mprec.c

Go to the documentation of this file.

/****************************************************************
 *
 * The author of this software is David M. Gay.
 *
 * Copyright (c) 1991 by AT&T.
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose without fee is hereby granted, provided that this entire notice
 * is included in all copies of any software which is or includes a copy
 * or modification of this software and in all copies of the supporting
 * documentation for such software.
 *
 * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
 * WARRANTY.  IN PARTICULAR, NEITHER THE AUTHOR NOR AT&T MAKES ANY
 * REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
 * OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
 *
 ***************************************************************/

/* Please send bug reports to
    David M. Gay
    AT&T Bell Laboratories, Room 2C-463
    600 Mountain Avenue
    Murray Hill, NJ 07974-2070
    U.S.A.
    dmg@research.att.com or research!dmg
 */

/* strtod for IEEE-, VAX-, and IBM-arithmetic machines.
 *
 * This strtod returns a nearest machine number to the input decimal
 * string (or sets errno to ERANGE).  With IEEE arithmetic, ties are
 * broken by the IEEE round-even rule.  Otherwise ties are broken by
 * biased rounding (add half and chop).
 *
 * Inspired loosely by William D. Clinger's paper "How to Read Floating
 * Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101].
 *
 * Modifications:
 *
 *  1. We only require IEEE, IBM, or VAX double-precision
 *      arithmetic (not IEEE double-extended).
 *  2. We get by with floating-point arithmetic in a case that
 *      Clinger missed -- when we're computing d * 10^n
 *      for a small integer d and the integer n is not too
 *      much larger than 22 (the maximum integer k for which
 *      we can represent 10^k exactly), we may be able to
 *      compute (d*10^k) * 10^(e-k) with just one roundoff.
 *  3. Rather than a bit-at-a-time adjustment of the binary
 *      result in the hard case, we use floating-point
 *      arithmetic to determine the adjustment to within
 *      one bit; only in really hard cases do we need to
 *      compute a second residual.
 *  4. Because of 3., we don't need a large table of powers of 10
 *      for ten-to-e (just some small tables, e.g. of 10^k
 *      for 0 <= k <= 22).
 */

/*
 * #define IEEE_8087 for IEEE-arithmetic machines where the least
 *  significant byte has the lowest address.
 * #define IEEE_MC68k for IEEE-arithmetic machines where the most
 *  significant byte has the lowest address.
 * #define Sudden_Underflow for IEEE-format machines without gradual
 *  underflow (i.e., that flush to zero on underflow).
 * #define IBM for IBM mainframe-style floating-point arithmetic.
 * #define VAX for VAX-style floating-point arithmetic.
 * #define Unsigned_Shifts if >> does treats its left operand as unsigned.
 * #define No_leftright to omit left-right logic in fast floating-point
 *  computation of dtoa.
 * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3.
 * #define RND_PRODQUOT to use rnd_prod and rnd_quot (assembly routines
 *  that use extended-precision instructions to compute rounded
 *  products and quotients) with IBM.
 * #define ROUND_BIASED for IEEE-format with biased rounding.
 * #define Inaccurate_Divide for IEEE-format with correctly rounded
 *  products but inaccurate quotients, e.g., for Intel i860.
 * #define Just_16 to store 16 bits per 32-bit long when doing high-precision
 *  integer arithmetic.  Whether this speeds things up or slows things
 *  down depends on the machine and the number being converted.
 */

/*#include <_ansi.h>*/
#include <assert.h>
#include <stdlib.h>
#include <string.h>
/* #include <reent.h> */
#include "mprec.h"

/* reent.c knows this value */
/* #define _Kmax 15 */

#define _reent _Jv_reent
#define _Bigint _Jv_Bigint

#define _REENT_CHECK_MP(x)
#define _REENT_MP_FREELIST(x) ((x)->_freelist)
#define _REENT_MP_P5S(x) ((x)->_p5s)

typedef unsigned long __ULong;
typedef long __Long;

static void *
mprec_calloc (void *ignore, size_t x1, size_t x2)
{
  char *result = (char *) malloc (x1 * x2);
  memset (result, 0, x1 * x2);
  return result;
}

