accept hexadecimal notation.
**
** Returns:
**
**    -1    Not even a prefix of the input text looks like an integer
**     0    Successful transformation.  Fits in a 64-bit signed integer.
**     1    Excess non-space text after the integer value
**     2    Integer too large for a 64-bit signed integer or is malformed
**     3    Special case of 9223372036854775808
**
** length is the number of bytes in the string (bytes, not characters).
** The string is not necessarily zero-terminated.  The encoding is
** given by enc.
*/
int sqlite3Atoi64(const char *zNum, i64 *pNum, int length, u8 enc){
  int incr;
  u64 u = 0;
  int neg = 0; /* assume positive */
  int i;
  int c = 0;
  int nonNum = 0;  /* True if input contains UTF16 with high byte non-zero */
  int rc;          /* Baseline return code */
  const char *zStart;
  const char *zEnd = zNum + length;
  assert( enc==SQLITE_UTF8 || enc==SQLITE_UTF16LE || enc==SQLITE_UTF16BE );
  if( enc==SQLITE_UTF8 ){
    incr = 1;
  }else{
    incr = 2;
    length &= ~1;
    assert( SQLITE_UTF16LE==2 && SQLITE_UTF16BE==3 );
    for(i=3-enc; i<length && zNum[i]==0; i+=2){}
    nonNum = i<length;
    zEnd = &zNum[i^1];
    zNum += (enc&1);
  }
  while( zNum<zEnd && sqlite3Isspace(*zNum) ) zNum+=incr;
  if( zNum<zEnd ){
    if( *zNum=='-' ){
      neg = 1;
      zNum+=incr;
    }else if( *zNum=='+' ){
      zNum+=incr;
    }
  }
  zStart = zNum;
  while( zNum<zEnd && zNum[0]=='0' ){ zNum+=incr; } /* Skip leading zeros. */
  for(i=0; &zNum[i]<zEnd && (c=zNum[i])>='0' && c<='9'; i+=incr){
    u = u*10 + c - '0';
  }
  testcase( i==18*incr );
  testcase( i==19*incr );
  testcase( i==20*incr );
  if( u>LARGEST_INT64 ){
    /* This test and assignment is needed only to suppress UB warnings
    ** from clang and -fsanitize=undefined.  This test and assignment make
    ** the code a little larger and slower, and no harm comes from omitting
    ** them, but we must appaise the undefined-behavior pharisees. */
    *pNum = neg ? SMALLEST_INT64 : LARGEST_INT64;
  }else if( neg ){
    *pNum = -(i64)u;
  }else{
    *pNum = (i64)u;
  }
  rc = 0;
  if( i==0 && zStart==zNum ){    /* No digits */
    rc = -1;
  }else if( nonNum ){            /* UTF16 with high-order bytes non-zero */
    rc = 1;
  }else if( &zNum[i]<zEnd ){     /* Extra bytes at the end */
    int jj = i;
    do{
      if( !sqlite3Isspace(zNum[jj]) ){
        rc = 1;          /* Extra non-space text after the integer */
        break;
      }
      jj += incr;
    }while( &zNum[jj]<zEnd );
  }
  if( i<19*incr ){
    /* Less than 19 digits, so we know that it fits in 64 bits */
    assert( u<=LARGEST_INT64 );
    return rc;
  }else{
    /* zNum is a 19-digit numbers.  Compare it against 9223372036854775808. */
    c = i>19*incr ? 1 : compare2pow63(zNum, incr);
    if( c<0 ){
      /* zNum is less than 9223372036854775808 so it fits */
      assert( u<=LARGEST_INT64 );
      return rc;
    }else{
      *pNum = neg ? SMALLEST_INT64 : LARGEST_INT64;
      if( c>0 ){
        /* zNum is greater than 9223372036854775808 so it overflows */
        return 2;
      }else{
        /* zNum is exactly 9223372036854775808.  Fits if negative.  The
        ** special case 2 overflow if positive */
        assert( u-1==LARGEST_INT64 );
        return neg ? rc : 3;
      }
    }
  }
}

