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dtoa.c

/****************************************************************
 *
 * The author of this software is David M. Gay.
 *
 * Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
 *
 * 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 LUCENT 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 (dmg at acm dot org,
 * with " at " changed at "@" and " dot " changed to ".").  */

/* On a machine with IEEE extended-precision registers, it is
 * necessary to specify double-precision (53-bit) rounding precision
 * before invoking strtod or dtoa.  If the machine uses (the equivalent
 * of) Intel 80x87 arithmetic, the call
 *    _control87(PC_53, MCW_PC);
 * does this with many compilers.  Whether this or another call is
 * appropriate depends on the compiler; for this to work, it may be
 * necessary to #include "float.h" or another system-dependent header
 * file.
 */

/* 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 Long int on machines with 32-bit ints and 64-bit longs.
 * #define IBM for IBM mainframe-style floating-point arithmetic.
 * #define VAX for VAX-style floating-point arithmetic (D_floating).
 * #define No_leftright to omit left-right logic in fast floating-point
 *    computation of dtoa.
 * #define Honor_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
 *    and strtod and dtoa should round accordingly.
 * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
 *    and Honor_FLT_ROUNDS is not #defined.
 * #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 NO_LONG_LONG on machines that do not have a "long long"
 *    integer type (of >= 64 bits).  On such machines, you can
 *    #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.  If long long is available and the name is
 *    something other than "long long", #define Llong to be the name,
 *    and if "unsigned Llong" does not work as an unsigned version of
 *    Llong, #define #ULLong to be the corresponding unsigned type.
 * #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.
 * #define Omit_Private_Memory to omit logic (added Jan. 1998) for making
 *    memory allocations from a private pool of memory when possible.
 *    When used, the private pool is PRIVATE_MEM bytes long:  2304 bytes,
 *    unless #defined to be a different length.  This default length
 *    suffices to get rid of MALLOC calls except for unusual cases,
 *    such as decimal-to-binary conversion of a very long string of
 *    digits.  The longest string dtoa can return is about 751 bytes
 *    long.  For conversions by strtod of strings of 800 digits and
 *    all dtoa conversions in single-threaded executions with 8-byte
 *    pointers, PRIVATE_MEM >= 7400 appears to suffice; with 4-byte
 *    pointers, PRIVATE_MEM >= 7112 appears adequate.
 * #define INFNAN_CHECK on IEEE systems to cause strtod to check for
 *    Infinity and NaN (case insensitively).  On some systems (e.g.,
 *    some HP systems), it may be necessary to #define NAN_WORD0
 *    appropriately -- to the most significant word of a quiet NaN.
 *    (On HP Series 700/800 machines, -DNAN_WORD0=0x7ff40000 works.)
 *    When INFNAN_CHECK is #defined and No_Hex_NaN is not #defined,
 *    strtod also accepts (case insensitively) strings of the form
 *    NaN(x), where x is a string of hexadecimal digits and spaces;
 *    if there is only one string of hexadecimal digits, it is taken
 *    for the 52 fraction bits of the resulting NaN; if there are two
 *    or more strings of hex digits, the first is for the high 20 bits,
 *    the second and subsequent for the low 32 bits, with intervening
 *    white space ignored; but if this results in none of the 52
 *    fraction bits being on (an IEEE Infinity symbol), then NAN_WORD0
 *    and NAN_WORD1 are used instead.
 * #define MULTIPLE_THREADS if the system offers preemptively scheduled
 *    multiple threads.  In this case, you must provide (or suitably
 *    #define) two locks, acquired by ACQUIRE_DTOA_LOCK(n) and freed
 *    by FREE_DTOA_LOCK(n) for n = 0 or 1.  (The second lock, accessed
 *    in pow5mult, ensures lazy evaluation of only one copy of high
 *    powers of 5; omitting this lock would introduce a small
 *    probability of wasting memory, but would otherwise be harmless.)
 *    You must also invoke freedtoa(s) to free the value s returned by
 *    dtoa.  You may do so whether or not MULTIPLE_THREADS is #defined.
 * #define NO_IEEE_Scale to disable new (Feb. 1997) logic in strtod that
 *    avoids underflows on inputs whose result does not underflow.
 *    If you #define NO_IEEE_Scale on a machine that uses IEEE-format
 *    floating-point numbers and flushes underflows to zero rather
 *    than implementing gradual underflow, then you must also #define
 *    Sudden_Underflow.
 * #define YES_ALIAS to permit aliasing certain double values with
 *    arrays of ULongs.  This leads to slightly better code with
 *    some compilers and was always used prior to 19990916, but it
 *    is not strictly legal and can cause trouble with aggressively
 *    optimizing compilers (e.g., gcc 2.95.1 under -O2).
 * #define USE_LOCALE to use the current locale's decimal_point value.
 * #define SET_INEXACT if IEEE arithmetic is being used and extra
 *    computation should be done to set the inexact flag when the
 *    result is inexact and avoid setting inexact when the result
 *    is exact.  In this case, dtoa.c must be compiled in
 *    an environment, perhaps provided by #include "dtoa.c" in a
 *    suitable wrapper, that defines two functions,
 *          int get_inexact(void);
 *          void clear_inexact(void);
 *    such that get_inexact() returns a nonzero value if the
 *    inexact bit is already set, and clear_inexact() sets the
 *    inexact bit to 0.  When SET_INEXACT is #defined, strtod
 *    also does extra computations to set the underflow and overflow
 *    flags when appropriate (i.e., when the result is tiny and
 *    inexact or when it is a numeric value rounded to +-infinity).
 * #define NO_ERRNO if strtod should not assign errno = ERANGE when
 *    the result overflows to +-Infinity or underflows to 0.
 */

#ifndef Long
#define Long long
#endif
#ifndef ULong
typedef unsigned Long ULong;
#endif

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

#include "stdlib.h"
#include "string.h"

#ifdef USE_LOCALE
#include "locale.h"
#endif

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

#ifndef Omit_Private_Memory
#ifndef PRIVATE_MEM
#define PRIVATE_MEM 2304
#endif
#define PRIVATE_mem ((PRIVATE_MEM+sizeof(double)-1)/sizeof(double))
static double private_mem[PRIVATE_mem], *pmem_next = private_mem;
#endif