_Bigint *
_DEFUN (Balloc, (ptr, k), struct _reent * ptr _AND int k)
{
  int x;
  _Bigint *rv;
  int new_k = k + 1;

  _REENT_CHECK_MP (ptr);
  if (_REENT_MP_FREELIST (ptr) == NULL)
    {
      /* Allocate a list of pointers to the mprec objects */
      _REENT_MP_FREELIST (ptr) = (struct _Bigint **) mprec_calloc (ptr, sizeof (struct _Bigint *), new_k);
      if (_REENT_MP_FREELIST (ptr) == NULL)
    {
      return NULL;
    }
      ptr->_max_k = new_k;
    }
  else if (new_k > ptr->_max_k)
    {
      struct _Bigint **new_list = (struct _Bigint **) realloc (ptr->_freelist,
                                   new_k * sizeof (struct _Bigint *));
      memset (&new_list[ptr->_max_k], 0, (new_k - ptr->_max_k) * sizeof (struct _Bigint *));
      ptr->_freelist = new_list;
      ptr->_max_k = new_k;

    }

  assert (k <= ptr->_max_k);

  if ((rv = _REENT_MP_FREELIST (ptr)[k]) != 0)
    {
      _REENT_MP_FREELIST (ptr)[k] = rv->_next;
    }
  else
    {
      x = 1 << k;
      /* Allocate an mprec Bigint and stick in in the freelist */
      rv = (_Bigint *) mprec_calloc (ptr, 1, sizeof (_Bigint) + (x - 1) * sizeof (rv->_x));
      if (rv == NULL)
    return NULL;
      rv->_k = k;
      rv->_maxwds = x;
    }
  rv->_sign = rv->_wds = 0;
  return rv;
}

void
_DEFUN (Bfree, (ptr, v), struct _reent *ptr _AND _Bigint * v)
{
  _REENT_CHECK_MP (ptr);
  if (v)
    {
      v->_next = _REENT_MP_FREELIST (ptr)[v->_k];
      _REENT_MP_FREELIST (ptr)[v->_k] = v;
    }
}

_Bigint *
_DEFUN (multadd, (ptr, b, m, a), struct _reent *ptr _AND _Bigint * b _AND int m _AND int a)
{
  int i, wds;
  __ULong *x, y;
#ifdef Pack_32
  __ULong xi, z;
#endif
  _Bigint *b1;

  wds = b->_wds;
  x = b->_x;
  i = 0;
  do
    {
#ifdef Pack_32
      xi = *x;
      y = (xi & 0xffff) * m + a;
      z = (xi >> 16) * m + (y >> 16);
      a = (int) (z >> 16);
      *x++ = (z << 16) + (y & 0xffff);
#else
      y = *x * m + a;
      a = (int) (y >> 16);
      *x++ = y & 0xffff;
#endif
    }
  while (++i < wds);
  if (a)
    {
      if (wds >= b->_maxwds)
    {
      b1 = Balloc (ptr, b->_k + 1);
      Bcopy (b1, b);
      Bfree (ptr, b);
      b = b1;
    }
      b->_x[wds++] = a;
      b->_wds = wds;
    }
  return b;
}