/*
** Transform a UTF-8 integer literal, in either decimal or hexadecimal,
** into a 64-bit signed integer.  This routine accepts hexadecimal literals,
** whereas sqlite3Atoi64() does not.
**
** Returns:
**
**     0    Successful transformation.  Fits in a 64-bit signed integer.
**     1    Excess text after the integer value
**     2    Integer too large for a 64-bit signed integer or is malformed
**     3    Special case of 9223372036854775808
*/
int sqlite3DecOrHexToI64(const char *z, i64 *pOut){
#ifndef SQLITE_OMIT_HEX_INTEGER
  if( z[0]=='0'
   && (z[1]=='x' || z[1]=='X')
  ){
    u64 u = 0;
    int i, k;
    for(i=2; z[i]=='0'; i++){}
    for(k=i; sqlite3Isxdigit(z[k]); k++){
      u = u*16 + sqlite3HexToInt(z[k]);
    }
    memcpy(pOut, &u, 8);
    return (z[k]==0 && k-i<=16) ? 0 : 2;
  }else
#endif /* SQLITE_OMIT_HEX_INTEGER */
  {
    return sqlite3Atoi64(z, pOut, sqlite3Strlen30(z), SQLITE_UTF8);
  }
}

/*
** If zNum represents an integer that will fit in 32-bits, then set
** *pValue to that integer and return true.  Otherwise return false.
**
** This routine accepts both decimal and hexadecimal notation for integers.
**
** Any non-numeric characters that following zNum are ignored.
** This is different from sqlite3Atoi64() which requires the
** input number to be zero-terminated.
*/
int sqlite3GetInt32(const char *zNum, int *pValue){
  sqlite_int64 v = 0;
  int i, c;
  int neg = 0;
  if( zNum[0]=='-' ){
    neg = 1;
    zNum++;
  }else if( zNum[0]=='+' ){
    zNum++;
  }
#ifndef SQLITE_OMIT_HEX_INTEGER
  else if( zNum[0]=='0'
        && (zNum[1]=='x' || zNum[1]=='X')
        && sqlite3Isxdigit(zNum[2])
  ){
    u32 u = 0;
    zNum += 2;
    while( zNum[0]=='0' ) zNum++;
    for(i=0; sqlite3Isxdigit(zNum[i]) && i<8; i++){
      u = u*16 + sqlite3HexToInt(zNum[i]);
    }
    if( (u&0x80000000)==0 && sqlite3Isxdigit(zNum[i])==0 ){
      memcpy(pValue, &u, 4);
      return 1;
    }else{
      return 0;
    }
  }
#endif
  if( !sqlite3Isdigit(zNum[0]) ) return 0;
  while( zNum[0]=='0' ) zNum++;
  for(i=0; i<11 && (c = zNum[i] - '0')>=0 && c<=9; i++){
    v = v*10 + c;
  }

  /* The longest decimal representation of a 32 bit integer is 10 digits:
  **
  **             1234567890
  **     2^31 -> 2147483648
  */
  testcase( i==10 );
  if( i>10 ){
    return 0;
  }
  testcase( v-neg==2147483647 );
  if( v-neg>2147483647 ){
    return 0;
  }
  if( neg ){
    v = -v;
  }
  *pValue = (int)v;
  return 1;
}

/*
** Return a 32-bit integer value extracted from a string.  If the
** string is not an integer, just return 0.
*/
int sqlite3Atoi(const char *z){
  int x = 0;
  sqlite3GetInt32(z, &x);
  return x;
}

/*
** Try to convert z into an unsigned 32-bit integer.  Return true on
** success and false if there is an error.
**
** Only decimal notation is accepted.
*/
int sqlite3GetUInt32(const char *z, u32 *pI){
  u64 v = 0;
  int i;
  for(i=0; sqlite3Isdigit(z[i]); i++){
    v = v*10 + z[i] - '0';
    if( v>4294967296LL ){ *pI = 0; return 0; }
  }
  if( i==0 || z[i]!=0 ){ *pI = 0; return 0; }
  *pI = (u32)v;
  return 1;
}