#undef IEEE_Arith
#undef Avoid_Underflow
#ifdef IEEE_MC68k
#define IEEE_Arith
#endif
#ifdef IEEE_8087
#define IEEE_Arith
#endif

#include "errno.h"

#ifdef Bad_float_h

#ifdef IEEE_Arith
#define DBL_DIG 15
#define DBL_MAX_10_EXP 308
#define DBL_MAX_EXP 1024
#define FLT_RADIX 2
#endif /*IEEE_Arith*/

#ifdef IBM
#define DBL_DIG 16
#define DBL_MAX_10_EXP 75
#define DBL_MAX_EXP 63
#define FLT_RADIX 16
#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 DBL_MAX 1.7014118346046923e+38
#endif

#ifndef LONG_MAX
#define LONG_MAX 2147483647
#endif

#else /* ifndef Bad_float_h */
#include "float.h"
#endif /* Bad_float_h */

#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

#if defined(IEEE_8087) + defined(IEEE_MC68k) + defined(VAX) + defined(IBM) != 1
Exactly one of IEEE_8087, IEEE_MC68k, VAX, or IBM should be defined.
#endif

typedef union { double d; ULong L[2]; } U;

#ifdef YES_ALIAS
#define dval(x) x
#ifdef IEEE_8087
#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
#else
#ifdef IEEE_8087
#define word0(x) ((U*)&x)->L[1]
#define word1(x) ((U*)&x)->L[0]
#else
#define word0(x) ((U*)&x)->L[0]
#define word1(x) ((U*)&x)->L[1]
#endif
#define dval(x) ((U*)&x)->d
#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_8087) + defined(VAX)
#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) */

#ifdef IEEE_Arith
#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 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
#ifndef NO_IEEE_Scale
#define Avoid_Underflow
#ifdef Flush_Denorm     /* debugging option */
#undef Sudden_Underflow
#endif
#endif

#ifndef Flt_Rounds
#ifdef FLT_ROUNDS
#define Flt_Rounds FLT_ROUNDS
#else
#define Flt_Rounds 1
#endif
#endif /*Flt_Rounds*/

#ifdef Honor_FLT_ROUNDS
#define Rounding rounding
#undef Check_FLT_ROUNDS
#define Check_FLT_ROUNDS
#else
#define Rounding Flt_Rounds
#endif

#else /* ifndef IEEE_Arith */
#undef Check_FLT_ROUNDS
#undef Honor_FLT_ROUNDS
#undef SET_INEXACT
#undef  Sudden_Underflow
#define Sudden_Underflow
#ifdef IBM
#undef Flt_Rounds
#define Flt_Rounds 0
#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 */
#undef Flt_Rounds
#define Flt_Rounds 1
#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 /* IBM, VAX */
#endif /* IEEE_Arith */

#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 Pack_32
#define Pack_32
#endif

#ifdef KR_headers
#define FFFFFFFF ((((unsigned long)0xffff)<<16)|(unsigned long)0xffff)
#else
#define FFFFFFFF 0xffffffffUL
#endif

#ifdef NO_LONG_LONG
#undef ULLong
#ifdef Just_16
#undef Pack_32
/* 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.
 */
#endif
#else /* long long available */
#ifndef Llong
#define Llong long long
#endif
#ifndef ULLong
#define ULLong unsigned Llong
#endif
#endif /* NO_LONG_LONG */

#ifndef MULTIPLE_THREADS
#define ACQUIRE_DTOA_LOCK(n)  /*nothing*/
#define FREE_DTOA_LOCK(n)     /*nothing*/
#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);
#endif

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

 typedef struct Bigint Bigint;

 static Bigint *freelist[Kmax+1];

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

      ACQUIRE_DTOA_LOCK(0);
        /* The k > Kmax case does not need ACQUIRE_DTOA_LOCK(0). */
        /* but this case seems very unlikely. */
      if (k <= Kmax && (rv = freelist[k])) {
            freelist[k] = rv->next;
            }
      else {
            x = 1 << k;
#ifdef Omit_Private_Memory
            rv = (Bigint *)MALLOC(sizeof(Bigint) + (x-1)*sizeof(ULong));
#else
            len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
                  /sizeof(double);
            if (k <= Kmax && pmem_next - private_mem + len <= PRIVATE_mem) {
                  rv = (Bigint*)pmem_next;
                  pmem_next += len;
                  }
            else
                  rv = (Bigint*)MALLOC(len*sizeof(double));
#endif
            rv->k = k;
            rv->maxwds = x;
            }
      FREE_DTOA_LOCK(0);
      rv->sign = rv->wds = 0;
      return rv;
      }

 static void
Bfree
#ifdef KR_headers
      (v) Bigint *v;
#else
      (Bigint *v)
#endif
{
      if (v) {
                if (v->k > Kmax)
                        free((void*)v);
                else {
                  ACQUIRE_DTOA_LOCK(0);
                  v->next = freelist[v->k];
                  freelist[v->k] = v;
                  FREE_DTOA_LOCK(0);
                        }
            }
      }

#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;
#ifdef ULLong
      ULong *x;
      ULLong carry, y;
#else
      ULong carry, *x, y;
#ifdef Pack_32
      ULong xi, z;
#endif
#endif
      Bigint *b1;

