[apple/libc.git] / stdlib.subproj / strtod.c

/*
 * Copyright (c) 1999 Apple Computer, Inc. All rights reserved.
 *
 * @APPLE_LICENSE_HEADER_START@
 * 
 * The contents of this file constitute Original Code as defined in and
 * are subject to the Apple Public Source License Version 1.1 (the
 * "License").  You may not use this file except in compliance with the
 * License.  Please obtain a copy of the License at
 * http://www.apple.com/publicsource and read it before using this file.
 * 
 * This Original Code and all software distributed under the License are
 * distributed on an "AS IS" basis, WITHOUT WARRANTY OF ANY KIND, EITHER
 * EXPRESS OR IMPLIED, AND APPLE HEREBY DISCLAIMS ALL SUCH WARRANTIES,
 * INCLUDING WITHOUT LIMITATION, ANY WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE OR NON-INFRINGEMENT.  Please see the
 * License for the specific language governing rights and limitations
 * under the License.
 * 
 * @APPLE_LICENSE_HEADER_END@
 */
/****************************************************************
 *
 * 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_LITTLE_ENDIAN for IEEE-arithmetic machines where the least
 *      significant byte has the lowest address.
 * #define IEEE_BIG_ENDIAN for IEEE-arithmetic machines where the most
 *      significant byte has the lowest address.
 * #define Long int on machines with 32-bit ints and 64-bit longs.
 * #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.
 * #define KR_headers for old-style C function headers.
 * #define Bad_float_h if your system lacks a float.h or if it does not
 *      define some or all of DBL_DIG, DBL_MAX_10_EXP, DBL_MAX_EXP,
 *      FLT_RADIX, FLT_ROUNDS, and DBL_MAX.
 * #define MALLOC your_malloc, where your_malloc(n) acts like malloc(n)
 *      if memory is available and otherwise does something you deem
 *      appropriate.  If MALLOC is undefined, malloc will be invoked
 *      directly -- and assumed always to succeed.
 */

#if defined(LIBC_SCCS) && !defined(lint)
static char *rcsid = "$OpenBSD: strtod.c,v 1.9 1997/03/25 17:07:30 rahnds Exp $";
#endif /* LIBC_SCCS and not lint */

#if defined(__m68k__) || defined(__sparc__) || defined(__i386__) || \
    defined(__mips__) || defined(__ns32k__) || defined(__alpha__) || \
    defined(__powerpc__) || defined(__m88k__) || defined(__ppc__)
#include <sys/types.h>
#if BYTE_ORDER == BIG_ENDIAN
#define IEEE_BIG_ENDIAN
#else
#define IEEE_LITTLE_ENDIAN
#endif
#endif

#ifdef __arm32__
/*
 * Although the CPU is little endian the FP has different
 * byte and word endianness. The byte order is still little endian
 * but the word order is big endian.
 */
#define IEEE_BIG_ENDIAN
#endif

#ifdef vax
#define VAX
#endif

#define Long    int32_t
#define ULong   u_int32_t

#ifdef DEBUG
#include "stdio.h"
#define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
#endif

#ifdef __cplusplus
#include "malloc.h"
#include "memory.h"
#else
#ifndef KR_headers
#include "stdlib.h"
#include "string.h"
#include "locale.h"
#else
#include "malloc.h"
#include "memory.h"
#endif
#endif

#ifdef MALLOC
#ifdef KR_headers
extern char *MALLOC();
#else
extern void *MALLOC(size_t);
#endif
#else
#define MALLOC malloc
#endif

#include "ctype.h"
#include "errno.h"

#ifdef Bad_float_h
#undef __STDC__
#ifdef IEEE_BIG_ENDIAN
#define IEEE_ARITHMETIC
#endif
#ifdef IEEE_LITTLE_ENDIAN
#define IEEE_ARITHMETIC
#endif

#ifdef IEEE_ARITHMETIC
#define DBL_DIG 15
#define DBL_MAX_10_EXP 308
#define DBL_MAX_EXP 1024
#define FLT_RADIX 2
#define FLT_ROUNDS 1
#define DBL_MAX 1.7976931348623157e+308
#endif

#ifdef IBM
#define DBL_DIG 16
#define DBL_MAX_10_EXP 75
#define DBL_MAX_EXP 63
#define FLT_RADIX 16
#define FLT_ROUNDS 0
#define DBL_MAX 7.2370055773322621e+75
#endif

#ifdef VAX
#define DBL_DIG 16
#define DBL_MAX_10_EXP 38
#define DBL_MAX_EXP 127
#define FLT_RADIX 2
#define FLT_ROUNDS 1
#define DBL_MAX 1.7014118346046923e+38
#endif

#ifndef LONG_MAX
#define LONG_MAX 2147483647
#endif
#else
#include "float.h"
#endif
#ifndef __MATH_H__
#include "math.h"
#endif

#ifdef __cplusplus
extern "C" {
#endif

#ifndef CONST
#ifdef KR_headers
#define CONST /* blank */
#else
#define CONST const
#endif
#endif

#ifdef Unsigned_Shifts
#define Sign_Extend(a,b) if (b < 0) a |= 0xffff0000;
#else
#define Sign_Extend(a,b) /*no-op*/
#endif

#if defined(IEEE_LITTLE_ENDIAN) + defined(IEEE_BIG_ENDIAN) + defined(VAX) + \
    defined(IBM) != 1
Exactly one of IEEE_LITTLE_ENDIAN IEEE_BIG_ENDIAN, VAX, or
IBM should be defined.
#endif

#ifdef IEEE_LITTLE_ENDIAN
#define word0(x) ((ULong *)&x)[1]
#define word1(x) ((ULong *)&x)[0]
#else
#define word0(x) ((ULong *)&x)[0]
#define word1(x) ((ULong *)&x)[1]
#endif

/* The following definition of Storeinc is appropriate for MIPS processors.
 * An alternative that might be better on some machines is
 * #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
 */
#if defined(IEEE_LITTLE_ENDIAN) + defined(VAX) + defined(__arm32__)
#define Storeinc(a,b,c) (((unsigned short *)a)[1] = (unsigned short)b, \
((unsigned short *)a)[0] = (unsigned short)c, a++)
#else
#define Storeinc(a,b,c) (((unsigned short *)a)[0] = (unsigned short)b, \
((unsigned short *)a)[1] = (unsigned short)c, a++)
#endif

/* #define P DBL_MANT_DIG */
/* Ten_pmax = floor(P*log(2)/log(5)) */
/* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
/* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
/* Int_max = floor(P*log(FLT_RADIX)/log(10) - 1) */