/*
** The variable-length integer encoding is as follows:
**
** KEY:
**         A = 0xxxxxxx    7 bits of data and one flag bit
**         B = 1xxxxxxx    7 bits of data and one flag bit
**         C = xxxxxxxx    8 bits of data
**
**  7 bits - A
** 14 bits - BA
** 21 bits - BBA
** 28 bits - BBBA
** 35 bits - BBBBA
** 42 bits - BBBBBA
** 49 bits - BBBBBBA
** 56 bits - BBBBBBBA
** 64 bits - BBBBBBBBC
*/

/*
** Write a 64-bit variable-length integer to memory starting at p[0].
** The length of data write will be between 1 and 9 bytes.  The number
** of bytes written is returned.
**
** A variable-length integer consists of the lower 7 bits of each byte
** for all bytes that have the 8th bit set and one byte with the 8th
** bit clear.  Except, if we get to the 9th byte, it stores the full
** 8 bits and is the last byte.
*/
static int SQLITE_NOINLINE putVarint64(unsigned char *p, u64 v){
  int i, j, n;
  u8 buf[10];
  if( v & (((u64)0xff000000)<<32) ){
    p[8] = (u8)v;
    v >>= 8;
    for(i=7; i>=0; i--){
      p[i] = (u8)((v & 0x7f) | 0x80);
      v >>= 7;
    }
    return 9;
  }    
  n = 0;
  do{
    buf[n++] = (u8)((v & 0x7f) | 0x80);
    v >>= 7;
  }while( v!=0 );
  buf[0] &= 0x7f;
  assert( n<=9 );
  for(i=0, j=n-1; j>=0; j--, i++){
    p[i] = buf[j];
  }
  return n;
}
int sqlite3PutVarint(unsigned char *p, u64 v){
  if( v<=0x7f ){
    p[0] = v&0x7f;
    return 1;
  }
  if( v<=0x3fff ){
    p[0] = ((v>>7)&0x7f)|0x80;
    p[1] = v&0x7f;
    return 2;
  }
  return putVarint64(p,v);
}

/*
** Bitmasks used by sqlite3GetVarint().  These precomputed constants
** are defined here rather than simply putting the constant expressions
** inline in order to work around bugs in the RVT compiler.
**
** SLOT_2_0     A mask for  (0x7f<<14) | 0x7f
**
** SLOT_4_2_0   A mask for  (0x7f<<28) | SLOT_2_0
*/
#define SLOT_2_0     0x001fc07f
#define SLOT_4_2_0   0xf01fc07f


/*
** Read a 64-bit variable-length integer from memory starting at p[0].
** Return the number of bytes read.  The value is stored in *v.
*/
u8 sqlite3GetVarint(const unsigned char *p, u64 *v){
  u32 a,b,s;

  if( ((signed char*)p)[0]>=0 ){
    *v = *p;
    return 1;
  }
  if( ((signed char*)p)[1]>=0 ){
    *v = ((u32)(p[0]&0x7f)<<7) | p[1];
    return 2;
  }

  /* Verify that constants are precomputed correctly */
  assert( SLOT_2_0 == ((0x7f<<14) | (0x7f)) );
  assert( SLOT_4_2_0 == ((0xfU<<28) | (0x7f<<14) | (0x7f)) );

  a = ((u32)p[0])<<14;
  b = p[1];
  p += 2;
  a |= *p;
  /* a: p0<<14 | p2 (unmasked) */
  if (!(a&0x80))
  {
    a &= SLOT_2_0;
    b &= 0x7f;
    b = b<<7;
    a |= b;
    *v = a;
    return 3;
  }

  /* CSE1 from below */
  a &= SLOT_2_0;
  p++;
  b = b<<14;
  b |= *p;
  /* b: p1<<14 | p3 (unmasked) */
  if (!(b&0x80))
  {
    b &= SLOT_2_0;
    /* moved CSE1 up */
    /* a &= (0x7f<<14)|(0x7f); */
    a = a<<7;
    a |= b;
    *v = a;
    return 4;
  }