      wds = b->wds;
      x = b->x;
      i = 0;
      carry = a;
      do {
#ifdef ULLong
            y = *x * (ULLong)m + carry;
            carry = y >> 32;
            *x++ = y & FFFFFFFF;
#else
#ifdef Pack_32
            xi = *x;
            y = (xi & 0xffff) * m + carry;
            z = (xi >> 16) * m + (y >> 16);
            carry = z >> 16;
            *x++ = (z << 16) + (y & 0xffff);
#else
            y = *x * m + carry;
            carry = y >> 16;
            *x++ = y & 0xffff;
#endif
#endif
            }
            while(++i < wds);
      if (carry) {
            if (wds >= b->maxwds) {
                  b1 = Balloc(b->k+1);
                  Bcopy(b1, b);
                  Bfree(b);
                  b = b1;
                  }
            b->x[wds++] = carry;
            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)
                  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 *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
      ULong y;
#ifdef ULLong
      ULLong carry, z;
#else
      ULong carry, z;
#ifdef Pack_32
      ULong z2;
#endif
#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 ULLong
      for(; xb < xbe; xc0++) {
            if ((y = *xb++)) {
                  x = xa;
                  xc = xc0;
                  carry = 0;
                  do {
                        z = *x++ * (ULLong)y + *xc + carry;
                        carry = z >> 32;
                        *xc++ = z & FFFFFFFF;
                        }
                        while(x < xae);
                  *xc = carry;
                  }
            }
#else
#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
#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 */
#ifdef MULTIPLE_THREADS
            ACQUIRE_DTOA_LOCK(1);
            if (!(p5 = p5s)) {
                  p5 = p5s = i2b(625);
                  p5->next = 0;
                  }
            FREE_DTOA_LOCK(1);
#else
            p5 = p5s = i2b(625);
            p5->next = 0;
#endif
            }
      for(;;) {
            if (k & 1) {
                  b1 = mult(b, p5);
                  Bfree(b);
                  b = b1;
                  }
            if (!(k >>= 1))
                  break;
            if (!(p51 = p5->next)) {
#ifdef MULTIPLE_THREADS
                  ACQUIRE_DTOA_LOCK(1);
                  if (!(p51 = p5->next)) {
                        p51 = p5->next = mult(p5,p5);
                        p51->next = 0;
                        }
                  FREE_DTOA_LOCK(1);
#else
                  p51 = p5->next = mult(p5,p5);
                  p51->next = 0;
#endif
                  }
            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;
      ULong *xa, *xae, *xb, *xbe, *xc;
#ifdef ULLong
      ULLong borrow, y;
#else
      ULong borrow, y;
#ifdef Pack_32
      ULong z;
#endif
#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 ULLong
      do {
            y = (ULLong)*xa++ - *xb++ - borrow;
            borrow = y >> 32 & (ULong)1;
            *xc++ = y & FFFFFFFF;
            }
            while(xb < xbe);
      while(xa < xae) {
            y = *xa++ - borrow;
            borrow = y >> 32 & (ULong)1;
            *xc++ = y & FFFFFFFF;
            }
#else
#ifdef Pack_32
      do {
            y = (*xa & 0xffff) - (*xb & 0xffff) - borrow;
            borrow = (y & 0x10000) >> 16;
            z = (*xa++ >> 16) - (*xb++ >> 16) - borrow;
            borrow = (z & 0x10000) >> 16;
            Storeinc(xc, z, y);
            }
            while(xb < xbe);
      while(xa < xae) {
            y = (*xa & 0xffff) - borrow;
            borrow = (y & 0x10000) >> 16;
            z = (*xa++ >> 16) - borrow;
            borrow = (z & 0x10000) >> 16;
            Storeinc(xc, z, y);
            }
#else
      do {
            y = *xa++ - *xb++ - borrow;
            borrow = (y & 0x10000) >> 16;
            *xc++ = y & 0xffff;
            }
            while(xb < xbe);
      while(xa < xae) {
            y = *xa++ - borrow;
            borrow = (y & 0x10000) >> 16;
            *xc++ = y & 0xffff;
            }
#endif
#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 Avoid_Underflow
#ifndef Sudden_Underflow
      if (L > 0) {
#endif
#endif
#ifdef IBM
            L |= Exp_msk1 >> 4;
#endif
            word0(a) = L;
            word1(a) = 0;
#ifndef Avoid_Underflow
#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
#endif
      return dval(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 dval(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, k;
      ULong *x, y, z;
#ifndef Sudden_Underflow
      int i;
#endif
#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;
#ifndef Sudden_Underflow
            i =
#endif
                b->wds = (x[1] = z) ? 2 : 1;
            }
      else {
               /* This assertion fails for "1e-500" and other very 
                * small numbers.  It provides the right result (0) 
                * though. This assert has also been removed from KJS's
                * version of dtoa.c.
                *
                * #ifdef DEBUG
                *     if (!z) Bug("zero z in b2d");
                * #endif
                */
            k = lo0bits(&z);
            x[0] = z;
#ifndef Sudden_Underflow
            i =
#endif
                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;

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

 static CONST double
#ifdef IEEE_Arith
bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 };
static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128,
#ifdef Avoid_Underflow
            9007199254740992.*9007199254740992.e-256
            /* = 2^106 * 1e-53 */
#else
            1e-256
#endif
            };
/* The factor of 2^53 in tinytens[4] helps us avoid setting the underflow */
/* flag unnecessarily.  It leads to a song and dance at the end of strtod. */
#define Scale_Bit 0x10
#define n_bigtens 5
#else
#ifdef IBM
bigtens[] = { 1e16, 1e32, 1e64 };
static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64 };
#define n_bigtens 3
#else
bigtens[] = { 1e16, 1e32 };
static CONST double tinytens[] = { 1e-16, 1e-32 };
#define n_bigtens 2
#endif
#endif

#ifndef IEEE_Arith
#undef INFNAN_CHECK
#endif

#ifdef INFNAN_CHECK

#ifndef NAN_WORD0
#define NAN_WORD0 0x7ff80000
#endif

#ifndef NAN_WORD1
#define NAN_WORD1 0
#endif

 static int
match
#ifdef KR_headers
      (sp, t) char **sp, *t;
#else
      (CONST char **sp, char *t)
#endif
{
      int c, d;
      CONST char *s = *sp;

      while(d = *t++) {
            if ((c = *++s) >= 'A' && c <= 'Z')
                  c += 'a' - 'A';
            if (c != d)
                  return 0;
            }
      *sp = s + 1;
      return 1;
      }