#if defined(IEEE_LITTLE_ENDIAN) + defined(IEEE_BIG_ENDIAN)
#define Exp_shift  20
#define Exp_shift1 20
#define Exp_msk1    0x100000
#define Exp_msk11   0x100000
#define Exp_mask  0x7ff00000
#define P 53
#define Bias 1023
#define IEEE_Arith
#define Emin (-1022)
#define Exp_1  0x3ff00000
#define Exp_11 0x3ff00000
#define Ebits 11
#define Frac_mask  0xfffff
#define Frac_mask1 0xfffff
#define Ten_pmax 22
#define Bletch 0x10
#define Bndry_mask  0xfffff
#define Bndry_mask1 0xfffff
#define LSB 1
#define Sign_bit 0x80000000
#define Log2P 1
#define Tiny0 0
#define Tiny1 1
#define Quick_max 14
#define Int_max 14
#define Infinite(x) (word0(x) == 0x7ff00000) /* sufficient test for here */
#else
#undef  Sudden_Underflow
#define Sudden_Underflow
#ifdef IBM
#define Exp_shift  24
#define Exp_shift1 24
#define Exp_msk1   0x1000000
#define Exp_msk11  0x1000000
#define Exp_mask  0x7f000000
#define P 14
#define Bias 65
#define Exp_1  0x41000000
#define Exp_11 0x41000000
#define Ebits 8 /* exponent has 7 bits, but 8 is the right value in b2d */
#define Frac_mask  0xffffff
#define Frac_mask1 0xffffff
#define Bletch 4
#define Ten_pmax 22
#define Bndry_mask  0xefffff
#define Bndry_mask1 0xffffff
#define LSB 1
#define Sign_bit 0x80000000
#define Log2P 4
#define Tiny0 0x100000
#define Tiny1 0
#define Quick_max 14
#define Int_max 15
#else /* VAX */
#define Exp_shift  23
#define Exp_shift1 7
#define Exp_msk1    0x80
#define Exp_msk11   0x800000
#define Exp_mask  0x7f80
#define P 56
#define Bias 129
#define Exp_1  0x40800000
#define Exp_11 0x4080
#define Ebits 8
#define Frac_mask  0x7fffff
#define Frac_mask1 0xffff007f
#define Ten_pmax 24
#define Bletch 2
#define Bndry_mask  0xffff007f
#define Bndry_mask1 0xffff007f
#define LSB 0x10000
#define Sign_bit 0x8000
#define Log2P 1
#define Tiny0 0x80
#define Tiny1 0
#define Quick_max 15
#define Int_max 15
#endif
#endif

#ifndef IEEE_Arith
#define ROUND_BIASED
#endif

#ifdef RND_PRODQUOT
#define rounded_product(a,b) a = rnd_prod(a, b)
#define rounded_quotient(a,b) a = rnd_quot(a, b)
#ifdef KR_headers
extern double rnd_prod(), rnd_quot();
#else
extern double rnd_prod(double, double), rnd_quot(double, double);
#endif
#else
#define rounded_product(a,b) a *= b
#define rounded_quotient(a,b) a /= b
#endif

#define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
#define Big1 0xffffffff

#ifndef Just_16
/* When Pack_32 is not defined, we store 16 bits per 32-bit Long.
 * This makes some inner loops simpler and sometimes saves work
 * during multiplications, but it often seems to make things slightly
 * slower.  Hence the default is now to store 32 bits per Long.
 */
#ifndef Pack_32
#define Pack_32
#endif
#endif

#define Kmax 15

#ifdef __cplusplus
extern "C" double strtod(const char *s00, char **se);
extern "C" char *__dtoa(double d, int mode, int ndigits,
                        int *decpt, int *sign, char **rve, char **resultp);
#endif

 struct
Bigint {
        struct Bigint *next;
        int k, maxwds, sign, wds;
        ULong x[1];
        };

 typedef struct Bigint Bigint;

 static Bigint *
Balloc
#ifdef KR_headers
        (k) int k;
#else
        (int k)
#endif
{
        int x;
        Bigint *rv;

        x = 1 << k;
        rv = (Bigint *)malloc(sizeof(Bigint) + (x-1)*sizeof(Long));
        rv->k = k;
        rv->maxwds = x;
        rv->sign = rv->wds = 0;
        return rv;
}

 static void
Bfree
#ifdef KR_headers
        (v) Bigint *v;
#else
        (Bigint *v)
#endif
{
        free(v);
}

#define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
y->wds*sizeof(Long) + 2*sizeof(int))

 static Bigint *
multadd
#ifdef KR_headers
        (b, m, a) Bigint *b; int m, a;
#else
        (Bigint *b, int m, int a)       /* multiply by m and add a */
#endif
{
        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(b->k+1);
                        Bcopy(b1, b);
                        Bfree(b);
                        b = b1;
                        }
                b->x[wds++] = a;
                b->wds = wds;
                }
        return b;
        }

 static Bigint *
s2b
#ifdef KR_headers
        (s, nd0, nd, y9) CONST char *s; int nd0, nd; ULong y9;
#else
        (CONST char *s, int nd0, int nd, ULong y9)
#endif
{
        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(k);
        b->x[0] = y9;
        b->wds = 1;
#else
        b = Balloc(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(b, 10, *s++ - '0');
                        while(++i < nd0);
                s++;
                }
        else
                s += 10;
        for(; i < nd; i++)
                b = multadd(b, 10, *s++ - '0');
        return b;
        }

 static int
hi0bits
#ifdef KR_headers
        (x) register ULong x;
#else
        (register ULong x)
#endif
{
        register 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;
        }

 static int
lo0bits
#ifdef KR_headers
        (y) ULong *y;
#else
        (ULong *y)
#endif
{
        register int k;
        register 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;
        }

 static Bigint *
i2b
#ifdef KR_headers
        (i) int i;
#else
        (int i)
#endif
{
        Bigint *b;

        b = Balloc(1);
        b->x[0] = i;
        b->wds = 1;
        return b;
        }

 static Bigint *
mult
#ifdef KR_headers
        (a, b) Bigint *a, *b;
#else
        (Bigint *a, Bigint *b)
#endif
{
        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(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) {
                        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) {
                        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;
        }

 static Bigint *p5s;

 static Bigint *
pow5mult
#ifdef KR_headers
        (b, k) Bigint *b; int k;
#else
        (Bigint *b, int k)
#endif
{
        Bigint *b1, *p5, *p51;
        int i;
        static int p05[3] = { 5, 25, 125 };

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

        if (!(k >>= 2))
                return b;
        if (!(p5 = p5s)) {
                /* first time */
                p5 = p5s = i2b(625);
                p5->next = 0;
                }
        for(;;) {
                if (k & 1) {
                        b1 = mult(b, p5);
                        Bfree(b);
                        b = b1;
                        }
                if (!(k >>= 1))
                        break;
                if (!(p51 = p5->next)) {
                        p51 = p5->next = mult(p5,p5);
                        p51->next = 0;
                        }
                p5 = p51;
                }
        return b;
        }

 static Bigint *
lshift
#ifdef KR_headers
        (b, k) Bigint *b; int k;
#else
        (Bigint *b, int k)
#endif
{
        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(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)
                        ++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(b);
        return b1;
        }

 static int
cmp
#ifdef KR_headers
        (a, b) Bigint *a, *b;
#else
        (Bigint *a, Bigint *b)
#endif
{
        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;
        }

 static Bigint *
diff
#ifdef KR_headers
        (a, b) Bigint *a, *b;
#else
        (Bigint *a, Bigint *b)
#endif
{
        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(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(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;
        }

 static double
ulp
#ifdef KR_headers
        (x) double x;
#else
        (double x)
#endif
{
        register Long L;
        double a;