  /* a: p0<<14 | p2 (masked) */
  /* b: p1<<14 | p3 (unmasked) */
  /* 1:save off p0<<21 | p1<<14 | p2<<7 | p3 (masked) */
  /* moved CSE1 up */
  /* a &= (0x7f<<14)|(0x7f); */
  b &= SLOT_2_0;
  s = a;
  /* s: p0<<14 | p2 (masked) */

  p++;
  a = a<<14;
  a |= *p;
  /* a: p0<<28 | p2<<14 | p4 (unmasked) */
  if (!(a&0x80))
  {
    /* we can skip these cause they were (effectively) done above
    ** while calculating s */
    /* a &= (0x7f<<28)|(0x7f<<14)|(0x7f); */
    /* b &= (0x7f<<14)|(0x7f); */
    b = b<<7;
    a |= b;
    s = s>>18;
    *v = ((u64)s)<<32 | a;
    return 5;
  }

  /* 2:save off p0<<21 | p1<<14 | p2<<7 | p3 (masked) */
  s = s<<7;
  s |= b;
  /* s: p0<<21 | p1<<14 | p2<<7 | p3 (masked) */

  p++;
  b = b<<14;
  b |= *p;
  /* b: p1<<28 | p3<<14 | p5 (unmasked) */
  if (!(b&0x80))
  {
    /* we can skip this cause it was (effectively) done above in calc'ing s */
    /* b &= (0x7f<<28)|(0x7f<<14)|(0x7f); */
    a &= SLOT_2_0;
    a = a<<7;
    a |= b;
    s = s>>18;
    *v = ((u64)s)<<32 | a;
    return 6;
  }

  p++;
  a = a<<14;
  a |= *p;
  /* a: p2<<28 | p4<<14 | p6 (unmasked) */
  if (!(a&0x80))
  {
    a &= SLOT_4_2_0;
    b &= SLOT_2_0;
    b = b<<7;
    a |= b;
    s = s>>11;
    *v = ((u64)s)<<32 | a;
    return 7;
  }

  /* CSE2 from below */
  a &= SLOT_2_0;
  p++;
  b = b<<14;
  b |= *p;
  /* b: p3<<28 | p5<<14 | p7 (unmasked) */
  if (!(b&0x80))
  {
    b &= SLOT_4_2_0;
    /* moved CSE2 up */
    /* a &= (0x7f<<14)|(0x7f); */
    a = a<<7;
    a |= b;
    s = s>>4;
    *v = ((u64)s)<<32 | a;
    return 8;
  }

  p++;
  a = a<<15;
  a |= *p;
  /* a: p4<<29 | p6<<15 | p8 (unmasked) */

  /* moved CSE2 up */
  /* a &= (0x7f<<29)|(0x7f<<15)|(0xff); */
  b &= SLOT_2_0;
  b = b<<8;
  a |= b;

  s = s<<4;
  b = p[-4];
  b &= 0x7f;
  b = b>>3;
  s |= b;

  *v = ((u64)s)<<32 | a;

  return 9;
}

/*
** Read a 32-bit variable-length integer from memory starting at p[0].
** Return the number of bytes read.  The value is stored in *v.
**
** If the varint stored in p[0] is larger than can fit in a 32-bit unsigned
** integer, then set *v to 0xffffffff.
**
** A MACRO version, getVarint32, is provided which inlines the 
** single-byte case.  All code should use the MACRO version as 
** this function assumes the single-byte case has already been handled.
*/
u8 sqlite3GetVarint32(const unsigned char *p, u32 *v){
  u32 a,b;

  /* The 1-byte case.  Overwhelmingly the most common.  Handled inline
  ** by the getVarin32() macro */
  a = *p;
  /* a: p0 (unmasked) */
#ifndef getVarint32
  if (!(a&0x80))
  {
    /* Values between 0 and 127 */
    *v = a;
    return 1;
  }
#endif