#ifndef No_Hex_NaN
 static void
hexnan
#ifdef KR_headers
      (rvp, sp) double *rvp; CONST char **sp;
#else
      (double *rvp, CONST char **sp)
#endif
{
      ULong c, x[2];
      CONST char *s;
      int havedig, udx0, xshift;

      x[0] = x[1] = 0;
      havedig = xshift = 0;
      udx0 = 1;
      s = *sp;
      while(c = *(CONST unsigned char*)++s) {
            if (c >= '0' && c <= '9')
                  c -= '0';
            else if (c >= 'a' && c <= 'f')
                  c += 10 - 'a';
            else if (c >= 'A' && c <= 'F')
                  c += 10 - 'A';
            else if (c <= ' ') {
                  if (udx0 && havedig) {
                        udx0 = 0;
                        xshift = 1;
                        }
                  continue;
                  }
            else if (/*(*/ c == ')' && havedig) {
                  *sp = s + 1;
                  break;
                  }
            else
                  return;     /* invalid form: don't change *sp */
            havedig = 1;
            if (xshift) {
                  xshift = 0;
                  x[0] = x[1];
                  x[1] = 0;
                  }
            if (udx0)
                  x[0] = (x[0] << 4) | (x[1] >> 28);
            x[1] = (x[1] << 4) | c;
            }
      if ((x[0] &= 0xfffff) || x[1]) {
            word0(*rvp) = Exp_mask | x[0];
            word1(*rvp) = x[1];
            }
      }
#endif /*No_Hex_NaN*/
#endif /* INFNAN_CHECK */

 double
strtod
#ifdef KR_headers
      (s00, se) CONST char *s00; char **se;
#else
      (CONST char *s00, char **se)
#endif
{
#ifdef Avoid_Underflow
      int scale;
#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 = NULL, *bb1, *bd = NULL, *bd0, *bs = NULL, *delta = NULL;
#ifdef SET_INEXACT
      int inexact, oldinexact;
#endif
#ifdef Honor_FLT_ROUNDS
      int rounding;
#endif
#ifdef USE_LOCALE
      CONST char *s2;
#endif

      sign = nz0 = nz = 0;
      dval(rv) = 0.;
      for(s = s00;;s++) switch(*s) {
            case '-':
                  sign = 1;
                  /* no break */
            case '+':
                  if (*++s)
                        goto break2;
                  /* no break */
            case 0:
                  goto ret0;
            case '\t':
            case '\n':
            case '\v':
            case '\f':
            case '\r':
            case ' ':
                  continue;
            default:
                  goto break2;
            }
 break2:
      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;
#ifdef USE_LOCALE
      s1 = localeconv()->decimal_point;
      if (c == *s1) {
            c = '.';
            if (*++s1) {
                  s2 = s;
                  for(;;) {
                        if (*++s2 != *s1) {
                              c = 0;
                              break;
                              }
                        if (!*++s1) {
                              s = s2;
                              break;
                              }
                        }
                  }
            }
#endif
      if (c == '.') {
            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) {
                  goto ret0;
                  }
            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) {
#ifdef INFNAN_CHECK
                  /* Check for Nan and Infinity */
                  switch(c) {
                    case 'i':
                    case 'I':
                        if (match(&s,"nf")) {
                              --s;
                              if (!match(&s,"inity"))
                                    ++s;
                              word0(rv) = 0x7ff00000;
                              word1(rv) = 0;
                              goto ret;
                              }
                        break;
                    case 'n':
                    case 'N':
                        if (match(&s, "an")) {
                              word0(rv) = NAN_WORD0;
                              word1(rv) = NAN_WORD1;
#ifndef No_Hex_NaN
                              if (*s == '(') /*)*/
                                    hexnan(&rv, &s);
#endif
                              goto ret;
                              }
                    }
#endif /* INFNAN_CHECK */
 ret0:
                  s = s00;
                  sign = 0;
                  }
            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;
      dval(rv) = y;
      if (k > 9) {
#ifdef SET_INEXACT
            if (k > DBL_DIG)
                  oldinexact = get_inexact();
#endif
            dval(rv) = tens[k - 9] * dval(rv) + z;
            }
      bd0 = 0;
      if (nd <= DBL_DIG
#ifndef RND_PRODQUOT
#ifndef Honor_FLT_ROUNDS
            && Flt_Rounds == 1
#endif
#endif
                  ) {
            if (!e)
                  goto ret;
            if (e > 0) {
                  if (e <= Ten_pmax) {
#ifdef VAX
                        goto vax_ovfl_check;
#else
#ifdef Honor_FLT_ROUNDS
                        /* round correctly FLT_ROUNDS = 2 or 3 */
                        if (sign) {
                              rv = -rv;
                              sign = 0;
                              }
#endif
                        /* rv = */ rounded_product(dval(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.
                         */
#ifdef Honor_FLT_ROUNDS
                        /* round correctly FLT_ROUNDS = 2 or 3 */
                        if (sign) {
                              rv = -rv;
                              sign = 0;
                              }
#endif
                        e -= i;
                        dval(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(dval(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(dval(rv), tens[e]);
#endif
                        goto ret;
                        }
                  }
#ifndef Inaccurate_Divide
            else if (e >= -Ten_pmax) {
#ifdef Honor_FLT_ROUNDS
                  /* round correctly FLT_ROUNDS = 2 or 3 */
                  if (sign) {
                        rv = -rv;
                        sign = 0;
                        }
#endif
                  /* rv = */ rounded_quotient(dval(rv), tens[-e]);
                  goto ret;
                  }
#endif
            }
      e1 += nd - k;

#ifdef IEEE_Arith
#ifdef SET_INEXACT
      inexact = 1;
      if (k <= DBL_DIG)
            oldinexact = get_inexact();
#endif
#ifdef Avoid_Underflow
      scale = 0;
#endif
#ifdef Honor_FLT_ROUNDS
      if ((rounding = Flt_Rounds) >= 2) {
            if (sign)
                  rounding = rounding == 2 ? 0 : 2;
            else
                  if (rounding != 2)
                        rounding = 0;
            }
#endif
#endif /*IEEE_Arith*/