        L = (word0(x) & Exp_mask) - (P-1)*Exp_msk1;
#ifndef Sudden_Underflow
        if (L > 0) {
#endif
#ifdef IBM
                L |= Exp_msk1 >> 4;
#endif
                word0(a) = L;
                word1(a) = 0;
#ifndef Sudden_Underflow
                }
        else {
                L = -L >> Exp_shift;
                if (L < Exp_shift) {
                        word0(a) = 0x80000 >> L;
                        word1(a) = 0;
                        }
                else {
                        word0(a) = 0;
                        L -= Exp_shift;
                        word1(a) = L >= 31 ? 1 : 1 << 31 - L;
                        }
                }
#endif
        return a;
        }

 static double
b2d
#ifdef KR_headers
        (a, e) Bigint *a; int *e;
#else
        (Bigint *a, int *e)
#endif
{
        ULong *xa, *xa0, w, y, z;
        int k;
        double 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;
                d1 = y << (32-Ebits) + k | w >> Ebits - k;
                goto ret_d;
                }
        z = xa > xa0 ? *--xa : 0;
        if (k -= Ebits) {
                d0 = Exp_1 | y << k | z >> 32 - k;
                y = xa > xa0 ? *--xa : 0;
                d1 = z << k | y >> 32 - k;
                }
        else {
                d0 = Exp_1 | y;
                d1 = z;
                }
#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;
        }

 static Bigint *
d2b
#ifdef KR_headers
        (d, e, bits) double d; int *e, *bits;
#else
        (double d, int *e, int *bits)
#endif
{
        Bigint *b;
        int de, i, k;
        ULong *x, y, z;
#ifdef VAX
        ULong d0, d1;
        d0 = word0(d) >> 16 | word0(d) << 16;
        d1 = word1(d) >> 16 | word1(d) << 16;
#else
#define d0 word0(d)
#define d1 word1(d)
#endif

#ifdef Pack_32
        b = Balloc(1);
#else
        b = Balloc(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))
                z |= Exp_msk1;
#endif
#ifdef Pack_32
        if (y = d1) {
                if (k = lo0bits(&y)) {
                        x[0] = y | z << 32 - k;
                        z >>= k;
                        }
                else
                        x[0] = y;
                i = b->wds = (x[1] = z) ? 2 : 1;
                }
        else {
#ifdef DEBUG
                if (!z)
                        Bug("Zero passed to d2b");
#endif
                k = lo0bits(&z);
                x[0] = z;
                i = b->wds = 1;
                k += 32;
                }
#else
        if (y = d1) {
                if (k = lo0bits(&y))
                        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 - (P-1) << 2) + k;
                *bits = 4*P + 8 - k - hi0bits(word0(d) & Frac_mask);
#else
                *e = de - Bias - (P-1) + k;
                *bits = P - k;
#endif
#ifndef Sudden_Underflow
                }
        else {
                *e = de - Bias - (P-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

 static double
ratio
#ifdef KR_headers
        (a, b) Bigint *a, *b;
#else
        (Bigint *a, Bigint *b)
#endif
{
        double da, db;
        int k, ka, kb;

        da = b2d(a, &ka);
        db = 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 *= 1 << k;
                }
        else {
                k = -k;
                word0(db) += (k >> 2)*Exp_msk1;
                if (k &= 3)
                        db *= 1 << k;
                }
#else
        if (k > 0)
                word0(da) += k*Exp_msk1;
        else {
                k = -k;
                word0(db) += k*Exp_msk1;
                }
#endif
        return da / db;
        }

static 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
#ifdef VAX
                , 1e23, 1e24
#endif
                };

#ifdef IEEE_Arith
static CONST double bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 };
static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128, 1e-256 };
#define n_bigtens 5
#else
#ifdef IBM
static CONST double bigtens[] = { 1e16, 1e32, 1e64 };
static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64 };
#define n_bigtens 3
#else
static CONST double bigtens[] = { 1e16, 1e32 };
static CONST double tinytens[] = { 1e-16, 1e-32 };
#define n_bigtens 2
#endif
#endif

 double
strtod
#ifdef KR_headers
        (s00, se) CONST char *s00; char **se;
#else
        (CONST char *s00, char **se)
#endif
{
        int bb2, bb5, bbe, bd2, bd5, bbbits, bs2, c, dsign,
                 e, e1, esign, i, j, k, nd, nd0, nf, nz, nz0, sign;
        CONST char *s, *s0, *s1;
        double aadj, aadj1, adj, rv, rv0;
        Long L;
        ULong y, z;
        Bigint *bb, *bb1, *bd, *bd0, *bs, *delta;

#ifndef KR_headers
        CONST char decimal_point = localeconv()->decimal_point[0];
#else
        CONST char decimal_point = '.';
#endif

        sign = nz0 = nz = 0;
        rv = 0.;


        for(s = s00; isspace((unsigned char) *s); s++)
                ;

        if (*s == '-') {
                sign = 1;
                s++;
        } else if (*s == '+') {
                s++;
        }

        if (*s == '\0') {
                s = s00;
                goto ret;
        }

        if (*s == '0') {
                nz0 = 1;
                while(*++s == '0') ;
                if (!*s)
                        goto ret;
                }
        s0 = s;
        y = z = 0;
        for(nd = nf = 0; (c = *s) >= '0' && c <= '9'; nd++, s++)
                if (nd < 9)
                        y = 10*y + c - '0';
                else if (nd < 16)
                        z = 10*z + c - '0';
        nd0 = nd;
        if (c == decimal_point) {
                c = *++s;
                if (!nd) {
                        for(; c == '0'; c = *++s)
                                nz++;
                        if (c > '0' && c <= '9') {
                                s0 = s;
                                nf += nz;
                                nz = 0;
                                goto have_dig;
                                }
                        goto dig_done;
                        }
                for(; c >= '0' && c <= '9'; c = *++s) {
 have_dig:
                        nz++;
                        if (c -= '0') {
                                nf += nz;
                                for(i = 1; i < nz; i++)
                                        if (nd++ < 9)
                                                y *= 10;
                                        else if (nd <= DBL_DIG + 1)
                                                z *= 10;
                                if (nd++ < 9)
                                        y = 10*y + c;
                                else if (nd <= DBL_DIG + 1)
                                        z = 10*z + c;
                                nz = 0;
                                }
                        }
                }
 dig_done:
        e = 0;
        if (c == 'e' || c == 'E') {
                if (!nd && !nz && !nz0) {
                        s = s00;
                        goto ret;
                        }
                s00 = s;
                esign = 0;
                switch(c = *++s) {
                        case '-':
                                esign = 1;
                        case '+':
                                c = *++s;
                        }
                if (c >= '0' && c <= '9') {
                        while(c == '0')
                                c = *++s;
                        if (c > '0' && c <= '9') {
                                L = c - '0';
                                s1 = s;
                                while((c = *++s) >= '0' && c <= '9')
                                        L = 10*L + c - '0';
                                if (s - s1 > 8 || L > 19999)
                                        /* Avoid confusion from exponents
                                         * so large that e might overflow.
                                         */
                                        e = 19999; /* safe for 16 bit ints */
                                else
                                        e = (int)L;
                                if (esign)
                                        e = -e;
                                }
                        else
                                e = 0;
                        }
                else
                        s = s00;
                }
        if (!nd) {
                if (!nz && !nz0)
                        s = s00;
                goto ret;
                }
        e1 = e -= nf;

