stdlib.subproj/strtod.c

   1 /*
   2  * Copyright (c) 1999 Apple Computer, Inc. All rights reserved.
   3  *
   4  * @APPLE_LICENSE_HEADER_START@
   5  *
   6  * The contents of this file constitute Original Code as defined in and
   7  * are subject to the Apple Public Source License Version 1.1 (the
   8  * "License").  You may not use this file except in compliance with the
   9  * License.  Please obtain a copy of the License at
  10  * http://www.apple.com/publicsource and read it before using this file.
  11  *
  12  * This Original Code and all software distributed under the License are
  13  * distributed on an "AS IS" basis, WITHOUT WARRANTY OF ANY KIND, EITHER
  14  * EXPRESS OR IMPLIED, AND APPLE HEREBY DISCLAIMS ALL SUCH WARRANTIES,
  15  * INCLUDING WITHOUT LIMITATION, ANY WARRANTIES OF MERCHANTABILITY,
  16  * FITNESS FOR A PARTICULAR PURPOSE OR NON-INFRINGEMENT.  Please see the
  17  * License for the specific language governing rights and limitations
  18  * under the License.
  19  *
  20  * @APPLE_LICENSE_HEADER_END@
  21  */
  22 /****************************************************************
  23  *
  24  * The author of this software is David M. Gay.
  25  *
  26  * Copyright (c) 1991 by AT&T.
  27  *
  28  * Permission to use, copy, modify, and distribute this software for any
  29  * purpose without fee is hereby granted, provided that this entire notice
  30  * is included in all copies of any software which is or includes a copy
  31  * or modification of this software and in all copies of the supporting
  32  * documentation for such software.
  33  *
  34  * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
  35  * WARRANTY.  IN PARTICULAR, NEITHER THE AUTHOR NOR AT&T MAKES ANY
  36  * REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
  37  * OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
  38  *
  39  ***************************************************************/
  40
  41 /* Please send bug reports to
  42         David M. Gay
  43         AT&T Bell Laboratories, Room 2C-463
  44         600 Mountain Avenue
  45         Murray Hill, NJ 07974-2070
  46         U.S.A.
  47         dmg@research.att.com or research!dmg
  48  */
  49
  50 /* strtod for IEEE-, VAX-, and IBM-arithmetic machines.
  51  *
  52  * This strtod returns a nearest machine number to the input decimal
  53  * string (or sets errno to ERANGE).  With IEEE arithmetic, ties are
  54  * broken by the IEEE round-even rule.  Otherwise ties are broken by
  55  * biased rounding (add half and chop).
  56  *
  57  * Inspired loosely by William D. Clinger's paper "How to Read Floating
  58  * Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101].
  59  *
  60  * Modifications:
  61  *
  62  *      1. We only require IEEE, IBM, or VAX double-precision
  63  *              arithmetic (not IEEE double-extended).
  64  *      2. We get by with floating-point arithmetic in a case that
  65  *              Clinger missed -- when we're computing d * 10^n
  66  *              for a small integer d and the integer n is not too
  67  *              much larger than 22 (the maximum integer k for which
  68  *              we can represent 10^k exactly), we may be able to
  69  *              compute (d*10^k) * 10^(e-k) with just one roundoff.
  70  *      3. Rather than a bit-at-a-time adjustment of the binary
  71  *              result in the hard case, we use floating-point
  72  *              arithmetic to determine the adjustment to within
  73  *              one bit; only in really hard cases do we need to
  74  *              compute a second residual.
  75  *      4. Because of 3., we don't need a large table of powers of 10
  76  *              for ten-to-e (just some small tables, e.g. of 10^k
  77  *              for 0 <= k <= 22).
  78  */
  79
  80 /*
  81  * #define IEEE_LITTLE_ENDIAN for IEEE-arithmetic machines where the least
  82  *      significant byte has the lowest address.
  83  * #define IEEE_BIG_ENDIAN for IEEE-arithmetic machines where the most
  84  *      significant byte has the lowest address.
  85  * #define Long int on machines with 32-bit ints and 64-bit longs.
  86  * #define Sudden_Underflow for IEEE-format machines without gradual
  87  *      underflow (i.e., that flush to zero on underflow).
  88  * #define IBM for IBM mainframe-style floating-point arithmetic.
  89  * #define VAX for VAX-style floating-point arithmetic.
  90  * #define Unsigned_Shifts if >> does treats its left operand as unsigned.
  91  * #define No_leftright to omit left-right logic in fast floating-point
  92  *      computation of dtoa.
  93  * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3.
  94  * #define RND_PRODQUOT to use rnd_prod and rnd_quot (assembly routines
  95  *      that use extended-precision instructions to compute rounded
  96  *      products and quotients) with IBM.
  97  * #define ROUND_BIASED for IEEE-format with biased rounding.
  98  * #define Inaccurate_Divide for IEEE-format with correctly rounded
  99  *      products but inaccurate quotients, e.g., for Intel i860.