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

      if (e1 > 0) {
            if ((i = e1 & 15))
                  dval(rv) *= tens[i];
            if (e1 &= ~15) {
                  if (e1 > DBL_MAX_10_EXP) {
 ovfl:
#ifndef NO_ERRNO
                        errno = ERANGE;
#endif
                        /* Can't trust HUGE_VAL */
#ifdef IEEE_Arith
#ifdef Honor_FLT_ROUNDS
                        switch(rounding) {
                          case 0: /* toward 0 */
                          case 3: /* toward -infinity */
                              word0(rv) = Big0;
                              word1(rv) = Big1;
                              break;
                          default:
                              word0(rv) = Exp_mask;
                              word1(rv) = 0;
                          }
#else /*Honor_FLT_ROUNDS*/
                        word0(rv) = Exp_mask;
                        word1(rv) = 0;
#endif /*Honor_FLT_ROUNDS*/
#ifdef SET_INEXACT
                        /* set overflow bit */
                        dval(rv0) = 1e300;
                        dval(rv0) *= dval(rv0);
#endif
#else /*IEEE_Arith*/
                        word0(rv) = Big0;
                        word1(rv) = Big1;
#endif /*IEEE_Arith*/
                        if (bd0)
                              goto retfree;
                        goto ret;
                        }
                  e1 >>= 4;
                  for(j = 0; e1 > 1; j++, e1 >>= 1)
                        if (e1 & 1)
                              dval(rv) *= bigtens[j];
            /* The last multiplication could overflow. */
                  word0(rv) -= P*Exp_msk1;
                  dval(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))
                  dval(rv) /= tens[i];
            if (e1 >>= 4) {
                  if (e1 >= 1 << n_bigtens)
                        goto undfl;
#ifdef Avoid_Underflow
                  if (e1 & Scale_Bit)
                        scale = 2*P;
                  for(j = 0; e1 > 0; j++, e1 >>= 1)
                        if (e1 & 1)
                              dval(rv) *= tinytens[j];
                  if (scale && (j = 2*P + 1 - ((word0(rv) & Exp_mask)
                                    >> Exp_shift)) > 0) {
                        /* scaled rv is denormal; zap j low bits */
                        if (j >= 32) {
                              word1(rv) = 0;
                              if (j >= 53)
                               word0(rv) = (P+2)*Exp_msk1;
                              else
                               word0(rv) &= 0xffffffff << (j-32);
                              }
                        else
                              word1(rv) &= 0xffffffff << j;
                        }
#else
                  for(j = 0; e1 > 1; j++, e1 >>= 1)
                        if (e1 & 1)
                              dval(rv) *= tinytens[j];
                  /* The last multiplication could underflow. */
                  dval(rv0) = dval(rv);
                  dval(rv) *= tinytens[j];
                  if (!dval(rv)) {
                        dval(rv) = 2.*dval(rv0);
                        dval(rv) *= tinytens[j];
#endif
                        if (!dval(rv)) {
 undfl:
                              dval(rv) = 0.;
#ifndef NO_ERRNO
                              errno = ERANGE;
#endif
                              if (bd0)
                                    goto retfree;
                              goto ret;
                              }
#ifndef Avoid_Underflow
                        word0(rv) = Tiny0;
                        word1(rv) = Tiny1;
                        /* The refinement below will clean
                         * this approximation up.
                         */
                        }
#endif
                  }
            }

      /* 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(dval(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 Honor_FLT_ROUNDS
            if (rounding != 1)
                  bs2++;
#endif
#ifdef Avoid_Underflow
            j = bbe - scale;
            i = j + bbbits - 1;     /* logb(rv) */
            if (i < Emin)     /* denormal */
                  j += P - Emin;
            else
                  j = P + 1 - bbbits;
#else /*Avoid_Underflow*/
#ifdef Sudden_Underflow
#ifdef IBM
            j = 1 + 4*P - 3 - bbbits + ((bbe + bbbits - 1) & 3);
#else
            j = P + 1 - bbbits;
#endif
#else /*Sudden_Underflow*/
            j = bbe;
            i = j + bbbits - 1;     /* logb(rv) */
            if (i < Emin)     /* denormal */
                  j += P - Emin;
            else
                  j = P + 1 - bbbits;
#endif /*Sudden_Underflow*/
#endif /*Avoid_Underflow*/
            bb2 += j;
            bd2 += j;
#ifdef Avoid_Underflow
            bd2 += scale;
#endif
            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);
#ifdef Honor_FLT_ROUNDS
            if (rounding != 1) {
                  if (i < 0) {
                        /* Error is less than an ulp */
                        if (!delta->x[0] && delta->wds <= 1) {
                              /* exact */
#ifdef SET_INEXACT
                              inexact = 0;
#endif
                              break;
                              }
                        if (rounding) {
                              if (dsign) {
                                    adj = 1.;
                                    goto apply_adj;
                                    }
                              }
                        else if (!dsign) {
                              adj = -1.;
                              if (!word1(rv)
                               && !(word0(rv) & Frac_mask)) {
                                    y = word0(rv) & Exp_mask;
#ifdef Avoid_Underflow
                                    if (!scale || y > 2*P*Exp_msk1)
#else
                                    if (y)
#endif
                                      {
                                      delta = lshift(delta,Log2P);
                                      if (cmp(delta, bs) <= 0)
                                          adj = -0.5;
                                      }
                                    }
 apply_adj:
#ifdef Avoid_Underflow
                              if (scale && (y = word0(rv) & Exp_mask)
                                    <= 2*P*Exp_msk1)
                                word0(adj) += (2*P+1)*Exp_msk1 - y;
#else
#ifdef Sudden_Underflow
                              if ((word0(rv) & Exp_mask) <=
                                          P*Exp_msk1) {
                                    word0(rv) += P*Exp_msk1;
                                    dval(rv) += adj*ulp(dval(rv));
                                    word0(rv) -= P*Exp_msk1;
                                    }
                              else
#endif /*Sudden_Underflow*/
#endif /*Avoid_Underflow*/
                              dval(rv) += adj*ulp(dval(rv));
                              }
                        break;
                        }
                  adj = ratio(delta, bs);
                  if (adj < 1.)
                        adj = 1.;
                  if (adj <= 0x7ffffffe) {
                        /* adj = rounding ? ceil(adj) : floor(adj); */
                        y = adj;
                        if (y != adj) {
                              if (!((rounding>>1) ^ dsign))
                                    y++;
                              adj = y;
                              }
                        }
#ifdef Avoid_Underflow
                  if (scale && (y = word0(rv) & Exp_mask) <= 2*P*Exp_msk1)
                        word0(adj) += (2*P+1)*Exp_msk1 - y;
#else
#ifdef Sudden_Underflow
                  if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
                        word0(rv) += P*Exp_msk1;
                        adj *= ulp(dval(rv));
                        if (dsign)
                              dval(rv) += adj;
                        else
                              dval(rv) -= adj;
                        word0(rv) -= P*Exp_msk1;
                        goto cont;
                        }
#endif /*Sudden_Underflow*/
#endif /*Avoid_Underflow*/
                  adj *= ulp(dval(rv));
                  if (dsign)
                        dval(rv) += adj;
                  else
                        dval(rv) -= adj;
                  goto cont;
                  }
#endif /*Honor_FLT_ROUNDS*/