        /* Now we have nd0 digits, starting at s0, followed by a
         * decimal point, followed by nd-nd0 digits.  The number we're
         * after is the integer represented by those digits times
         * 10**e */

        if (!nd0)
                nd0 = nd;
        k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1;
        rv = y;
        if (k > 9)
                rv = tens[k - 9] * rv + z;
        bd0 = 0;
        if (nd <= DBL_DIG
#ifndef RND_PRODQUOT
                && FLT_ROUNDS == 1
#endif
                        ) {
                if (!e)
                        goto ret;
                if (e > 0) {
                        if (e <= Ten_pmax) {
#ifdef VAX
                                goto vax_ovfl_check;
#else
                                /* rv = */ rounded_product(rv, tens[e]);
                                goto ret;
#endif
                                }
                        i = DBL_DIG - nd;
                        if (e <= Ten_pmax + i) {
                                /* A fancier test would sometimes let us do
                                 * this for larger i values.
                                 */
                                e -= i;
                                rv *= tens[i];
#ifdef VAX
                                /* VAX exponent range is so narrow we must
                                 * worry about overflow here...
                                 */
 vax_ovfl_check:
                                word0(rv) -= P*Exp_msk1;
                                /* rv = */ rounded_product(rv, tens[e]);
                                if ((word0(rv) & Exp_mask)
                                 > Exp_msk1*(DBL_MAX_EXP+Bias-1-P))
                                        goto ovfl;
                                word0(rv) += P*Exp_msk1;
#else
                                /* rv = */ rounded_product(rv, tens[e]);
#endif
                                goto ret;
                                }
                        }
#ifndef Inaccurate_Divide
                else if (e >= -Ten_pmax) {
                        /* rv = */ rounded_quotient(rv, tens[-e]);
                        goto ret;
                        }
#endif
                }
        e1 += nd - k;

        /* Get starting approximation = rv * 10**e1 */

        if (e1 > 0) {
                if (i = e1 & 15)
                        rv *= tens[i];
                if (e1 &= ~15) {
                        if (e1 > DBL_MAX_10_EXP) {
 ovfl:
                                errno = ERANGE;
#ifdef __STDC__
                                rv = HUGE_VAL;
#else
                                /* Can't trust HUGE_VAL */
#ifdef IEEE_Arith
                                word0(rv) = Exp_mask;
                                word1(rv) = 0;
#else
                                word0(rv) = Big0;
                                word1(rv) = Big1;
#endif
#endif
                                if (bd0)
                                        goto retfree;
                                goto ret;
                                }
                        if (e1 >>= 4) {
                                for(j = 0; e1 > 1; j++, e1 >>= 1)
                                        if (e1 & 1)
                                                rv *= bigtens[j];
                        /* The last multiplication could overflow. */
                                word0(rv) -= P*Exp_msk1;
                                rv *= bigtens[j];
                                if ((z = word0(rv) & Exp_mask)
                                 > Exp_msk1*(DBL_MAX_EXP+Bias-P))
                                        goto ovfl;
                                if (z > Exp_msk1*(DBL_MAX_EXP+Bias-1-P)) {
                                        /* set to largest number */
                                        /* (Can't trust DBL_MAX) */
                                        word0(rv) = Big0;
                                        word1(rv) = Big1;
                                        }
                                else
                                        word0(rv) += P*Exp_msk1;
                                }

                        }
                }
        else if (e1 < 0) {
                e1 = -e1;
                if (i = e1 & 15)
                        rv /= tens[i];
                if (e1 &= ~15) {
                        e1 >>= 4;
                        if (e1 >= 1 << n_bigtens)
                                goto undfl;
                        for(j = 0; e1 > 1; j++, e1 >>= 1)
                                if (e1 & 1)
                                        rv *= tinytens[j];
                        /* The last multiplication could underflow. */
                        rv0 = rv;
                        rv *= tinytens[j];
                        if (!rv) {
                                rv = 2.*rv0;
                                rv *= tinytens[j];
                                if (!rv) {
 undfl:
                                        rv = 0.;
                                        errno = ERANGE;
                                        if (bd0)
                                                goto retfree;
                                        goto ret;
                                        }
                                word0(rv) = Tiny0;
                                word1(rv) = Tiny1;
                                /* The refinement below will clean
                                 * this approximation up.
                                 */
                                }
                        }
                }

        /* Now the hard part -- adjusting rv to the correct value.*/

        /* Put digits into bd: true value = bd * 10^e */

        bd0 = s2b(s0, nd0, nd, y);

        for(;;) {
                bd = Balloc(bd0->k);
                Bcopy(bd, bd0);
                bb = d2b(rv, &bbe, &bbbits);    /* rv = bb * 2^bbe */
                bs = i2b(1);