  /* The 2-byte case */
  p++;
  b = *p;
  /* b: p1 (unmasked) */
  if (!(b&0x80))
  {
    /* Values between 128 and 16383 */
    a &= 0x7f;
    a = a<<7;
    *v = a | b;
    return 2;
  }

  /* The 3-byte case */
  p++;
  a = a<<14;
  a |= *p;
  /* a: p0<<14 | p2 (unmasked) */
  if (!(a&0x80))
  {
    /* Values between 16384 and 2097151 */
    a &= (0x7f<<14)|(0x7f);
    b &= 0x7f;
    b = b<<7;
    *v = a | b;
    return 3;
  }

  /* A 32-bit varint is used to store size information in btrees.
  ** Objects are rarely larger than 2MiB limit of a 3-byte varint.
  ** A 3-byte varint is sufficient, for example, to record the size
  ** of a 1048569-byte BLOB or string.
  **
  ** We only unroll the first 1-, 2-, and 3- byte cases.  The very
  ** rare larger cases can be handled by the slower 64-bit varint
  ** routine.
  */
#if 1
  {
    u64 v64;
    u8 n;

    n = sqlite3GetVarint(p-2, &v64);
    assert( n>3 && n<=9 );
    if( (v64 & SQLITE_MAX_U32)!=v64 ){
      *v = 0xffffffff;
    }else{
      *v = (u32)v64;
    }
    return n;
  }

#else
  /* For following code (kept for historical record only) shows an
  ** unrolling for the 3- and 4-byte varint cases.  This code is
  ** slightly faster, but it is also larger and much harder to test.
  */
  p++;
  b = b<<14;
  b |= *p;
  /* b: p1<<14 | p3 (unmasked) */
  if (!(b&0x80))
  {
    /* Values between 2097152 and 268435455 */
    b &= (0x7f<<14)|(0x7f);
    a &= (0x7f<<14)|(0x7f);
    a = a<<7;
    *v = a | b;
    return 4;
  }

  p++;
  a = a<<14;
  a |= *p;
  /* a: p0<<28 | p2<<14 | p4 (unmasked) */
  if (!(a&0x80))
  {
    /* Values  between 268435456 and 34359738367 */
    a &= SLOT_4_2_0;
    b &= SLOT_4_2_0;
    b = b<<7;
    *v = a | b;
    return 5;
  }

  /* We can only reach this point when reading a corrupt database
  ** file.  In that case we are not in any hurry.  Use the (relatively
  ** slow) general-purpose sqlite3GetVarint() routine to extract the
  ** value. */
  {
    u64 v64;
    u8 n;

    p -= 4;
    n = sqlite3GetVarint(p, &v64);
    assert( n>5 && n<=9 );
    *v = (u32)v64;
    return n;
  }
#endif
}

/*
** Return the number of bytes that will be needed to store the given
** 64-bit integer.
*/
int sqlite3VarintLen(u64 v){
  int i;
  for(i=1; (v >>= 7)!=0; i++){ assert( i<10 ); }
  return i;
}


/*
** Read or write a four-byte big-endian integer value.
*/
u32 sqlite3Get4byte(const u8 *p){
#if SQLITE_BYTEORDER==4321
  u32 x;
  memcpy(&x,p,4);
  return x;
#elif SQLITE_BYTEORDER==1234 && GCC_VERSION>=4003000
  u32 x;
  memcpy(&x,p,4);
  return __builtin_bswap32(x);
#elif SQLITE_BYTEORDER==1234 && MSVC_VERSION>=1300
  u32 x;
  memcpy(&x,p,4);
  return _byteswap_ulong(x);
#else
  testcase( p[0]&0x80 );
  return ((unsigned)p[0]<<24) | (p[1]<<16) | (p[2]<<8) | p[3];
#endif
}
void sqlite3Put4byte(unsigned char *p, u32 v){
#if SQLITE_BYTEORDER==4321
  memcpy(p,&v,4);
#elif SQLITE_BYTEORDER==1234 && GCC_VERSION>=4003000
  u32 x = __builtin_bswap32(v);
  memcpy(p,&x,4);
#elif SQLITE_BYTEORDER==1234 && MSVC_VERSION>=1300
  u32 x = _byteswap_ulong(v);
  memcpy(p,&x,4);
#else
  p[0] = (u8)(v>>24);
  p[1] = (u8)(v>>16);
  p[2] = (u8)(v>>8);
  p[3] = (u8)v;
#endif
}