 100  * #define Just_16 to store 16 bits per 32-bit Long when doing high-precision
 101  *      integer arithmetic.  Whether this speeds things up or slows things
 102  *      down depends on the machine and the number being converted.
 103  * #define KR_headers for old-style C function headers.
 104  * #define Bad_float_h if your system lacks a float.h or if it does not
 105  *      define some or all of DBL_DIG, DBL_MAX_10_EXP, DBL_MAX_EXP,
 106  *      FLT_RADIX, FLT_ROUNDS, and DBL_MAX.
 107  * #define MALLOC your_malloc, where your_malloc(n) acts like malloc(n)
 108  *      if memory is available and otherwise does something you deem
 109  *      appropriate.  If MALLOC is undefined, malloc will be invoked
 110  *      directly -- and assumed always to succeed.
 111  */
 112
 113 #if defined(LIBC_SCCS) && !defined(lint)
 114 static char *rcsid = "$OpenBSD: strtod.c,v 1.9 1997/03/25 17:07:30 rahnds Exp $";
 115 #endif /* LIBC_SCCS and not lint */
 116
 117 #if defined(__m68k__) || defined(__sparc__) || defined(__i386__) || \
 118     defined(__mips__) || defined(__ns32k__) || defined(__alpha__) || \
 119     defined(__powerpc__) || defined(__m88k__) || defined(__ppc__)
 120 #include <sys/types.h>
 121 #if BYTE_ORDER == BIG_ENDIAN
 122 #define IEEE_BIG_ENDIAN
 123 #else
 124 #define IEEE_LITTLE_ENDIAN
 125 #endif
 126 #endif
 127
 128 #ifdef __arm32__
 129 /*
 130  * Although the CPU is little endian the FP has different
 131  * byte and word endianness. The byte order is still little endian
 132  * but the word order is big endian.
 133  */
 134 #define IEEE_BIG_ENDIAN
 135 #endif
 136
 137 #ifdef vax
 138 #define VAX
 139 #endif
 140
 141 #define Long    int32_t
 142 #define ULong   u_int32_t
 143
 144 #ifdef DEBUG
 145 #include "stdio.h"
 146 #define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
 147 #endif
 148
 149 #ifdef __cplusplus
 150 #include "malloc.h"
 151 #include "memory.h"
 152 #else
 153 #ifndef KR_headers
 154 #include "stdlib.h"
 155 #include "string.h"
 156 #include "locale.h"
 157 #else
 158 #include "malloc.h"
 159 #include "memory.h"
 160 #endif
 161 #endif
 162
 163 #ifdef MALLOC
 164 #ifdef KR_headers
 165 extern char *MALLOC();
 166 #else
 167 extern void *MALLOC(size_t);
 168 #endif
 169 #else
 170 #define MALLOC malloc
 171 #endif
 172
 173 #include "ctype.h"
 174 #include "errno.h"
 175
 176 #ifdef Bad_float_h
 177 #undef __STDC__
 178 #ifdef IEEE_BIG_ENDIAN
 179 #define IEEE_ARITHMETIC
 180 #endif
 181 #ifdef IEEE_LITTLE_ENDIAN
 182 #define IEEE_ARITHMETIC
 183 #endif
 184
 185 #ifdef IEEE_ARITHMETIC
 186 #define DBL_DIG 15
 187 #define DBL_MAX_10_EXP 308
 188 #define DBL_MAX_EXP 1024
 189 #define FLT_RADIX 2
 190 #define FLT_ROUNDS 1
 191 #define DBL_MAX 1.7976931348623157e+308
 192 #endif
 193
 194 #ifdef IBM
 195 #define DBL_DIG 16
 196 #define DBL_MAX_10_EXP 75
 197 #define DBL_MAX_EXP 63
 198 #define FLT_RADIX 16
 199 #define FLT_ROUNDS 0
 200 #define DBL_MAX 7.2370055773322621e+75
 201 #endif
 202
 203 #ifdef VAX
 204 #define DBL_DIG 16
 205 #define DBL_MAX_10_EXP 38
 206 #define DBL_MAX_EXP 127
 207 #define FLT_RADIX 2
 208 #define FLT_ROUNDS 1
 209 #define DBL_MAX 1.7014118346046923e+38
 210 #endif
 211
 212 #ifndef LONG_MAX
 213 #define LONG_MAX 2147483647
 214 #endif
 215 #else
 216 #include "float.h"
 217 #endif
 218 #ifndef __MATH_H__
 219 #include "math.h"
 220 #endif
 221
 222 #ifdef __cplusplus
 223 extern "C" {
 224 #endif
 225
 226 #ifndef CONST
 227 #ifdef KR_headers
 228 #define CONST /* blank */
 229 #else
 230 #define CONST const
 231 #endif
 232 #endif
 233
 234 #ifdef Unsigned_Shifts
 235 #define Sign_Extend(a,b) if (b < 0) a |= 0xffff0000;
 236 #else
 237 #define Sign_Extend(a,b) /*no-op*/
 238 #endif
 239
 240 #if defined(IEEE_LITTLE_ENDIAN) + defined(IEEE_BIG_ENDIAN) + defined(VAX) + \