            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
#ifdef IEEE_Arith
#ifdef Avoid_Underflow
                   || (word0(rv) & Exp_mask) <= (2*P+1)*Exp_msk1
#else
                   || (word0(rv) & Exp_mask) <= Exp_msk1
#endif
#endif
                        ) {
#ifdef SET_INEXACT
                        if (!delta->x[0] && delta->wds <= 1)
                              inexact = 0;
#endif
                        break;
                        }
                  if (!delta->x[0] && delta->wds <= 1) {
                        /* exact result */
#ifdef SET_INEXACT
                        inexact = 0;
#endif
                        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) == (
#ifdef Avoid_Underflow
                  (scale && (y = word0(rv) & Exp_mask) <= 2*P*Exp_msk1)
            ? (0xffffffff & (0xffffffff << (2*P+1-(y>>Exp_shift)))) :
#endif
                                       0xffffffff)) {
                              /*boundary case -- increment exponent*/
                              word0(rv) = (word0(rv) & Exp_mask)
                                    + Exp_msk1
#ifdef IBM
                                    | Exp_msk1 >> 4
#endif
                                    ;
                              word1(rv) = 0;
#ifdef Avoid_Underflow
                              dsign = 0;
#endif
                              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
#ifdef Avoid_Underflow
                        if (L <= (scale ? (2*P+1)*Exp_msk1 : Exp_msk1))
#else
                        if (L <= Exp_msk1)
#endif /*Avoid_Underflow*/
#endif /*IBM*/
                              goto undfl;
                        L -= Exp_msk1;
#else /*Sudden_Underflow}{*/
#ifdef Avoid_Underflow
                        if (scale) {
                              L = word0(rv) & Exp_mask;
                              if (L <= (2*P+1)*Exp_msk1) {
                                    if (L > (P+2)*Exp_msk1)
                                          /* round even ==> */
                                          /* accept rv */
                                          break;
                                    /* rv = smallest denormal */
                                    goto undfl;
                                    }
                              }
#endif /*Avoid_Underflow*/
                        L = (word0(rv) & Exp_mask) - Exp_msk1;
#endif /*Sudden_Underflow}}*/
                        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)
                        dval(rv) += ulp(dval(rv));
#ifndef ROUND_BIASED
                  else {
                        dval(rv) -= ulp(dval(rv));
#ifndef Sudden_Underflow
                        if (!dval(rv))
                              goto undfl;
#endif
                        }
#ifdef Avoid_Underflow
                  dsign = 1 - dsign;
#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(Rounding) {
                        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 /*Check_FLT_ROUNDS*/
                  }
            y = word0(rv) & Exp_mask;

            /* Check for overflow */

            if (y == Exp_msk1*(DBL_MAX_EXP+Bias-1)) {
                  dval(rv0) = dval(rv);
                  word0(rv) -= P*Exp_msk1;
                  adj = aadj1 * ulp(dval(rv));
                  dval(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 Avoid_Underflow
                  if (scale && y <= 2*P*Exp_msk1) {
                        if (aadj <= 0x7fffffff) {
                              if ((z = aadj) <= 0)
                                    z = 1;
                              aadj = z;
                              aadj1 = dsign ? aadj : -aadj;
                              }
                        word0(aadj1) += (2*P+1)*Exp_msk1 - y;
                        }
                  adj = aadj1 * ulp(dval(rv));
                  dval(rv) += adj;
#else
#ifdef Sudden_Underflow
                  if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
                        dval(rv0) = dval(rv);
                        word0(rv) += P*Exp_msk1;
                        adj = aadj1 * ulp(dval(rv));
                        dval(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(dval(rv));
                        dval(rv) += adj;
                        }
#else /*Sudden_Underflow*/
                  /* 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(dval(rv));
                  dval(rv) += adj;
#endif /*Sudden_Underflow*/
#endif /*Avoid_Underflow*/
                  }
            z = word0(rv) & Exp_mask;
#ifndef SET_INEXACT
#ifdef Avoid_Underflow
            if (!scale)
#endif
            if (y == z) {
                  /* Can we stop now? */
                  L = (Long)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;
                  }
#endif
 cont:
            Bfree(bb);
            Bfree(bd);
            Bfree(bs);
            Bfree(delta);
            }
#ifdef SET_INEXACT
      if (inexact) {
            if (!oldinexact) {
                  word0(rv0) = Exp_1 + (70 << Exp_shift);
                  word1(rv0) = 0;
                  dval(rv0) += 1.;
                  }
            }
      else if (!oldinexact)
            clear_inexact();
#endif
#ifdef Avoid_Underflow
      if (scale) {
            word0(rv0) = Exp_1 - 2*P*Exp_msk1;
            word1(rv0) = 0;
            dval(rv) *= dval(rv0);
#ifndef NO_ERRNO
            /* try to avoid the bug of testing an 8087 register value */
            if (word0(rv) == 0 && word1(rv) == 0)
                  errno = ERANGE;
#endif
            }
#endif /* Avoid_Underflow */
#ifdef SET_INEXACT
      if (inexact && !(word0(rv) & Exp_mask)) {
            /* set underflow bit */
            dval(rv0) = 1e-300;
            dval(rv0) *= dval(rv0);
            }
#endif
 retfree:
      Bfree(bb);
      Bfree(bd);
      Bfree(bs);
      Bfree(bd0);
      Bfree(delta);
 ret:
      if (se)
            *se = (char *)s;
      return sign ? -dval(rv) : dval(rv);
      }