#if defined (ENABLE_UNUSED_FUNCTION)
_Bigint *
_DEFUN (s2b, (ptr, s, nd0, nd, y9), struct _reent * ptr _AND _CONST char *s _AND int nd0 _AND int nd _AND __ULong y9)
{
  _Bigint *b;
  int i, k;
  __Long x, y;

  x = (nd + 8) / 9;
  for (k = 0, y = 1; x > y; y <<= 1, k++);
#ifdef Pack_32
  b = Balloc (ptr, k);
  b->_x[0] = y9;
  b->_wds = 1;
#else
  b = Balloc (ptr, k + 1);
  b->_x[0] = y9 & 0xffff;
  b->_wds = (b->_x[1] = y9 >> 16) ? 2 : 1;
#endif

  i = 9;
  if (9 < nd0)
    {
      s += 9;
      do
    b = multadd (ptr, b, 10, *s++ - '0');
      while (++i < nd0);
      s++;
    }
  else
    s += 10;
  for (; i < nd; i++)
    b = multadd (ptr, b, 10, *s++ - '0');
  return b;
}

#endif /* ENABLE_UNUSED_FUNCTION */
int
_DEFUN (hi0bits, (x), __ULong x)
{
  int k = 0;

  if (!(x & 0xffff0000))
    {
      k = 16;
      x <<= 16;
    }
  if (!(x & 0xff000000))
    {
      k += 8;
      x <<= 8;
    }
  if (!(x & 0xf0000000))
    {
      k += 4;
      x <<= 4;
    }
  if (!(x & 0xc0000000))
    {
      k += 2;
      x <<= 2;
    }
  if (!(x & 0x80000000))
    {
      k++;
      if (!(x & 0x40000000))
    return 32;
    }
  return k;
}

int
_DEFUN (lo0bits, (y), __ULong * y)
{
  int k;
  __ULong x = *y;

  if (x & 7)
    {
      if (x & 1)
    return 0;
      if (x & 2)
    {
      *y = x >> 1;
      return 1;
    }
      *y = x >> 2;
      return 2;
    }
  k = 0;
  if (!(x & 0xffff))
    {
      k = 16;
      x >>= 16;
    }
  if (!(x & 0xff))
    {
      k += 8;
      x >>= 8;
    }
  if (!(x & 0xf))
    {
      k += 4;
      x >>= 4;
    }
  if (!(x & 0x3))
    {
      k += 2;
      x >>= 2;
    }
  if (!(x & 1))
    {
      k++;
      x >>= 1;
      if (!x & 1)
    return 32;
    }
  *y = x;
  return k;
}

_Bigint *
_DEFUN (i2b, (ptr, i), struct _reent * ptr _AND int i)
{
  _Bigint *b;

  b = Balloc (ptr, 1);
  b->_x[0] = i;
  b->_wds = 1;
  return b;
}

_Bigint *
_DEFUN (mult, (ptr, a, b), struct _reent * ptr _AND _Bigint * a _AND _Bigint * b)
{
  _Bigint *c;
  int k, wa, wb, wc;
  __ULong carry, y, z;
  __ULong *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
#ifdef Pack_32
  __ULong z2;
#endif

  if (a->_wds < b->_wds)
    {
      c = a;
      a = b;
      b = c;
    }
  k = a->_k;
  wa = a->_wds;
  wb = b->_wds;
  wc = wa + wb;
  if (wc > a->_maxwds)
    k++;
  c = Balloc (ptr, k);
  for (x = c->_x, xa = x + wc; x < xa; x++)
    *x = 0;
  xa = a->_x;
  xae = xa + wa;
  xb = b->_x;
  xbe = xb + wb;
  xc0 = c->_x;
#ifdef Pack_32
  for (; xb < xbe; xb++, xc0++)
    {
      if ((y = *xb & 0xffff) != 0)
    {
      x = xa;
      xc = xc0;
      carry = 0;
      do
        {
          z = (*x & 0xffff) * y + (*xc & 0xffff) + carry;
          carry = z >> 16;
          z2 = (*x++ >> 16) * y + (*xc >> 16) + carry;
          carry = z2 >> 16;
          Storeinc (xc, z2, z);
        }
      while (x < xae);
      *xc = carry;
    }
      if ((y = *xb >> 16) != 0)
    {
      x = xa;
      xc = xc0;
      carry = 0;
      z2 = *xc;
      do
        {
          z = (*x & 0xffff) * y + (*xc >> 16) + carry;
          carry = z >> 16;
          Storeinc (xc, z, z2);
          z2 = (*x++ >> 16) * y + (*xc & 0xffff) + carry;
          carry = z2 >> 16;
        }
      while (x < xae);
      *xc = z2;
    }
    }
#else
  for (; xb < xbe; xc0++)
    {
      if ((y = *xb++))
    {
      x = xa;
      xc = xc0;
      carry = 0;
      do
        {
          z = *x++ * y + *xc + carry;
          carry = z >> 16;
          *xc++ = z & 0xffff;
        }
      while (x < xae);
      *xc = carry;
    }
    }
#endif
  for (xc0 = c->_x, xc = xc0 + wc; wc > 0 && !*--xc; --wc);
  c->_wds = wc;
  return c;
}