                if (e >= 0) {
                        bb2 = bb5 = 0;
                        bd2 = bd5 = e;
                        }
                else {
                        bb2 = bb5 = -e;
                        bd2 = bd5 = 0;
                        }
                if (bbe >= 0)
                        bb2 += bbe;
                else
                        bd2 -= bbe;
                bs2 = bb2;
#ifdef Sudden_Underflow
#ifdef IBM
                j = 1 + 4*P - 3 - bbbits + ((bbe + bbbits - 1) & 3);
#else
                j = P + 1 - bbbits;
#endif
#else
                i = bbe + bbbits - 1;   /* logb(rv) */
                if (i < Emin)   /* denormal */
                        j = bbe + (P-Emin);
                else
                        j = P + 1 - bbbits;
#endif
                bb2 += j;
                bd2 += j;
                i = bb2 < bd2 ? bb2 : bd2;
                if (i > bs2)
                        i = bs2;
                if (i > 0) {
                        bb2 -= i;
                        bd2 -= i;
                        bs2 -= i;
                        }
                if (bb5 > 0) {
                        bs = pow5mult(bs, bb5);
                        bb1 = mult(bs, bb);
                        Bfree(bb);
                        bb = bb1;
                        }
                if (bb2 > 0)
                        bb = lshift(bb, bb2);
                if (bd5 > 0)
                        bd = pow5mult(bd, bd5);
                if (bd2 > 0)
                        bd = lshift(bd, bd2);
                if (bs2 > 0)
                        bs = lshift(bs, bs2);
                delta = diff(bb, bd);
                dsign = delta->sign;
                delta->sign = 0;
                i = cmp(delta, bs);
                if (i < 0) {
                        /* Error is less than half an ulp -- check for
                         * special case of mantissa a power of two.
                         */
                        if (dsign || word1(rv) || word0(rv) & Bndry_mask)
                                break;
                        delta = lshift(delta,Log2P);
                        if (cmp(delta, bs) > 0)
                                goto drop_down;
                        break;
                        }
                if (i == 0) {
                        /* exactly half-way between */
                        if (dsign) {
                                if ((word0(rv) & Bndry_mask1) == Bndry_mask1
                                 &&  word1(rv) == 0xffffffff) {
                                        /*boundary case -- increment exponent*/
                                        word0(rv) = (word0(rv) & Exp_mask)
                                                + Exp_msk1
#ifdef IBM
                                                | Exp_msk1 >> 4
#endif
                                                ;
                                        word1(rv) = 0;
                                        break;
                                        }
                                }
                        else if (!(word0(rv) & Bndry_mask) && !word1(rv)) {
 drop_down:
                                /* boundary case -- decrement exponent */
#ifdef Sudden_Underflow
                                L = word0(rv) & Exp_mask;
#ifdef IBM
                                if (L <  Exp_msk1)
#else
                                if (L <= Exp_msk1)
#endif
                                        goto undfl;
                                L -= Exp_msk1;
#else
                                L = (word0(rv) & Exp_mask) - Exp_msk1;
#endif
                                word0(rv) = L | Bndry_mask1;
                                word1(rv) = 0xffffffff;
#ifdef IBM
                                goto cont;
#else
                                break;
#endif
                                }
#ifndef ROUND_BIASED
                        if (!(word1(rv) & LSB))
                                break;
#endif
                        if (dsign)
                                rv += ulp(rv);
#ifndef ROUND_BIASED
                        else {
                                rv -= ulp(rv);
#ifndef Sudden_Underflow
                                if (!rv)
                                        goto undfl;
#endif
                                }
#endif
                        break;
                        }
                if ((aadj = ratio(delta, bs)) <= 2.) {
                        if (dsign)
                                aadj = aadj1 = 1.;
                        else if (word1(rv) || word0(rv) & Bndry_mask) {
#ifndef Sudden_Underflow
                                if (word1(rv) == Tiny1 && !word0(rv))
                                        goto undfl;
#endif
                                aadj = 1.;
                                aadj1 = -1.;
                                }
                        else {
                                /* special case -- power of FLT_RADIX to be */
                                /* rounded down... */

                                if (aadj < 2./FLT_RADIX)
                                        aadj = 1./FLT_RADIX;
                                else
                                        aadj *= 0.5;
                                aadj1 = -aadj;
                                }
                        }
                else {
                        aadj *= 0.5;
                        aadj1 = dsign ? aadj : -aadj;
#ifdef Check_FLT_ROUNDS
                        switch(FLT_ROUNDS) {
                                case 2: /* towards +infinity */
                                        aadj1 -= 0.5;
                                        break;
                                case 0: /* towards 0 */
                                case 3: /* towards -infinity */
                                        aadj1 += 0.5;
                                }
#else
                        if (FLT_ROUNDS == 0)
                                aadj1 += 0.5;
#endif
                        }
                y = word0(rv) & Exp_mask;

                /* Check for overflow */

                if (y == Exp_msk1*(DBL_MAX_EXP+Bias-1)) {
                        rv0 = rv;
                        word0(rv) -= P*Exp_msk1;
                        adj = aadj1 * ulp(rv);
                        rv += adj;
                        if ((word0(rv) & Exp_mask) >=
                                        Exp_msk1*(DBL_MAX_EXP+Bias-P)) {
                                if (word0(rv0) == Big0 && word1(rv0) == Big1)
                                        goto ovfl;
                                word0(rv) = Big0;
                                word1(rv) = Big1;
                                goto cont;
                                }
                        else
                                word0(rv) += P*Exp_msk1;
                        }
                else {
#ifdef Sudden_Underflow
                        if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
                                rv0 = rv;
                                word0(rv) += P*Exp_msk1;
                                adj = aadj1 * ulp(rv);
                                rv += adj;
#ifdef IBM
                                if ((word0(rv) & Exp_mask) <  P*Exp_msk1)
#else
                                if ((word0(rv) & Exp_mask) <= P*Exp_msk1)
#endif
                                        {
                                        if (word0(rv0) == Tiny0
                                         && word1(rv0) == Tiny1)
                                                goto undfl;
                                        word0(rv) = Tiny0;
                                        word1(rv) = Tiny1;
                                        goto cont;
                                        }
                                else
                                        word0(rv) -= P*Exp_msk1;
                                }
                        else {
                                adj = aadj1 * ulp(rv);
                                rv += adj;
                                }
#else
                        /* Compute adj so that the IEEE rounding rules will
                         * correctly round rv + adj in some half-way cases.
                         * If rv * ulp(rv) is denormalized (i.e.,
                         * y <= (P-1)*Exp_msk1), we must adjust aadj to avoid
                         * trouble from bits lost to denormalization;
                         * example: 1.2e-307 .
                         */
                        if (y <= (P-1)*Exp_msk1 && aadj >= 1.) {
                                aadj1 = (double)(int)(aadj + 0.5);
                                if (!dsign)
                                        aadj1 = -aadj1;
                                }
                        adj = aadj1 * ulp(rv);
                        rv += adj;
#endif
                        }
                z = word0(rv) & Exp_mask;
                if (y == z) {
                        /* Can we stop now? */
                        L = aadj;
                        aadj -= L;
                        /* The tolerances below are conservative. */
                        if (dsign || word1(rv) || word0(rv) & Bndry_mask) {
                                if (aadj < .4999999 || aadj > .5000001)
                                        break;
                                }
                        else if (aadj < .4999999/FLT_RADIX)
                                break;
                        }
 cont:
                Bfree(bb);
                Bfree(bd);
                Bfree(bs);
                Bfree(delta);
                }
 retfree:
        Bfree(bb);
        Bfree(bd);
        Bfree(bs);
        Bfree(bd0);
        Bfree(delta);
 ret:
        if (se)
                *se = (char *)s;
        return sign ? -rv : rv;
        }