 static int
quorem
#ifdef KR_headers
      (b, S) Bigint *b, *S;
#else
      (Bigint *b, Bigint *S)
#endif
{
      int n;
      ULong *bx, *bxe, q, *sx, *sxe;
#ifdef ULLong
      ULLong borrow, carry, y, ys;
#else
      ULong borrow, carry, y, ys;
#ifdef Pack_32
      ULong si, z, zs;
#endif
#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 ULLong
                  ys = *sx++ * (ULLong)q + carry;
                  carry = ys >> 32;
                  y = *bx - (ys & FFFFFFFF) - borrow;
                  borrow = y >> 32 & (ULong)1;
                  *bx++ = y & FFFFFFFF;
#else
#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 & 0x10000) >> 16;
                  z = (*bx >> 16) - (zs & 0xffff) - borrow;
                  borrow = (z & 0x10000) >> 16;
                  Storeinc(bx, z, y);
#else
                  ys = *sx++ * q + carry;
                  carry = ys >> 16;
                  y = *bx - (ys & 0xffff) - borrow;
                  borrow = (y & 0x10000) >> 16;
                  *bx++ = y & 0xffff;
#endif
#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 ULLong
                  ys = *sx++ + carry;
                  carry = ys >> 32;
                  y = *bx - (ys & FFFFFFFF) - borrow;
                  borrow = y >> 32 & (ULong)1;
                  *bx++ = y & FFFFFFFF;
#else
#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 & 0x10000) >> 16;
                  z = (*bx >> 16) - (zs & 0xffff) - borrow;
                  borrow = (z & 0x10000) >> 16;
                  Storeinc(bx, z, y);
#else
                  ys = *sx++ + carry;
                  carry = ys >> 16;
                  y = *bx - (ys & 0xffff) - borrow;
                  borrow = (y & 0x10000) >> 16;
                  *bx++ = y & 0xffff;
#endif
#endif
                  }
                  while(sx <= sxe);
            bx = b->x;
            bxe = bx + n;
            if (!*bxe) {
                  while(--bxe > bx && !*bxe)
                        --n;
                  b->wds = n;
                  }
            }
      return q;
      }

#ifndef MULTIPLE_THREADS
 static char *dtoa_result;
#endif

 static char *
#ifdef KR_headers
rv_alloc(i) int i;
#else
rv_alloc(int i)
#endif
{
      int j, k, *r;

      j = sizeof(ULong);
      for(k = 0;
            sizeof(Bigint) - sizeof(ULong) - sizeof(int) + j <= i;
            j <<= 1)
                  k++;
      r = (int*)Balloc(k);
      *r = k;
      return
#ifndef MULTIPLE_THREADS
      dtoa_result =
#endif
            (char *)(r+1);
      }

 static char *
#ifdef KR_headers
nrv_alloc(s, rve, n) char *s, **rve; int n;
#else
nrv_alloc(const char *s, char **rve, int n)
#endif
{
      char *rv, *t;

      t = rv = rv_alloc(n);
      while ((*t = *s++)) t++;
      if (rve)
            *rve = t;
      return rv;
      }

/* freedtoa(s) must be used to free values s returned by dtoa
 * when MULTIPLE_THREADS is #defined.  It should be used in all cases,
 * but for consistency with earlier versions of dtoa, it is optional
 * when MULTIPLE_THREADS is not defined.
 */

 void
#ifdef KR_headers
freedtoa(s) char *s;
#else
freedtoa(char *s)
#endif
{
      Bigint *b = (Bigint *)((int *)s - 1);
      b->maxwds = 1 << (b->k = *(int*)b);
      Bfree(b);
#ifndef MULTIPLE_THREADS
      if (s == dtoa_result)
            dtoa_result = 0;
#endif
      }

/* 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. 112-126].
 *
 * 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;
#else
      (double d, int mode, int ndigits, int *decpt, int *sign, char **rve)
#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.
            4,5 ==> similar to 2 and 3, respectively, but (in
                  round-nearest mode) with the tests of mode 0 to
                  possibly return a shorter string that rounds to d.
                  With IEEE arithmetic and compilation with
                  -DHonor_FLT_ROUNDS, modes 4 and 5 behave the same
                  as modes 2 and 3 when FLT_ROUNDS != 1.
            6-9 ==> Debugging modes similar to mode - 4:  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, bias_round_up;
      Long L;
#ifndef Sudden_Underflow
      int denorm;
      ULong x;
#endif
      Bigint *b, *b1, *delta, *mlo, *mhi, *S;
      double d2, ds, eps;
      char *s, *s0;
#ifdef Honor_FLT_ROUNDS
      int rounding;
#endif
#ifdef SET_INEXACT
      int inexact, oldinexact;
#endif

        /* In mode 2 and 3 we bias rounding up when there are ties. */
        bias_round_up = mode == 2 || mode == 3;

        ilim = ilim1 = 0; /* to avoid Google3 compiler warnings */

#ifndef MULTIPLE_THREADS
      if (dtoa_result) {
            freedtoa(dtoa_result);
            dtoa_result = 0;
            }
#endif

      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;
#ifdef IEEE_Arith
            if (!word1(d) && !(word0(d) & 0xfffff))
                  return nrv_alloc("Infinity", rve, 8);
#endif
            return nrv_alloc("NaN", rve, 3);
            }
#endif
#ifdef IBM
      dval(d) += 0; /* normalize */
#endif
      if (!dval(d)) {
            *decpt = 1;
            return nrv_alloc("0", rve, 1);
            }

#ifdef SET_INEXACT
      try_quick = oldinexact = get_inexact();
      inexact = 1;
#endif
#ifdef Honor_FLT_ROUNDS
      if ((rounding = Flt_Rounds) >= 2) {
            if (*sign)
                  rounding = rounding == 2 ? 0 : 2;
            else
                  if (rounding != 2)
                        rounding = 0;
            }
#endif

      b = d2b(dval(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
            dval(d2) = dval(d);
            word0(d2) &= Frac_mask1;
            word0(d2) |= Exp_11;
#ifdef IBM
            if (j = 11 - hi0bits(word0(d2) & Frac_mask))
                  dval(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);
            dval(d2) = x;
            word0(d2) -= 31*Exp_msk1; /* adjust exponent */
            i -= (Bias + (P-1) - 1) + 1;
            denorm = 1;
            }
#endif
      ds = (dval(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 (dval(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;