/*
** Translate a single byte of Hex into an integer.
** This routine only works if h really is a valid hexadecimal
** character:  0..9a..fA..F
*/
u8 sqlite3HexToInt(int h){
  assert( (h>='0' && h<='9') ||  (h>='a' && h<='f') ||  (h>='A' && h<='F') );
#ifdef SQLITE_ASCII
  h += 9*(1&(h>>6));
#endif
#ifdef SQLITE_EBCDIC
  h += 9*(1&~(h>>4));
#endif
  return (u8)(h & 0xf);
}

#if !defined(SQLITE_OMIT_BLOB_LITERAL)
/*
** Convert a BLOB literal of the form "x'hhhhhh'" into its binary
** value.  Return a pointer to its binary value.  Space to hold the
** binary value has been obtained from malloc and must be freed by
** the calling routine.
*/
void *sqlite3HexToBlob(sqlite3 *db, const char *z, int n){
  char *zBlob;
  int i;

  zBlob = (char *)sqlite3DbMallocRawNN(db, n/2 + 1);
  n--;
  if( zBlob ){
    for(i=0; i<n; i+=2){
      zBlob[i/2] = (sqlite3HexToInt(z[i])<<4) | sqlite3HexToInt(z[i+1]);
    }
    zBlob[i/2] = 0;
  }
  return zBlob;
}
#endif /* !SQLITE_OMIT_BLOB_LITERAL */

/*
** Log an error that is an API call on a connection pointer that should
** not have been used.  The "type" of connection pointer is given as the
** argument.  The zType is a word like "NULL" or "closed" or "invalid".
*/
static void logBadConnection(const char *zType){
  sqlite3_log(SQLITE_MISUSE, 
     "API call with %s database connection pointer",
     zType
  );
}

/*
** Check to make sure we have a valid db pointer.  This test is not
** foolproof but it does provide some measure of protection against
** misuse of the interface such as passing in db pointers that are
** NULL or which have been previously closed.  If this routine returns
** 1 it means that the db pointer is valid and 0 if it should not be
** dereferenced for any reason.  The calling function should invoke
** SQLITE_MISUSE immediately.
**
** sqlite3SafetyCheckOk() requires that the db pointer be valid for
** use.  sqlite3SafetyCheckSickOrOk() allows a db pointer that failed to
** open properly and is not fit for general use but which can be
** used as an argument to sqlite3_errmsg() or sqlite3_close().
*/
int sqlite3SafetyCheckOk(sqlite3 *db){
  u8 eOpenState;
  if( db==0 ){
    logBadConnection("NULL");
    return 0;
  }
  eOpenState = db->eOpenState;
  if( eOpenState!=SQLITE_STATE_OPEN ){
    if( sqlite3SafetyCheckSickOrOk(db) ){
      testcase( sqlite3GlobalConfig.xLog!=0 );
      logBadConnection("unopened");
    }
    return 0;
  }else{
    return 1;
  }
}
int sqlite3SafetyCheckSickOrOk(sqlite3 *db){
  u8 eOpenState;
  eOpenState = db->eOpenState;
  if( eOpenState!=SQLITE_STATE_SICK &&
      eOpenState!=SQLITE_STATE_OPEN &&
      eOpenState!=SQLITE_STATE_BUSY ){
    testcase( sqlite3GlobalConfig.xLog!=0 );
    logBadConnection("invalid");
    return 0;
  }else{
    return 1;
  }
}

