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#ifndef SQLITE_OMIT_VIRTUALTABLE
/*
** Convert a double into a LogEst
** In other words, compute an approximation for 10*log2(x).
*/
SQLITE_PRIVATE 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 */

#if defined(SQLITE_ENABLE_STMT_SCANSTATUS) || \
    defined(SQLITE_ENABLE_STAT3_OR_STAT4) || \
    defined(SQLITE_EXPLAIN_ESTIMATED_ROWS)
/*
** Convert a LogEst into an integer.
**
** Note that this routine is only used when one or more of various
** non-standard compile-time options is enabled.
*/
SQLITE_PRIVATE u64 sqlite3LogEstToInt(LogEst x){
  u64 n;
  n = x%10;
  x /= 10;
  if( n>=5 ) n -= 2;
  else if( n>=1 ) n -= 1;
#if defined(SQLITE_ENABLE_STMT_SCANSTATUS) || \
    defined(SQLITE_EXPLAIN_ESTIMATED_ROWS)
  if( x>60 ) return (u64)LARGEST_INT64;
#else
  /* If only SQLITE_ENABLE_STAT3_OR_STAT4 is on, then the largest input
  ** possible to this routine is 310, resulting in a maximum x of 31 */
  assert( x<=60 );
#endif
  return x>=3 ? (n+8)<<(x-3) : (n+8)>>(3-x);
}
#endif /* defined SCANSTAT or STAT4 or ESTIMATED_ROWS */

/*
** 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.
*/
SQLITE_PRIVATE 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 */
    int nAlloc = (pIn ? pIn[0]*2 : 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
*/
SQLITE_PRIVATE const char *sqlite3VListNumToName(VList *pIn, int iVal){
  int i, mx;
  if( pIn==0 ) return 0;