#ifndef SET_INEXACT
#ifdef Check_FLT_ROUNDS
      try_quick = Rounding == 1;
#else
      try_quick = 1;
#endif
#endif /*SET_INEXACT*/

      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;
            }
      s = s0 = rv_alloc(i);

#ifdef Honor_FLT_ROUNDS
      if (mode > 1 && rounding != 1)
            leftright = 0;
#endif

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

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

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

                /* Limit looping by the number of digits to produce.
                 * Firefox had a crash bug because some plugins reduce
                 * the precision of double arithmetic.  With reduced
                 * precision "dval(d) -= L*ds" might be imprecise and
                 * d might not become zero and the loop might not
                 * terminate.
                 *
                 * See https://bugzilla.mozilla.org/show_bug.cgi?id=358569
                 */
            for(i = 1; i <= k+1; i++, dval(d) *= 10.) {
                  L = (Long)(dval(d) / ds);
                  dval(d) -= L*ds;
#ifdef Check_FLT_ROUNDS
                  /* If FLT_ROUNDS == 2, L will usually be high by 1 */
                  if (dval(d) < 0) {
                        L--;
                        dval(d) += ds;
                        }
#endif
                  *s++ = '0' + (int)L;
                  if (!dval(d)) {
#ifdef SET_INEXACT
                        inexact = 0;
#endif
                        break;
                        }
                  if (i == ilim) {
#ifdef Honor_FLT_ROUNDS
                        if (mode > 1)
                        switch(rounding) {
                          case 0: goto ret1;
                          case 2: goto bump_up;
                          }
#endif
                        dval(d) += dval(d);
                        if (dval(d) > ds || (dval(d) == ds && ((L & 1) || bias_round_up))) {
 bump_up:
                              while(*--s == '9')
                                    if (s == s0) {
                                          k++;
                                          *s = '0';
                                          break;
                                          }
                              ++*s++;
                              }
                        break;
                        }
                  }
            goto ret1;
            }

      m2 = b2;
      m5 = b5;
      mhi = mlo = 0;
      if (leftright) {
            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
            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. */

      spec_case = 0;
      if ((mode < 2 || leftright)
#ifdef Honor_FLT_ROUNDS
                  && rounding == 1
#endif
                        ) {
            if (!word1(d) && !(word0(d) & Bndry_mask)
#ifndef Sudden_Underflow
             && word0(d) & (Exp_mask & ~Exp_msk1)
#endif
                        ) {
                  /* The special case */
                  b2 += Log2P;
                  s2 += Log2P;
                  spec_case = 1;
                  }
            }

      /* 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 == 3 || mode == 5)) {
                S = multadd(S, 5, 0);
            if (ilim < 0 || cmp(b, S) < 0 || ((cmp(b, S) == 0) && !bias_round_up)) {
                  /* 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 != 1 && !(word1(d) & 1)
#ifdef Honor_FLT_ROUNDS
                        && rounding >= 1
#endif
                                                   ) {
                        if (dig == '9')
                              goto round_9_up;
                        if (j > 0)
                              dig++;
#ifdef SET_INEXACT
                        else if (!b->x[0] && b->wds <= 1)
                              inexact = 0;
#endif
                        *s++ = dig;
                        goto ret;
                        }
#endif
                  if (j < 0 || (j == 0 && mode != 1
#ifndef ROUND_BIASED
                                          && !(word1(d) & 1)
#endif
                              )) {
                        if (!b->x[0] && b->wds <= 1) {
#ifdef SET_INEXACT
                              inexact = 0;
#endif
                              goto accept_dig;
                              }
#ifdef Honor_FLT_ROUNDS
                        if (mode > 1)
                         switch(rounding) {
                          case 0: goto accept_dig;
                          case 2: goto keep_dig;
                          }
#endif /*Honor_FLT_ROUNDS*/
                        if (j1 > 0) {
                              b = lshift(b, 1);
                              j1 = cmp(b, S);
                              if ((j1 > 0 || (j1 == 0 && ((dig & 1) || bias_round_up)))
                                            && dig++ == '9')
                                    goto round_9_up;
                              }
 accept_dig:
                        *s++ = dig;
                        goto ret;
                        }
                  if (j1 > 0) {
#ifdef Honor_FLT_ROUNDS
                        if (!rounding)
                              goto accept_dig;
#endif
                        if (dig == '9') { /* possible if i == 1 */
 round_9_up:
                              *s++ = '9';
                              goto roundoff;
                              }
                        *s++ = dig + 1;
                        goto ret;
                        }
#ifdef Honor_FLT_ROUNDS
 keep_dig:
#endif
                  *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 (!b->x[0] && b->wds <= 1) {
#ifdef SET_INEXACT
                        inexact = 0;
#endif
                        goto ret;
                        }
                  if (i >= ilim)
                        break;
                  b = multadd(b, 10, 0);
                  }

      /* Round off last digit */

#ifdef Honor_FLT_ROUNDS
      switch(rounding) {
        case 0: goto trimzeros;
        case 2: goto roundoff;
        }
#endif
      b = lshift(b, 1);
      j = cmp(b, S);
      if (j > 0 || (j == 0 && ((dig & 1) || bias_round_up))) {
 roundoff:
            while(*--s == '9')
                  if (s == s0) {
                        k++;
                        *s++ = '1';
                        goto ret;
                        }
            ++*s++;
            }
      else {
/* trimzeros:  (never used) */
            while(*--s == '0');
            s++;
            }
 ret:
      Bfree(S);
      if (mhi) {
            if (mlo && mlo != mhi)
                  Bfree(mlo);
            Bfree(mhi);
            }
 ret1:
#ifdef SET_INEXACT
      if (inexact) {
            if (!oldinexact) {
                  word0(d) = Exp_1 + (70 << Exp_shift);
                  word1(d) = 0;
                  dval(d) += 1.;
                  }
            }
      else if (!oldinexact)
            clear_inexact();
#endif
      Bfree(b);
      *s = 0;
      *decpt = k + 1;
      if (rve)
            *rve = s;
      return s0;
      }
#ifdef __cplusplus
}
#endif

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