 241     defined(IBM) != 1
 242 Exactly one of IEEE_LITTLE_ENDIAN IEEE_BIG_ENDIAN, VAX, or
 243 IBM should be defined.
 244 #endif
 245
 246 #ifdef IEEE_LITTLE_ENDIAN
 247 #define word0(x) ((ULong *)&x)[1]
 248 #define word1(x) ((ULong *)&x)[0]
 249 #else
 250 #define word0(x) ((ULong *)&x)[0]
 251 #define word1(x) ((ULong *)&x)[1]
 252 #endif
 253
 254 /* The following definition of Storeinc is appropriate for MIPS processors.
 255  * An alternative that might be better on some machines is
 256  * #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
 257  */
 258 #if defined(IEEE_LITTLE_ENDIAN) + defined(VAX) + defined(__arm32__)
 259 #define Storeinc(a,b,c) (((unsigned short *)a)[1] = (unsigned short)b, \
 260 ((unsigned short *)a)[0] = (unsigned short)c, a++)
 261 #else
 262 #define Storeinc(a,b,c) (((unsigned short *)a)[0] = (unsigned short)b, \
 263 ((unsigned short *)a)[1] = (unsigned short)c, a++)
 264 #endif
 265
 266 /* #define P DBL_MANT_DIG */
 267 /* Ten_pmax = floor(P*log(2)/log(5)) */
 268 /* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
 269 /* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
 270 /* Int_max = floor(P*log(FLT_RADIX)/log(10) - 1) */
 271
 272 #if defined(IEEE_LITTLE_ENDIAN) + defined(IEEE_BIG_ENDIAN)
 273 #define Exp_shift  20
 274 #define Exp_shift1 20
 275 #define Exp_msk1    0x100000
 276 #define Exp_msk11   0x100000
 277 #define Exp_mask  0x7ff00000
 278 #define P 53
 279 #define Bias 1023
 280 #define IEEE_Arith
 281 #define Emin (-1022)
 282 #define Exp_1  0x3ff00000
 283 #define Exp_11 0x3ff00000
 284 #define Ebits 11
 285 #define Frac_mask  0xfffff
 286 #define Frac_mask1 0xfffff
 287 #define Ten_pmax 22
 288 #define Bletch 0x10
 289 #define Bndry_mask  0xfffff
 290 #define Bndry_mask1 0xfffff
 291 #define LSB 1
 292 #define Sign_bit 0x80000000
 293 #define Log2P 1
 294 #define Tiny0 0
 295 #define Tiny1 1
 296 #define Quick_max 14
 297 #define Int_max 14
 298 #define Infinite(x) (word0(x) == 0x7ff00000) /* sufficient test for here */
 299 #else
 300 #undef  Sudden_Underflow
 301 #define Sudden_Underflow
 302 #ifdef IBM
 303 #define Exp_shift  24
 304 #define Exp_shift1 24
 305 #define Exp_msk1   0x1000000
 306 #define Exp_msk11  0x1000000
 307 #define Exp_mask  0x7f000000
 308 #define P 14
 309 #define Bias 65
 310 #define Exp_1  0x41000000
 311 #define Exp_11 0x41000000
 312 #define Ebits 8 /* exponent has 7 bits, but 8 is the right value in b2d */
 313 #define Frac_mask  0xffffff
 314 #define Frac_mask1 0xffffff
 315 #define Bletch 4
 316 #define Ten_pmax 22
 317 #define Bndry_mask  0xefffff
 318 #define Bndry_mask1 0xffffff
 319 #define LSB 1
 320 #define Sign_bit 0x80000000
 321 #define Log2P 4
 322 #define Tiny0 0x100000
 323 #define Tiny1 0
 324 #define Quick_max 14
 325 #define Int_max 15
 326 #else /* VAX */
 327 #define Exp_shift  23
 328 #define Exp_shift1 7
 329 #define Exp_msk1    0x80
 330 #define Exp_msk11   0x800000
 331 #define Exp_mask  0x7f80
 332 #define P 56
 333 #define Bias 129
 334 #define Exp_1  0x40800000
 335 #define Exp_11 0x4080
 336 #define Ebits 8
 337 #define Frac_mask  0x7fffff
 338 #define Frac_mask1 0xffff007f
 339 #define Ten_pmax 24
 340 #define Bletch 2
 341 #define Bndry_mask  0xffff007f
 342 #define Bndry_mask1 0xffff007f
 343 #define LSB 0x10000
 344 #define Sign_bit 0x8000
 345 #define Log2P 1
 346 #define Tiny0 0x80
 347 #define Tiny1 0
 348 #define Quick_max 15
 349 #define Int_max 15
 350 #endif
 351 #endif
 352
 353 #ifndef IEEE_Arith
 354 #define ROUND_BIASED
 355 #endif
 356
 357 #ifdef RND_PRODQUOT
 358 #define rounded_product(a,b) a = rnd_prod(a, b)
 359 #define rounded_quotient(a,b) a = rnd_quot(a, b)
 360 #ifdef KR_headers
 361 extern double rnd_prod(), rnd_quot();
 362 #else
 363 extern double rnd_prod(double, double), rnd_quot(double, double);
 364 #endif
 365 #else
 366 #define rounded_product(a,b) a *= b
 367 #define rounded_quotient(a,b) a /= b
 368 #endif
 369
 370 #define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
 371 #define Big1 0xffffffff
 372
 373 #ifndef Just_16
 374 /* When Pack_32 is not defined, we store 16 bits per 32-bit Long.