 static int
quorem
#ifdef KR_headers
        (b, S) Bigint *b, *S;
#else
        (Bigint *b, Bigint *S)
#endif
{
        int n;
        Long borrow, y;
        ULong carry, q, ys;
        ULong *bx, *bxe, *sx, *sxe;
#ifdef Pack_32
        Long z;
        ULong si, zs;
#endif

        n = S->wds;
#ifdef DEBUG
        /*debug*/ if (b->wds > n)
        /*debug*/       Bug("oversize b in quorem");
#endif
        if (b->wds < n)
                return 0;
        sx = S->x;
        sxe = sx + --n;
        bx = b->x;
        bxe = bx + n;
        q = *bxe / (*sxe + 1);  /* ensure q <= true quotient */
#ifdef DEBUG
        /*debug*/ if (q > 9)
        /*debug*/       Bug("oversized quotient in quorem");
#endif
        if (q) {
                borrow = 0;
                carry = 0;
                do {
#ifdef Pack_32
                        si = *sx++;
                        ys = (si & 0xffff) * q + carry;
                        zs = (si >> 16) * q + (ys >> 16);
                        carry = zs >> 16;
                        y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
                        borrow = y >> 16;
                        Sign_Extend(borrow, y);
                        z = (*bx >> 16) - (zs & 0xffff) + borrow;
                        borrow = z >> 16;
                        Sign_Extend(borrow, z);
                        Storeinc(bx, z, y);
#else
                        ys = *sx++ * q + carry;
                        carry = ys >> 16;
                        y = *bx - (ys & 0xffff) + borrow;
                        borrow = y >> 16;
                        Sign_Extend(borrow, y);
                        *bx++ = y & 0xffff;
#endif
                        }
                        while(sx <= sxe);
                if (!*bxe) {
                        bx = b->x;
                        while(--bxe > bx && !*bxe)
                                --n;
                        b->wds = n;
                        }
                }
        if (cmp(b, S) >= 0) {
                q++;
                borrow = 0;
                carry = 0;
                bx = b->x;
                sx = S->x;
                do {
#ifdef Pack_32
                        si = *sx++;
                        ys = (si & 0xffff) + carry;
                        zs = (si >> 16) + (ys >> 16);
                        carry = zs >> 16;
                        y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
                        borrow = y >> 16;
                        Sign_Extend(borrow, y);
                        z = (*bx >> 16) - (zs & 0xffff) + borrow;
                        borrow = z >> 16;
                        Sign_Extend(borrow, z);
                        Storeinc(bx, z, y);
#else
                        ys = *sx++ + carry;
                        carry = ys >> 16;
                        y = *bx - (ys & 0xffff) + borrow;
                        borrow = y >> 16;
                        Sign_Extend(borrow, y);
                        *bx++ = y & 0xffff;
#endif
                        }
                        while(sx <= sxe);
                bx = b->x;
                bxe = bx + n;
                if (!*bxe) {
                        while(--bxe > bx && !*bxe)
                                --n;
                        b->wds = n;
                        }
                }
        return q;
        }

/* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
 *
 * Inspired by "How to Print Floating-Point Numbers Accurately" by
 * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 92-101].
 *
 * Modifications:
 *      1. Rather than iterating, we use a simple numeric overestimate
 *         to determine k = floor(log10(d)).  We scale relevant
 *         quantities using O(log2(k)) rather than O(k) multiplications.
 *      2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
 *         try to generate digits strictly left to right.  Instead, we
 *         compute with fewer bits and propagate the carry if necessary
 *         when rounding the final digit up.  This is often faster.
 *      3. Under the assumption that input will be rounded nearest,
 *         mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
 *         That is, we allow equality in stopping tests when the
 *         round-nearest rule will give the same floating-point value
 *         as would satisfaction of the stopping test with strict
 *         inequality.
 *      4. We remove common factors of powers of 2 from relevant
 *         quantities.
 *      5. When converting floating-point integers less than 1e16,
 *         we use floating-point arithmetic rather than resorting
 *         to multiple-precision integers.
 *      6. When asked to produce fewer than 15 digits, we first try
 *         to get by with floating-point arithmetic; we resort to
 *         multiple-precision integer arithmetic only if we cannot
 *         guarantee that the floating-point calculation has given
 *         the correctly rounded result.  For k requested digits and
 *         "uniformly" distributed input, the probability is
 *         something like 10^(k-15) that we must resort to the Long
 *         calculation.
 */

 char *
__dtoa
#ifdef KR_headers
        (d, mode, ndigits, decpt, sign, rve)
        double d; int mode, ndigits, *decpt, *sign; char **rve, char **resultp;
#else
        (double d, int mode, int ndigits, int *decpt, int *sign, char **rve, char **resultp)
#endif
{
 /*     Arguments ndigits, decpt, sign are similar to those
        of ecvt and fcvt; trailing zeros are suppressed from
        the returned string.  If not null, *rve is set to point
        to the end of the return value.  If d is +-Infinity or NaN,
        then *decpt is set to 9999.

        mode:
                0 ==> shortest string that yields d when read in
                        and rounded to nearest.
                1 ==> like 0, but with Steele & White stopping rule;
                        e.g. with IEEE P754 arithmetic , mode 0 gives
                        1e23 whereas mode 1 gives 9.999999999999999e22.
                2 ==> max(1,ndigits) significant digits.  This gives a
                        return value similar to that of ecvt, except
                        that trailing zeros are suppressed.
                3 ==> through ndigits past the decimal point.  This
                        gives a return value similar to that from fcvt,
                        except that trailing zeros are suppressed, and
                        ndigits can be negative.
                4-9 should give the same return values as 2-3, i.e.,
                        4 <= mode <= 9 ==> same return as mode
                        2 + (mode & 1).  These modes are mainly for
                        debugging; often they run slower but sometimes
                        faster than modes 2-3.
                4,5,8,9 ==> left-to-right digit generation.
                6-9 ==> don't try fast floating-point estimate
                        (if applicable).

                Values of mode other than 0-9 are treated as mode 0.