/*
** Attempt to add, substract, or multiply the 64-bit signed value iB against
** the other 64-bit signed integer at *pA and store the result in *pA.
** Return 0 on success.  Or if the operation would have resulted in an
** overflow, leave *pA unchanged and return 1.
*/
int sqlite3AddInt64(i64 *pA, i64 iB){
#if GCC_VERSION>=5004000 && !defined(__INTEL_COMPILER)
  return __builtin_add_overflow(*pA, iB, pA);
#else
  i64 iA = *pA;
  testcase( iA==0 ); testcase( iA==1 );
  testcase( iB==-1 ); testcase( iB==0 );
  if( iB>=0 ){
    testcase( iA>0 && LARGEST_INT64 - iA == iB );
    testcase( iA>0 && LARGEST_INT64 - iA == iB - 1 );
    if( iA>0 && LARGEST_INT64 - iA < iB ) return 1;
  }else{
    testcase( iA<0 && -(iA + LARGEST_INT64) == iB + 1 );
    testcase( iA<0 && -(iA + LARGEST_INT64) == iB + 2 );
    if( iA<0 && -(iA + LARGEST_INT64) > iB + 1 ) return 1;
  }
  *pA += iB;
  return 0; 
#endif
}
int sqlite3SubInt64(i64 *pA, i64 iB){
#if GCC_VERSION>=5004000 && !defined(__INTEL_COMPILER)
  return __builtin_sub_overflow(*pA, iB, pA);
#else
  testcase( iB==SMALLEST_INT64+1 );
  if( iB==SMALLEST_INT64 ){
    testcase( (*pA)==(-1) ); testcase( (*pA)==0 );
    if( (*pA)>=0 ) return 1;
    *pA -= iB;
    return 0;
  }else{
    return sqlite3AddInt64(pA, -iB);
  }
#endif
}
int sqlite3MulInt64(i64 *pA, i64 iB){
#if GCC_VERSION>=5004000 && !defined(__INTEL_COMPILER)
  return __builtin_mul_overflow(*pA, iB, pA);
#else
  i64 iA = *pA;
  if( iB>0 ){
    if( iA>LARGEST_INT64/iB ) return 1;
    if( iA<SMALLEST_INT64/iB ) return 1;
  }else if( iB<0 ){
    if( iA>0 ){
      if( iB<SMALLEST_INT64/iA ) return 1;
    }else if( iA<0 ){
      if( iB==SMALLEST_INT64 ) return 1;
      if( iA==SMALLEST_INT64 ) return 1;
      if( -iA>LARGEST_INT64/-iB ) return 1;
    }
  }
  *pA = iA*iB;
  return 0;
#endif
}

/*
** Compute the absolute value of a 32-bit signed integer, of possible.  Or 
** if the integer has a value of -2147483648, return +2147483647
*/
int sqlite3AbsInt32(int x){
  if( x>=0 ) return x;
  if( x==(int)0x80000000 ) return 0x7fffffff;
  return -x;
}

#ifdef SQLITE_ENABLE_8_3_NAMES
/*
** If SQLITE_ENABLE_8_3_NAMES is set at compile-time and if the database
** filename in zBaseFilename is a URI with the "8_3_names=1" parameter and
** if filename in z[] has a suffix (a.k.a. "extension") that is longer than
** three characters, then shorten the suffix on z[] to be the last three
** characters of the original suffix.
**
** If SQLITE_ENABLE_8_3_NAMES is set to 2 at compile-time, then always
** do the suffix shortening regardless of URI parameter.
**
** Examples:
**
**     test.db-journal    =>   test.nal
**     test.db-wal        =>   test.wal
**     test.db-shm        =>   test.shm
**     test.db-mj7f3319fa =>   test.9fa
*/
void sqlite3FileSuffix3(const char *zBaseFilename, char *z){
#if SQLITE_ENABLE_8_3_NAMES<2
  if( sqlite3_uri_boolean(zBaseFilename, "8_3_names", 0) )
#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 an
** approximation 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{
#if GCC_VERSION>=5004000
    int i = 60 - __builtin_clzll(x);
    y += i*10;
    x >>= i;
#else
    while( x>255 ){ y += 40; x >>= 4; }  /*OPTIMIZATION-IF-TRUE*/
    while( x>15 ){  y += 10; x >>= 1; }
#endif
  }
  return a[x&7] + y - 10;
}