 375  * This makes some inner loops simpler and sometimes saves work
 376  * during multiplications, but it often seems to make things slightly
 377  * slower.  Hence the default is now to store 32 bits per Long.
 378  */
 379 #ifndef Pack_32
 380 #define Pack_32
 381 #endif
 382 #endif
 383
 384 #define Kmax 15
 385
 386 #ifdef __cplusplus
 387 extern "C" double strtod(const char *s00, char **se);
 388 extern "C" char *__dtoa(double d, int mode, int ndigits,
 389                         int *decpt, int *sign, char **rve);
 390 #endif
 391
 392  struct
 393 Bigint {
 394         struct Bigint *next;
 395         int k, maxwds, sign, wds;
 396         ULong x[1];
 397         };
 398
 399  typedef struct Bigint Bigint;
 400
 401  static Bigint *freelist[Kmax+1];
 402
 403  static Bigint *
 404 Balloc
 405 #ifdef KR_headers
 406         (k) int k;
 407 #else
 408         (int k)
 409 #endif
 410 {
 411         int x;
 412         Bigint *rv;
 413
 414         if (rv = freelist[k]) {
 415                 freelist[k] = rv->next;
 416                 }
 417         else {
 418                 x = 1 << k;
 419                 rv = (Bigint *)MALLOC(sizeof(Bigint) + (x-1)*sizeof(Long));
 420                 rv->k = k;
 421                 rv->maxwds = x;
 422                 }
 423         rv->sign = rv->wds = 0;
 424         return rv;
 425         }
 426
 427  static void
 428 Bfree
 429 #ifdef KR_headers
 430         (v) Bigint *v;
 431 #else
 432         (Bigint *v)
 433 #endif
 434 {
 435         if (v) {
 436                 v->next = freelist[v->k];
 437                 freelist[v->k] = v;
 438                 }
 439         }
 440
 441 #define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
 442 y->wds*sizeof(Long) + 2*sizeof(int))
 443
 444  static Bigint *
 445 multadd
 446 #ifdef KR_headers
 447         (b, m, a) Bigint *b; int m, a;
 448 #else
 449         (Bigint *b, int m, int a)       /* multiply by m and add a */
 450 #endif
 451 {
 452         int i, wds;
 453         ULong *x, y;
 454 #ifdef Pack_32
 455         ULong xi, z;
 456 #endif
 457         Bigint *b1;
 458
 459         wds = b->wds;
 460         x = b->x;
 461         i = 0;
 462         do {
 463 #ifdef Pack_32
 464                 xi = *x;
 465                 y = (xi & 0xffff) * m + a;
 466                 z = (xi >> 16) * m + (y >> 16);
 467                 a = (int)(z >> 16);
 468                 *x++ = (z << 16) + (y & 0xffff);
 469 #else
 470                 y = *x * m + a;
 471                 a = (int)(y >> 16);
 472                 *x++ = y & 0xffff;
 473 #endif
 474                 }
 475                 while(++i < wds);
 476         if (a) {
 477                 if (wds >= b->maxwds) {
 478                         b1 = Balloc(b->k+1);
 479                         Bcopy(b1, b);
 480                         Bfree(b);
 481                         b = b1;
 482                         }
 483                 b->x[wds++] = a;
 484                 b->wds = wds;
 485                 }
 486         return b;
 487         }
 488
 489  static Bigint *
 490 s2b
 491 #ifdef KR_headers
 492         (s, nd0, nd, y9) CONST char *s; int nd0, nd; ULong y9;
 493 #else
 494         (CONST char *s, int nd0, int nd, ULong y9)
 495 #endif
 496 {
 497         Bigint *b;
 498         int i, k;
 499         Long x, y;
 500
 501         x = (nd + 8) / 9;
 502         for(k = 0, y = 1; x > y; y <<= 1, k++) ;
 503 #ifdef Pack_32
 504         b = Balloc(k);
 505         b->x[0] = y9;
 506         b->wds = 1;
 507 #else
 508         b = Balloc(k+1);
 509         b->x[0] = y9 & 0xffff;
 510         b->wds = (b->x[1] = y9 >> 16) ? 2 : 1;
 511 #endif
 512
 513         i = 9;
 514         if (9 < nd0) {
 515                 s += 9;
 516                 do b = multadd(b, 10, *s++ - '0');
 517                         while(++i < nd0);
 518                 s++;
 519                 }
 520         else
 521                 s += 10;
 522         for(; i < nd; i++)
 523                 b = multadd(b, 10, *s++ - '0');
 524         return b;
 525         }
 526
 527  static int
 528 hi0bits
 529 #ifdef KR_headers
 530         (x) register ULong x;
 531 #else
 532         (register ULong x)
 533 #endif
 534 {
 535         register int k = 0;
 536
 537         if (!(x & 0xffff0000)) {
 538                 k = 16;
 539                 x <<= 16;
 540                 }
 541         if (!(x & 0xff000000)) {
 542                 k += 8;
 543                 x <<= 8;