                Sufficient space is allocated to the return value
                to hold the suppressed trailing zeros.
        */

        int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
                j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
                spec_case, try_quick;
        Long L;
#ifndef Sudden_Underflow
        int denorm;
        ULong x;
#endif
        Bigint *b, *b1, *delta, *mlo, *mhi, *S;
        double d2, ds, eps;
        char *s, *s0;

        if (word0(d) & Sign_bit) {
                /* set sign for everything, including 0's and NaNs */
                *sign = 1;
                word0(d) &= ~Sign_bit;  /* clear sign bit */
                }
        else
                *sign = 0;

#if defined(IEEE_Arith) + defined(VAX)
#ifdef IEEE_Arith
        if ((word0(d) & Exp_mask) == Exp_mask)
#else
        if (word0(d)  == 0x8000)
#endif
                {
                /* Infinity or NaN */
                *decpt = 9999;
                s =
#ifdef IEEE_Arith
                        !word1(d) && !(word0(d) & 0xfffff) ? "Infinity" :
#endif
                                "NaN";
                if (rve)
                        *rve =
#ifdef IEEE_Arith
                                s[3] ? s + 8 :
#endif
                                                s + 3;
                return s;
                }
#endif
#ifdef IBM
        d += 0; /* normalize */
#endif
        if (!d) {
                *decpt = 1;
                s = "0";
                if (rve)
                        *rve = s + 1;
                return s;
                }

        b = d2b(d, &be, &bbits);
#ifdef Sudden_Underflow
        i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1));
#else
        if (i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1))) {
#endif
                d2 = d;
                word0(d2) &= Frac_mask1;
                word0(d2) |= Exp_11;
#ifdef IBM
                if (j = 11 - hi0bits(word0(d2) & Frac_mask))
                        d2 /= 1 << j;
#endif

                /* log(x)       ~=~ log(1.5) + (x-1.5)/1.5
                 * log10(x)      =  log(x) / log(10)
                 *              ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
                 * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
                 *
                 * This suggests computing an approximation k to log10(d) by
                 *
                 * k = (i - Bias)*0.301029995663981
                 *      + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
                 *
                 * We want k to be too large rather than too small.
                 * The error in the first-order Taylor series approximation
                 * is in our favor, so we just round up the constant enough
                 * to compensate for any error in the multiplication of
                 * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
                 * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
                 * adding 1e-13 to the constant term more than suffices.
                 * Hence we adjust the constant term to 0.1760912590558.
                 * (We could get a more accurate k by invoking log10,
                 *  but this is probably not worthwhile.)
                 */

                i -= Bias;
#ifdef IBM
                i <<= 2;
                i += j;
#endif
#ifndef Sudden_Underflow
                denorm = 0;
                }
        else {
                /* d is denormalized */

                i = bbits + be + (Bias + (P-1) - 1);
                x = i > 32  ? word0(d) << 64 - i | word1(d) >> i - 32
                            : word1(d) << 32 - i;
                d2 = x;
                word0(d2) -= 31*Exp_msk1; /* adjust exponent */
                i -= (Bias + (P-1) - 1) + 1;
                denorm = 1;
                }
#endif
        ds = (d2-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
        k = (int)ds;
        if (ds < 0. && ds != k)
                k--;    /* want k = floor(ds) */
        k_check = 1;
        if (k >= 0 && k <= Ten_pmax) {
                if (d < tens[k])
                        k--;
                k_check = 0;
                }
        j = bbits - i - 1;
        if (j >= 0) {
                b2 = 0;
                s2 = j;
                }
        else {
                b2 = -j;
                s2 = 0;
                }
        if (k >= 0) {
                b5 = 0;
                s5 = k;
                s2 += k;
                }
        else {
                b2 -= k;
                b5 = -k;
                s5 = 0;
                }
        if (mode < 0 || mode > 9)
                mode = 0;
        try_quick = 1;
        if (mode > 5) {
                mode -= 4;
                try_quick = 0;
                }
        leftright = 1;
        switch(mode) {
                case 0:
                case 1:
                        ilim = ilim1 = -1;
                        i = 18;
                        ndigits = 0;
                        break;
                case 2:
                        leftright = 0;
                        /* no break */
                case 4:
                        if (ndigits <= 0)
                                ndigits = 1;
                        ilim = ilim1 = i = ndigits;
                        break;
                case 3:
                        leftright = 0;
                        /* no break */
                case 5:
                        i = ndigits + k + 1;
                        ilim = i;
                        ilim1 = i - 1;
                        if (i <= 0)
                                i = 1;
                }
        *resultp = (char *) malloc(i + 1);
        s = s0 = *resultp;

        if (ilim >= 0 && ilim <= Quick_max && try_quick) {

                /* Try to get by with floating-point arithmetic. */

                i = 0;
                d2 = d;
                k0 = k;
                ilim0 = ilim;
                ieps = 2; /* conservative */
                if (k > 0) {
                        ds = tens[k&0xf];
                        j = k >> 4;
                        if (j & Bletch) {
                                /* prevent overflows */
                                j &= Bletch - 1;
                                d /= bigtens[n_bigtens-1];
                                ieps++;
                                }
                        for(; j; j >>= 1, i++)
                                if (j & 1) {
                                        ieps++;
                                        ds *= bigtens[i];
                                        }
                        d /= ds;
                        }
                else if (j1 = -k) {
                        d *= tens[j1 & 0xf];
                        for(j = j1 >> 4; j; j >>= 1, i++)
                                if (j & 1) {
                                        ieps++;
                                        d *= bigtens[i];
                                        }
                        }
                if (k_check && d < 1. && ilim > 0) {
                        if (ilim1 <= 0)
                                goto fast_failed;
                        ilim = ilim1;
                        k--;
                        d *= 10.;
                        ieps++;
                        }
                eps = ieps*d + 7.;
                word0(eps) -= (P-1)*Exp_msk1;
                if (ilim == 0) {
                        S = mhi = 0;
                        d -= 5.;
                        if (d > eps)
                                goto one_digit;
                        if (d < -eps)
                                goto no_digits;
                        goto fast_failed;
                        }
#ifndef No_leftright
                if (leftright) {
                        /* Use Steele & White method of only
                         * generating digits needed.
                         */
                        eps = 0.5/tens[ilim-1] - eps;
                        for(i = 0;;) {
                                L = d;
                                d -= L;
                                *s++ = '0' + (int)L;
                                if (d < eps)
                                        goto ret1;
                                if (1. - d < eps)
                                        goto bump_up;
                                if (++i >= ilim)
                                        break;
                                eps *= 10.;
                                d *= 10.;
                                }
                        }
                else {
#endif
                        /* Generate ilim digits, then fix them up. */
                        eps *= tens[ilim-1];
                        for(i = 1;; i++, d *= 10.) {
                                L = d;
                                d -= L;
                                *s++ = '0' + (int)L;
                                if (i == ilim) {
                                        if (d > 0.5 + eps)
                                                goto bump_up;
                                        else if (d < 0.5 - eps) {
                                                while(*--s == '0');
                                                s++;
                                                goto ret1;
                                                }
                                        break;
                                        }
                                }
#ifndef No_leftright
                        }
#endif
 fast_failed:
                s = s0;
                d = d2;
                k = k0;
                ilim = ilim0;
                }