/*
** 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;
}

/*
** Convert a LogEst into an integer.
*/
u64 sqlite3LogEstToInt(LogEst x){
  u64 n;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
  if( x>60 ) return (u64)LARGEST_INT64;
  return x>=3 ? (n+8)<<(x-3) : (n+8)>>(3-x);
}

/*
** Add a new name/number pair to a VList.  This might require that the
** VList object be reallocated, so return the new VList.  If an OOM
** error occurs, the original VList returned and the
** db->mallocFailed flag is set.
**
** A VList is really just an array of integers.  To destroy a VList,
** simply pass it to sqlite3DbFree().
**
** The first integer is the number of integers allocated for the whole
** VList.  The second integer is the number of integers actually used.
** Each name/number pair is encoded by subsequent groups of 3 or more
** integers.
**
** Each name/number pair starts with two integers which are the numeric
** value for the pair and the size of the name/number pair, respectively.
** The text name overlays one or more following integers.  The text name
** is always zero-terminated.
**
** Conceptually:
**
**    struct VList {
**      int nAlloc;   // Number of allocated slots 
**      int nUsed;    // Number of used slots 
**      struct VListEntry {
**        int iValue;    // Value for this entry
**        int nSlot;     // Slots used by this entry
**        // ... variable name goes here
**      } a[0];
**    }
**
** During code generation, pointers to the variable names within the
** VList are taken.  When that happens, nAlloc is set to zero as an 
** indication that the VList may never again be enlarged, since the
** accompanying realloc() would invalidate the pointers.
*/
VList *sqlite3VListAdd(
  sqlite3 *db,           /* The database connection used for malloc() */
  VList *pIn,            /* The input VList.  Might be NULL */
  const char *zName,     /* Name of symbol to add */
  int nName,             /* Bytes of text in zName */
  int iVal               /* Value to associate with zName */
){
  int nInt;              /* number of sizeof(int) objects needed for zName */
  char *z;               /* Pointer to where zName will be stored */
  int i;                 /* Index in pIn[] where zName is stored */

  nInt = nName/4 + 3;
  assert( pIn==0 || pIn[0]>=3 );  /* Verify ok to add new elements */
  if( pIn==0 || pIn[1]+nInt > pIn[0] ){
    /* Enlarge the allocation */
    sqlite3_int64 nAlloc = (pIn ? 2*(sqlite3_int64)pIn[0] : 10) + nInt;
    VList *pOut = sqlite3DbRealloc(db, pIn, nAlloc*sizeof(int));
    if( pOut==0 ) return pIn;
    if( pIn==0 ) pOut[1] = 2;
    pIn = pOut;
    pIn[0] = nAlloc;
  }
  i = pIn[1];
  pIn[i] = iVal;
  pIn[i+1] = nInt;
  z = (char*)&pIn[i+2];
  pIn[1] = i+nInt;
  assert( pIn[1]<=pIn[0] );
  memcpy(z, zName, nName);
  z[nName] = 0;
  return pIn;
}

/*
** Return a pointer to the name of a variable in the given VList that
** has the value iVal.  Or return a NULL if there is no such variable in
** the list
*/
const char *sqlite3VListNumToName(VList *pIn, int iVal){
  int i, mx;
  if( pIn==0 ) return 0;
  mx = pIn[1];
  i = 2;
  do{
    if( pIn[i]==iVal ) return (char*)&pIn[i+2];
    i += pIn[i+1];
  }while( i<mx );
  return 0;
}

/*
** Return the number of the variable named zName, if it is in VList.
** or return 0 if there is no such variable.
*/
int sqlite3VListNameToNum(VList *pIn, const char *zName, int nName){
  int i, mx;
  if( pIn==0 ) return 0;
  mx = pIn[1];
  i = 2;
  do{
    const char *z = (const char*)&pIn[i+2];
    if( strncmp(z,zName,nName)==0 && z[nName]==0 ) return pIn[i];
    i += pIn[i+1];
  }while( i<mx );
  return 0;
}