 544                 }
 545         if (!(x & 0xf0000000)) {
 546                 k += 4;
 547                 x <<= 4;
 548                 }
 549         if (!(x & 0xc0000000)) {
 550                 k += 2;
 551                 x <<= 2;
 552                 }
 553         if (!(x & 0x80000000)) {
 554                 k++;
 555                 if (!(x & 0x40000000))
 556                         return 32;
 557                 }
 558         return k;
 559         }
 560
 561  static int
 562 lo0bits
 563 #ifdef KR_headers
 564         (y) ULong *y;
 565 #else
 566         (ULong *y)
 567 #endif
 568 {
 569         register int k;
 570         register ULong x = *y;
 571
 572         if (x & 7) {
 573                 if (x & 1)
 574                         return 0;
 575                 if (x & 2) {
 576                         *y = x >> 1;
 577                         return 1;
 578                         }
 579                 *y = x >> 2;
 580                 return 2;
 581                 }
 582         k = 0;
 583         if (!(x & 0xffff)) {
 584                 k = 16;
 585                 x >>= 16;
 586                 }
 587         if (!(x & 0xff)) {
 588                 k += 8;
 589                 x >>= 8;
 590                 }
 591         if (!(x & 0xf)) {
 592                 k += 4;
 593                 x >>= 4;
 594                 }
 595         if (!(x & 0x3)) {
 596                 k += 2;
 597                 x >>= 2;
 598                 }
 599         if (!(x & 1)) {
 600                 k++;
 601                 x >>= 1;
 602                 if (!x & 1)
 603                         return 32;
 604                 }
 605         *y = x;
 606         return k;
 607         }
 608
 609  static Bigint *
 610 i2b
 611 #ifdef KR_headers
 612         (i) int i;
 613 #else
 614         (int i)
 615 #endif
 616 {
 617         Bigint *b;
 618
 619         b = Balloc(1);
 620         b->x[0] = i;
 621         b->wds = 1;
 622         return b;
 623         }
 624
 625  static Bigint *
 626 mult
 627 #ifdef KR_headers
 628         (a, b) Bigint *a, *b;
 629 #else
 630         (Bigint *a, Bigint *b)
 631 #endif
 632 {
 633         Bigint *c;
 634         int k, wa, wb, wc;
 635         ULong carry, y, z;
 636         ULong *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
 637 #ifdef Pack_32
 638         ULong z2;
 639 #endif
 640
 641         if (a->wds < b->wds) {
 642                 c = a;
 643                 a = b;
 644                 b = c;
 645                 }
 646         k = a->k;
 647         wa = a->wds;
 648         wb = b->wds;
 649         wc = wa + wb;
 650         if (wc > a->maxwds)
 651                 k++;
 652         c = Balloc(k);
 653         for(x = c->x, xa = x + wc; x < xa; x++)
 654                 *x = 0;
 655         xa = a->x;
 656         xae = xa + wa;
 657         xb = b->x;
 658         xbe = xb + wb;
 659         xc0 = c->x;
 660 #ifdef Pack_32
 661         for(; xb < xbe; xb++, xc0++) {
 662                 if (y = *xb & 0xffff) {
 663                         x = xa;
 664                         xc = xc0;
 665                         carry = 0;
 666                         do {
 667                                 z = (*x & 0xffff) * y + (*xc & 0xffff) + carry;
 668                                 carry = z >> 16;
 669                                 z2 = (*x++ >> 16) * y + (*xc >> 16) + carry;
 670                                 carry = z2 >> 16;
 671                                 Storeinc(xc, z2, z);
 672                                 }
 673                                 while(x < xae);
 674                         *xc = carry;
 675                         }
 676                 if (y = *xb >> 16) {
 677                         x = xa;
 678                         xc = xc0;
 679                         carry = 0;
 680                         z2 = *xc;
 681                         do {
 682                                 z = (*x & 0xffff) * y + (*xc >> 16) + carry;
 683                                 carry = z >> 16;
 684                                 Storeinc(xc, z, z2);
 685                                 z2 = (*x++ >> 16) * y + (*xc & 0xffff) + carry;
 686                                 carry = z2 >> 16;
 687                                 }
 688                                 while(x < xae);
 689                         *xc = z2;
 690                         }
 691                 }
 692 #else
 693         for(; xb < xbe; xc0++) {