        /* Do we have a "small" integer? */

        if (be >= 0 && k <= Int_max) {
                /* Yes. */
                ds = tens[k];
                if (ndigits < 0 && ilim <= 0) {
                        S = mhi = 0;
                        if (ilim < 0 || d <= 5*ds)
                                goto no_digits;
                        goto one_digit;
                        }
                for(i = 1;; i++) {
                        L = d / ds;
                        d -= L*ds;
#ifdef Check_FLT_ROUNDS
                        /* If FLT_ROUNDS == 2, L will usually be high by 1 */
                        if (d < 0) {
                                L--;
                                d += ds;
                                }
#endif
                        *s++ = '0' + (int)L;
                        if (i == ilim) {
                                d += d;
                                if (d > ds || d == ds && L & 1) {
 bump_up:
                                        while(*--s == '9')
                                                if (s == s0) {
                                                        k++;
                                                        *s = '0';
                                                        break;
                                                        }
                                        ++*s++;
                                        }
                                break;
                                }
                        if (!(d *= 10.))
                                break;
                        }
                goto ret1;
                }

        m2 = b2;
        m5 = b5;
        mhi = mlo = 0;
        if (leftright) {
                if (mode < 2) {
                        i =
#ifndef Sudden_Underflow
                                denorm ? be + (Bias + (P-1) - 1 + 1) :
#endif
#ifdef IBM
                                1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
#else
                                1 + P - bbits;
#endif
                        }
                else {
                        j = ilim - 1;
                        if (m5 >= j)
                                m5 -= j;
                        else {
                                s5 += j -= m5;
                                b5 += j;
                                m5 = 0;
                                }
                        if ((i = ilim) < 0) {
                                m2 -= i;
                                i = 0;
                                }
                        }
                b2 += i;
                s2 += i;
                mhi = i2b(1);
                }
        if (m2 > 0 && s2 > 0) {
                i = m2 < s2 ? m2 : s2;
                b2 -= i;
                m2 -= i;
                s2 -= i;
                }
        if (b5 > 0) {
                if (leftright) {
                        if (m5 > 0) {
                                mhi = pow5mult(mhi, m5);
                                b1 = mult(mhi, b);
                                Bfree(b);
                                b = b1;
                                }
                        if (j = b5 - m5)
                                b = pow5mult(b, j);
                        }
                else
                        b = pow5mult(b, b5);
                }
        S = i2b(1);
        if (s5 > 0)
                S = pow5mult(S, s5);

        /* Check for special case that d is a normalized power of 2. */

        if (mode < 2) {
                if (!word1(d) && !(word0(d) & Bndry_mask)
#ifndef Sudden_Underflow
                 && word0(d) & Exp_mask
#endif
                                ) {
                        /* The special case */
                        b2 += Log2P;
                        s2 += Log2P;
                        spec_case = 1;
                        }
                else
                        spec_case = 0;
                }

        /* Arrange for convenient computation of quotients:
         * shift left if necessary so divisor has 4 leading 0 bits.
         *
         * Perhaps we should just compute leading 28 bits of S once
         * and for all and pass them and a shift to quorem, so it
         * can do shifts and ors to compute the numerator for q.
         */
#ifdef Pack_32
        if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f)
                i = 32 - i;
#else
        if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf)
                i = 16 - i;
#endif
        if (i > 4) {
                i -= 4;
                b2 += i;
                m2 += i;
                s2 += i;
                }
        else if (i < 4) {
                i += 28;
                b2 += i;
                m2 += i;
                s2 += i;
                }
        if (b2 > 0)
                b = lshift(b, b2);
        if (s2 > 0)
                S = lshift(S, s2);
        if (k_check) {
                if (cmp(b,S) < 0) {
                        k--;
                        b = multadd(b, 10, 0);  /* we botched the k estimate */
                        if (leftright)
                                mhi = multadd(mhi, 10, 0);
                        ilim = ilim1;
                        }
                }
        if (ilim <= 0 && mode > 2) {
                if (ilim < 0 || cmp(b,S = multadd(S,5,0)) <= 0) {
                        /* no digits, fcvt style */
 no_digits:
                        k = -1 - ndigits;
                        goto ret;
                        }
 one_digit:
                *s++ = '1';
                k++;
                goto ret;
                }
        if (leftright) {
                if (m2 > 0)
                        mhi = lshift(mhi, m2);

                /* Compute mlo -- check for special case
                 * that d is a normalized power of 2.
                 */

                mlo = mhi;
                if (spec_case) {
                        mhi = Balloc(mhi->k);
                        Bcopy(mhi, mlo);
                        mhi = lshift(mhi, Log2P);
                        }

                for(i = 1;;i++) {
                        dig = quorem(b,S) + '0';
                        /* Do we yet have the shortest decimal string
                         * that will round to d?
                         */
                        j = cmp(b, mlo);
                        delta = diff(S, mhi);
                        j1 = delta->sign ? 1 : cmp(b, delta);
                        Bfree(delta);
#ifndef ROUND_BIASED
                        if (j1 == 0 && !mode && !(word1(d) & 1)) {
                                if (dig == '9')
                                        goto round_9_up;
                                if (j > 0)
                                        dig++;
                                *s++ = dig;
                                goto ret;
                                }
#endif
                        if (j < 0 || j == 0 && !mode
#ifndef ROUND_BIASED
                                                        && !(word1(d) & 1)
#endif
                                        ) {
                                if (j1 > 0) {
                                        b = lshift(b, 1);
                                        j1 = cmp(b, S);
                                        if ((j1 > 0 || j1 == 0 && dig & 1)
                                        && dig++ == '9')
                                                goto round_9_up;
                                        }
                                *s++ = dig;
                                goto ret;
                                }
                        if (j1 > 0) {
                                if (dig == '9') { /* possible if i == 1 */
 round_9_up:
                                        *s++ = '9';
                                        goto roundoff;
                                        }
                                *s++ = dig + 1;
                                goto ret;
                                }
                        *s++ = dig;
                        if (i == ilim)
                                break;
                        b = multadd(b, 10, 0);
                        if (mlo == mhi)
                                mlo = mhi = multadd(mhi, 10, 0);
                        else {
                                mlo = multadd(mlo, 10, 0);
                                mhi = multadd(mhi, 10, 0);
                                }
                        }
                }
        else
                for(i = 1;; i++) {
                        *s++ = dig = quorem(b,S) + '0';
                        if (i >= ilim)
                                break;
                        b = multadd(b, 10, 0);
                        }

        /* Round off last digit */

        b = lshift(b, 1);
        j = cmp(b, S);
        if (j > 0 || j == 0 && dig & 1) {
 roundoff:
                while(*--s == '9')
                        if (s == s0) {
                                k++;
                                *s++ = '1';
                                goto ret;
                                }
                ++*s++;
                }
        else {
                while(*--s == '0');
                s++;
                }
 ret:
        Bfree(S);
        if (mhi) {
                if (mlo && mlo != mhi)
                        Bfree(mlo);
                Bfree(mhi);
                }
 ret1:
        Bfree(b);
        if (s == s0) {                          /* don't return empty string */
                *s++ = '0';
                k = 0;
        }
        *s = 0;
        *decpt = k + 1;
        if (rve)
                *rve = s;
        return s0;
        }
#ifdef __cplusplus
}
#endif