_Bigint *
_DEFUN (pow5mult, (ptr, b, k), struct _reent * ptr _AND _Bigint * b _AND int k)
{
  _Bigint *b1, *p5, *p51;
  int i;
  static _CONST int p05[3] = { 5, 25, 125 };

  if ((i = k & 3) != 0)
    b = multadd (ptr, b, p05[i - 1], 0);

  if (!(k >>= 2))
    return b;
  _REENT_CHECK_MP (ptr);
  if (!(p5 = _REENT_MP_P5S (ptr)))
    {
      /* first time */
      p5 = _REENT_MP_P5S (ptr) = i2b (ptr, 625);
      p5->_next = 0;
    }
  for (;;)
    {
      if (k & 1)
    {
      b1 = mult (ptr, b, p5);
      Bfree (ptr, b);
      b = b1;
    }
      if (!(k >>= 1))
    break;
      if (!(p51 = p5->_next))
    {
      p51 = p5->_next = mult (ptr, p5, p5);
      p51->_next = 0;
    }
      p5 = p51;
    }
  return b;
}

_Bigint *
_DEFUN (lshift, (ptr, b, k), struct _reent * ptr _AND _Bigint * b _AND int k)
{
  int i, k1, n, n1;
  _Bigint *b1;
  __ULong *x, *x1, *xe, z;

#ifdef Pack_32
  n = k >> 5;
#else
  n = k >> 4;
#endif
  k1 = b->_k;
  n1 = n + b->_wds + 1;
  for (i = b->_maxwds; n1 > i; i <<= 1)
    k1++;
  b1 = Balloc (ptr, k1);
  x1 = b1->_x;
  for (i = 0; i < n; i++)
    *x1++ = 0;
  x = b->_x;
  xe = x + b->_wds;
#ifdef Pack_32
  if (k &= 0x1f)
    {
      k1 = 32 - k;
      z = 0;
      do
    {
      *x1++ = *x << k | z;
      z = *x++ >> k1;
    }
      while (x < xe);
      if ((*x1 = z) != 0)
    ++n1;
    }
#else
  if (k &= 0xf)
    {
      k1 = 16 - k;
      z = 0;
      do
    {
      *x1++ = (*x << k & 0xffff) | z;
      z = *x++ >> k1;
    }
      while (x < xe);
      if ((*x1 = z))
    ++n1;
    }
#endif
  else
    do
      *x1++ = *x++;
    while (x < xe);
  b1->_wds = n1 - 1;
  Bfree (ptr, b);
  return b1;
}

int
_DEFUN (cmp, (a, b), _Bigint * a _AND _Bigint * b)
{
  __ULong *xa, *xa0, *xb, *xb0;
  int i, j;

  i = a->_wds;
  j = b->_wds;
#ifdef DEBUG
  if (i > 1 && !a->_x[i - 1])
    Bug ("cmp called with a->_x[a->_wds-1] == 0");
  if (j > 1 && !b->_x[j - 1])
    Bug ("cmp called with b->_x[b->_wds-1] == 0");
#endif
  if (i -= j)
    return i;
  xa0 = a->_x;
  xa = xa0 + j;
  xb0 = b->_x;
  xb = xb0 + j;
  for (;;)
    {
      if (*--xa != *--xb)
    return *xa < *xb ? -1 : 1;
      if (xa <= xa0)
    break;
    }
  return 0;
}