 694                 if (y = *xb++) {
 695                         x = xa;
 696                         xc = xc0;
 697                         carry = 0;
 698                         do {
 699                                 z = *x++ * y + *xc + carry;
 700                                 carry = z >> 16;
 701                                 *xc++ = z & 0xffff;
 702                                 }
 703                                 while(x < xae);
 704                         *xc = carry;
 705                         }
 706                 }
 707 #endif
 708         for(xc0 = c->x, xc = xc0 + wc; wc > 0 && !*--xc; --wc) ;
 709         c->wds = wc;
 710         return c;
 711         }
 712
 713  static Bigint *p5s;
 714
 715  static Bigint *
 716 pow5mult
 717 #ifdef KR_headers
 718         (b, k) Bigint *b; int k;
 719 #else
 720         (Bigint *b, int k)
 721 #endif
 722 {
 723         Bigint *b1, *p5, *p51;
 724         int i;
 725         static int p05[3] = { 5, 25, 125 };
 726
 727         if (i = k & 3)
 728                 b = multadd(b, p05[i-1], 0);
 729
 730         if (!(k >>= 2))
 731                 return b;
 732         if (!(p5 = p5s)) {
 733                 /* first time */
 734                 p5 = p5s = i2b(625);
 735                 p5->next = 0;
 736                 }
 737         for(;;) {
 738                 if (k & 1) {
 739                         b1 = mult(b, p5);
 740                         Bfree(b);
 741                         b = b1;
 742                         }
 743                 if (!(k >>= 1))
 744                         break;
 745                 if (!(p51 = p5->next)) {
 746                         p51 = p5->next = mult(p5,p5);
 747                         p51->next = 0;
 748                         }
 749                 p5 = p51;
 750                 }
 751         return b;
 752         }
 753
 754  static Bigint *
 755 lshift
 756 #ifdef KR_headers
 757         (b, k) Bigint *b; int k;
 758 #else
 759         (Bigint *b, int k)
 760 #endif
 761 {
 762         int i, k1, n, n1;
 763         Bigint *b1;
 764         ULong *x, *x1, *xe, z;
 765
 766 #ifdef Pack_32
 767         n = k >> 5;
 768 #else
 769         n = k >> 4;
 770 #endif
 771         k1 = b->k;
 772         n1 = n + b->wds + 1;
 773         for(i = b->maxwds; n1 > i; i <<= 1)
 774                 k1++;
 775         b1 = Balloc(k1);
 776         x1 = b1->x;
 777         for(i = 0; i < n; i++)
 778                 *x1++ = 0;
 779         x = b->x;
 780         xe = x + b->wds;
 781 #ifdef Pack_32
 782         if (k &= 0x1f) {
 783                 k1 = 32 - k;
 784                 z = 0;
 785                 do {
 786                         *x1++ = *x << k | z;
 787                         z = *x++ >> k1;
 788                         }
 789                         while(x < xe);
 790                 if (*x1 = z)
 791                         ++n1;
 792                 }
 793 #else
 794         if (k &= 0xf) {
 795                 k1 = 16 - k;
 796                 z = 0;
 797                 do {
 798                         *x1++ = *x << k  & 0xffff | z;
 799                         z = *x++ >> k1;
 800                         }
 801                         while(x < xe);
 802                 if (*x1 = z)
 803                         ++n1;
 804                 }
 805 #endif
 806         else do
 807                 *x1++ = *x++;
 808                 while(x < xe);
 809         b1->wds = n1 - 1;
 810         Bfree(b);
 811         return b1;
 812         }
 813
 814  static int
 815 cmp
 816 #ifdef KR_headers
 817         (a, b) Bigint *a, *b;
 818 #else
 819         (Bigint *a, Bigint *b)
 820 #endif
 821 {
 822         ULong *xa, *xa0, *xb, *xb0;
 823         int i, j;
 824
 825         i = a->wds;
 826         j = b->wds;
 827 #ifdef DEBUG
 828         if (i > 1 && !a->x[i-1])
 829                 Bug("cmp called with a->x[a->wds-1] == 0");
 830         if (j > 1 && !b->x[j-1])
 831                 Bug("cmp called with b->x[b->wds-1] == 0");
 832 #endif
 833         if (i -= j)
 834                 return i;
 835         xa0 = a->x;
 836         xa = xa0 + j;
 837         xb0 = b->x;
 838         xb = xb0 + j;
 839         for(;;) {
 840                 if (*--xa != *--xb)
 841                         return *xa < *xb ? -1 : 1;
 842                 if (xa <= xa0)
 843                         break;
 844                 }
 845         return 0;
 846         }
 847
 848  static Bigint *
 849 diff
 850 #ifdef KR_headers
 851         (a, b) Bigint *a, *b;
 852 #else
 853         (Bigint *a, Bigint *b)
 854 #endif
 855 {
 856         Bigint *c;
 857         int i, wa, wb;
 858         Long borrow, y; /* We need signed shifts here. */
 859         ULong *xa, *xae, *xb, *xbe, *xc;
 860 #ifdef Pack_32
 861         Long z;
 862 #endif
 863
 864         i = cmp(a,b);
 865         if (!i) {
 866                 c = Balloc(0);
 867                 c->wds = 1;
 868                 c->x[0] = 0;
 869                 return c;
 870                 }
 871         if (i < 0) {
 872                 c = a;
 873                 a = b;
 874                 b = c;
 875                 i = 1;
 876                 }
 877         else
 878                 i = 0;
 879         c = Balloc(a->k);
 880         c->sign = i;
 881         wa = a->wds;
 882         xa = a->x;
 883         xae = xa + wa;
 884         wb = b->wds;
 885         xb = b->x;
 886         xbe = xb + wb;
 887         xc = c->x;
 888         borrow = 0;
 889 #ifdef Pack_32
 890         do {
 891                 y = (*xa & 0xffff) - (*xb & 0xffff) + borrow;
 892                 borrow = y >> 16;
 893                 Sign_Extend(borrow, y);
 894                 z = (*xa++ >> 16) - (*xb++ >> 16) + borrow;
 895                 borrow = z >> 16;
 896                 Sign_Extend(borrow, z);
 897                 Storeinc(xc, z, y);
 898                 }
 899                 while(xb < xbe);
 900         while(xa < xae) {
 901                 y = (*xa & 0xffff) + borrow;
 902                 borrow = y >> 16;
 903                 Sign_Extend(borrow, y);
 904                 z = (*xa++ >> 16) + borrow;
 905                 borrow = z >> 16;
 906                 Sign_Extend(borrow, z);
 907                 Storeinc(xc, z, y);
 908                 }
 909 #else
 910         do {
 911                 y = *xa++ - *xb++ + borrow;
 912                 borrow = y >> 16;
 913                 Sign_Extend(borrow, y);
 914                 *xc++ = y & 0xffff;
 915                 }
 916                 while(xb < xbe);
 917         while(xa < xae) {
 918                 y = *xa++ + borrow;
 919                 borrow = y >> 16;
 920                 Sign_Extend(borrow, y);
 921                 *xc++ = y & 0xffff;
 922                 }
 923 #endif
 924         while(!*--xc)
 925                 wa--;
 926         c->wds = wa;
 927         return c;
 928         }
 929
 930  static double
 931 ulp
 932 #ifdef KR_headers
 933         (x) double x;
 934 #else
 935         (double x)
 936 #endif
 937 {
 938         register Long L;
 939         double a;
 940
 941         L = (word0(x) & Exp_mask) - (P-1)*Exp_msk1;
 942 #ifndef Sudden_Underflow
 943         if (L > 0) {
 944 #endif
 945 #ifdef IBM
 946                 L |= Exp_msk1 >> 4;
 947 #endif
 948                 word0(a) = L;
 949                 word1(a) = 0;
 950 #ifndef Sudden_Underflow
 951                 }
 952         else {
 953                 L = -L >> Exp_shift;
 954                 if (L < Exp_shift) {
 955                         word0(a) = 0x80000 >> L;
 956                         word1(a) = 0;
 957                         }
 958                 else {
 959                         word0(a) = 0;
 960                         L -= Exp_shift;
 961                         word1(a) = L >= 31 ? 1 : 1 << 31 - L;
 962                         }
 963                 }
 964 #endif
 965         return a;
 966         }
 967
 968  static double
 969 b2d
 970 #ifdef KR_headers
 971         (a, e) Bigint *a; int *e;
 972 #else
 973         (Bigint *a, int *e)
 974 #endif
 975 {
 976         ULong *xa, *xa0, w, y, z;
 977         int k;
 978         double d;
 979 #ifdef VAX
 980         ULong d0, d1;
 981 #else
 982 #define d0 word0(d)
 983 #define d1 word1(d)
 984 #endif
 985
 986         xa0 = a->x;
 987         xa = xa0 + a->wds;
 988         y = *--xa;
 989 #ifdef DEBUG
 990         if (!y) Bug("zero y in b2d");
 991 #endif
 992         k = hi0bits(y);
 993         *e = 32 - k;
 994 #ifdef Pack_32
 995         if (k < Ebits) {
 996                 d0 = Exp_1 | y >> Ebits - k;
 997                 w = xa > xa0 ? *--xa : 0;
 998                 d1 = y << (32-Ebits) + k | w >> Ebits - k;
 999                 goto ret_d;
1000                 }
1001         z = xa > xa0 ? *--xa : 0;
1002         if (k -= Ebits) {
1003                 d0 = Exp_1 | y << k | z >> 32 - k;
1004                 y = xa > xa0 ? *--xa : 0;