_Bigint *
_DEFUN (diff, (ptr, a, b), struct _reent * ptr _AND _Bigint * a _AND _Bigint * b)
{
  _Bigint *c;
  int i, wa, wb;
  __Long borrow, y;     /* We need signed shifts here. */
  __ULong *xa, *xae, *xb, *xbe, *xc;
#ifdef Pack_32
  __Long z;
#endif

  i = cmp (a, b);
  if (!i)
    {
      c = Balloc (ptr, 0);
      c->_wds = 1;
      c->_x[0] = 0;
      return c;
    }
  if (i < 0)
    {
      c = a;
      a = b;
      b = c;
      i = 1;
    }
  else
    i = 0;
  c = Balloc (ptr, a->_k);
  c->_sign = i;
  wa = a->_wds;
  xa = a->_x;
  xae = xa + wa;
  wb = b->_wds;
  xb = b->_x;
  xbe = xb + wb;
  xc = c->_x;
  borrow = 0;
#ifdef Pack_32
  do
    {
      y = (*xa & 0xffff) - (*xb & 0xffff) + borrow;
      borrow = y >> 16;
      Sign_Extend (borrow, y);
      z = (*xa++ >> 16) - (*xb++ >> 16) + borrow;
      borrow = z >> 16;
      Sign_Extend (borrow, z);
      Storeinc (xc, z, y);
    }
  while (xb < xbe);
  while (xa < xae)
    {
      y = (*xa & 0xffff) + borrow;
      borrow = y >> 16;
      Sign_Extend (borrow, y);
      z = (*xa++ >> 16) + borrow;
      borrow = z >> 16;
      Sign_Extend (borrow, z);
      Storeinc (xc, z, y);
    }
#else
  do
    {
      y = *xa++ - *xb++ + borrow;
      borrow = y >> 16;
      Sign_Extend (borrow, y);
      *xc++ = y & 0xffff;
    }
  while (xb < xbe);
  while (xa < xae)
    {
      y = *xa++ + borrow;
      borrow = y >> 16;
      Sign_Extend (borrow, y);
      *xc++ = y & 0xffff;
    }
#endif
  while (!*--xc)
    wa--;
  c->_wds = wa;
  return c;
}

#if defined (ENABLE_UNUSED_FUNCTION)
double
_DEFUN (ulp, (_x), double _x)
{
  union double_union x, a;
  register int32_t L;

  x.d = _x;

  L = (word0 (x) & Exp_mask) - (PREC - 1) * Exp_msk1;
#ifndef Sudden_Underflow
  if (L > 0)
    {
#endif
#ifdef IBM
      L |= Exp_msk1 >> 4;
#endif
      word0 (a) = L;
#ifndef _DOUBLE_IS_32BITS
      word1 (a) = 0;
#endif

#ifndef Sudden_Underflow
    }
  else
    {
      L = -L >> Exp_shift;
      if (L < Exp_shift)
    {
      word0 (a) = 0x80000 >> L;
#ifndef _DOUBLE_IS_32BITS
      word1 (a) = 0;
#endif
    }
      else
    {
      word0 (a) = 0;
      L -= Exp_shift;
#ifndef _DOUBLE_IS_32BITS
      word1 (a) = L >= 31 ? 1 : 1 << (31 - L);
#endif
    }
    }
#endif
  return a.d;
}

double
_DEFUN (b2d, (a, e), _Bigint * a _AND int *e)
{
  __ULong *xa, *xa0, w, y, z;
  int k;
  union double_union d;
#ifdef VAX
  __ULong d0, d1;
#else
#define d0 word0(d)
#define d1 word1(d)
#endif

  xa0 = a->_x;
  xa = xa0 + a->_wds;
  y = *--xa;
#ifdef DEBUG
  if (!y)
    Bug ("zero y in b2d");
#endif
  k = hi0bits (y);
  *e = 32 - k;
#ifdef Pack_32
  if (k < Ebits)
    {
      d0 = Exp_1 | y >> (Ebits - k);
      w = xa > xa0 ? *--xa : 0;
#ifndef _DOUBLE_IS_32BITS
      d1 = y << ((32 - Ebits) + k) | w >> (Ebits - k);
#endif
      goto ret_d;
    }
  z = xa > xa0 ? *--xa : 0;
  if (k -= Ebits)
    {
      d0 = Exp_1 | y << k | z >> (32 - k);
      y = xa > xa0 ? *--xa : 0;
#ifndef _DOUBLE_IS_32BITS
      d1 = z << k | y >> (32 - k);
#endif
    }
  else
    {
      d0 = Exp_1 | y;
#ifndef _DOUBLE_IS_32BITS
      d1 = z;
#endif
    }
#else
  if (k < Ebits + 16)
    {
      z = xa > xa0 ? *--xa : 0;
      d0 = Exp_1 | y << (k - Ebits) | z >> (Ebits + 16 - k);
      w = xa > xa0 ? *--xa : 0;
      y = xa > xa0 ? *--xa : 0;
      d1 = z << (k + 16 - Ebits) | w << (k - Ebits) | y >> (16 + Ebits - k);
      goto ret_d;
    }
  z = xa > xa0 ? *--xa : 0;
  w = xa > xa0 ? *--xa : 0;
  k -= Ebits + 16;
  d0 = Exp_1 | y << (k + 16) | z << k | w >> (16 - k);
  y = xa > xa0 ? *--xa : 0;
  d1 = w << (k + 16) | y << k;
#endif
ret_d:
#ifdef VAX
  word0 (d) = d0 >> 16 | d0 << 16;
  word1 (d) = d1 >> 16 | d1 << 16;
#else
#undef d0
#undef d1
#endif
  return d.d;
}

#endif /* ENABLE_UNUSED_FUNCTION */
_Bigint *
_DEFUN (d2b, (ptr, _d, e, bits), struct _reent * ptr _AND double _d _AND int *e _AND int *bits)
{
  union double_union d;
  _Bigint *b;
  int de, i, k;
  __ULong *x, y, z;
#ifdef VAX
  __ULong d0, d1;
#endif
  d.d = _d;
#ifdef VAX
  d0 = word0 (d) >> 16 | word0 (d) << 16;
  d1 = word1 (d) >> 16 | word1 (d) << 16;
#else
#define d0 word0(d)
#define d1 word1(d)
  d.d = _d;
#endif

#ifdef Pack_32
  b = Balloc (ptr, 1);
#else
  b = Balloc (ptr, 2);
#endif
  x = b->_x;

  z = d0 & Frac_mask;
  d0 &= 0x7fffffff;     /* clear sign bit, which we ignore */
#ifdef Sudden_Underflow
  de = (int) (d0 >> Exp_shift);
#ifndef IBM
  z |= Exp_msk11;
#endif
#else
  if ((de = (int) (d0 >> Exp_shift)) != 0)
    z |= Exp_msk1;
#endif
#ifdef Pack_32
#ifndef _DOUBLE_IS_32BITS
  if (d1)
    {
      y = d1;
      k = lo0bits (&y);
      if (k)
    {
      x[0] = y | z << (32 - k);
      z >>= k;
    }
      else
    x[0] = y;
      i = b->_wds = (x[1] = z) ? 2 : 1;
    }
  else
#endif
    {
#ifdef DEBUG
      if (!z)
    Bug ("Zero passed to d2b");
#endif
      k = lo0bits (&z);
      x[0] = z;
      i = b->_wds = 1;
#ifndef _DOUBLE_IS_32BITS
      k += 32;
#endif
    }
#else
  if (d1)
    {
      y = d1;
      k = lo0bits (&y);
      if (k)
    if (k >= 16)
      {
        x[0] = y | (z << (32 - k) & 0xffff);
        x[1] = z >> (k - 16) & 0xffff;
        x[2] = z >> k;
        i = 2;
      }
    else
      {
        x[0] = y & 0xffff;
        x[1] = y >> 16 | (z << (16 - k) & 0xffff);
        x[2] = z >> k & 0xffff;
        x[3] = z >> (k + 16);
        i = 3;
      }
      else
    {
      x[0] = y & 0xffff;
      x[1] = y >> 16;
      x[2] = z & 0xffff;
      x[3] = z >> 16;
      i = 3;
    }
    }
  else
    {
#ifdef DEBUG
      if (!z)
    Bug ("Zero passed to d2b");
#endif
      k = lo0bits (&z);
      if (k >= 16)
    {
      x[0] = z;
      i = 0;
    }
      else
    {
      x[0] = z & 0xffff;
      x[1] = z >> 16;
      i = 1;
    }
      k += 32;
    }
  while (!x[i])
    --i;
  b->_wds = i + 1;
#endif
#ifndef Sudden_Underflow
  if (de)
    {
#endif
#ifdef IBM
      *e = (de - Bias - (PREC - 1) << 2) + k;
      *bits = 4 * PREC + 8 - k - hi0bits (word0 (d) & Frac_mask);
#else
      *e = de - Bias - (PREC - 1) + k;
      *bits = PREC - k;
#endif
#ifndef Sudden_Underflow
    }
  else
    {
      *e = de - Bias - (PREC - 1) + 1 + k;
#ifdef Pack_32
      *bits = 32 * i - hi0bits (x[i - 1]);
#else
      *bits = (i + 2) * 16 - hi0bits (x[i]);
#endif
    }
#endif
  return b;
}

#undef d0
#undef d1

#if defined (ENABLE_UNUSED_FUNCTION)
double
_DEFUN (ratio, (a, b), _Bigint * a _AND _Bigint * b)
{
  union double_union da, db;
  int k, ka, kb;

  da.d = b2d (a, &ka);
  db.d = b2d (b, &kb);
#ifdef Pack_32
  k = ka - kb + 32 * (a->_wds - b->_wds);
#else
  k = ka - kb + 16 * (a->_wds - b->_wds);
#endif
#ifdef IBM
  if (k > 0)
    {
      word0 (da) += (k >> 2) * Exp_msk1;
      if (k &= 3)
    da.d *= 1 << k;
    }
  else
    {
      k = -k;
      word0 (db) += (k >> 2) * Exp_msk1;
      if (k &= 3)
    db.d *= 1 << k;
    }
#else
  if (k > 0)
    word0 (da) += k * Exp_msk1;
  else
    {
      k = -k;
      word0 (db) += k * Exp_msk1;
    }
#endif
  return da.d / db.d;
}
#endif /* ENABLE_UNUSED_FUNCTION */

_CONST double tens[] = {
  1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
  1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
  1e20, 1e21, 1e22, 1e23, 1e24
};

#if !defined(_DOUBLE_IS_32BITS) && !defined(__v800)
_CONST double bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 };

_CONST double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128, 1e-256 };
#else
_CONST double bigtens[] = { 1e16, 1e32 };

_CONST double tinytens[] = { 1e-16, 1e-32 };
#endif

#if defined (ENABLE_UNUSED_FUNCTION)

double
_DEFUN (_mprec_log10, (dig), int dig)
{
  double v = 1.0;
  if (dig < 24)
    return tens[dig];
  while (dig > 0)
    {
      v *= 10;
      dig--;
    }
  return v;
}
#endif /* ENABLE_UNUSED_FUNCTION */