1005                 d1 = z << k | y >> 32 - k;
1006                 }
1007         else {
1008                 d0 = Exp_1 | y;
1009                 d1 = z;
1010                 }
1011 #else
1012         if (k < Ebits + 16) {
1013                 z = xa > xa0 ? *--xa : 0;
1014                 d0 = Exp_1 | y << k - Ebits | z >> Ebits + 16 - k;
1015                 w = xa > xa0 ? *--xa : 0;
1016                 y = xa > xa0 ? *--xa : 0;
1017                 d1 = z << k + 16 - Ebits | w << k - Ebits | y >> 16 + Ebits - k;
1018                 goto ret_d;
1019                 }
1020         z = xa > xa0 ? *--xa : 0;
1021         w = xa > xa0 ? *--xa : 0;
1022         k -= Ebits + 16;
1023         d0 = Exp_1 | y << k + 16 | z << k | w >> 16 - k;
1024         y = xa > xa0 ? *--xa : 0;
1025         d1 = w << k + 16 | y << k;
1026 #endif
1027  ret_d:
1028 #ifdef VAX
1029         word0(d) = d0 >> 16 | d0 << 16;
1030         word1(d) = d1 >> 16 | d1 << 16;
1031 #else
1032 #undef d0
1033 #undef d1
1034 #endif
1035         return d;
1036         }
1037
1038  static Bigint *
1039 d2b
1040 #ifdef KR_headers
1041         (d, e, bits) double d; int *e, *bits;
1042 #else
1043         (double d, int *e, int *bits)
1044 #endif
1045 {
1046         Bigint *b;
1047         int de, i, k;
1048         ULong *x, y, z;
1049 #ifdef VAX
1050         ULong d0, d1;
1051         d0 = word0(d) >> 16 | word0(d) << 16;
1052         d1 = word1(d) >> 16 | word1(d) << 16;
1053 #else
1054 #define d0 word0(d)
1055 #define d1 word1(d)
1056 #endif
1057
1058 #ifdef Pack_32
1059         b = Balloc(1);
1060 #else
1061         b = Balloc(2);
1062 #endif
1063         x = b->x;
1064
1065         z = d0 & Frac_mask;
1066         d0 &= 0x7fffffff;       /* clear sign bit, which we ignore */
1067 #ifdef Sudden_Underflow
1068         de = (int)(d0 >> Exp_shift);
1069 #ifndef IBM
1070         z |= Exp_msk11;
1071 #endif
1072 #else
1073         if (de = (int)(d0 >> Exp_shift))
1074                 z |= Exp_msk1;
1075 #endif
1076 #ifdef Pack_32
1077         if (y = d1) {
1078                 if (k = lo0bits(&y)) {
1079                         x[0] = y | z << 32 - k;
1080                         z >>= k;
1081                         }
1082                 else
1083                         x[0] = y;
1084                 i = b->wds = (x[1] = z) ? 2 : 1;
1085                 }
1086         else {
1087 #ifdef DEBUG
1088                 if (!z)
1089                         Bug("Zero passed to d2b");
1090 #endif
1091                 k = lo0bits(&z);
1092                 x[0] = z;
1093                 i = b->wds = 1;
1094                 k += 32;
1095                 }
1096 #else
1097         if (y = d1) {
1098                 if (k = lo0bits(&y))
1099                         if (k >= 16) {
1100                                 x[0] = y | z << 32 - k & 0xffff;
1101                                 x[1] = z >> k - 16 & 0xffff;
1102                                 x[2] = z >> k;
1103                                 i = 2;
1104                                 }
1105                         else {
1106                                 x[0] = y & 0xffff;
1107                                 x[1] = y >> 16 | z << 16 - k & 0xffff;
1108                                 x[2] = z >> k & 0xffff;
1109                                 x[3] = z >> k+16;
1110                                 i = 3;
1111                                 }
1112                 else {
1113                         x[0] = y & 0xffff;
1114                         x[1] = y >> 16;
1115                         x[2] = z & 0xffff;
1116                         x[3] = z >> 16;
1117                         i = 3;
1118                         }
1119                 }
1120         else {
1121 #ifdef DEBUG
1122                 if (!z)
1123                         Bug("Zero passed to d2b");
1124 #endif
1125                 k = lo0bits(&z);
1126                 if (k >= 16) {
1127                         x[0] = z;
1128                         i = 0;
1129                         }
1130                 else {
1131                         x[0] = z & 0xffff;
1132                         x[1] = z >> 16;
1133                         i = 1;
1134                         }
1135                 k += 32;
1136                 }
1137         while(!x[i])
1138                 --i;
1139         b->wds = i + 1;
1140 #endif
1141 #ifndef Sudden_Underflow
1142         if (de) {
1143 #endif
1144 #ifdef IBM
1145                 *e = (de - Bias - (P-1) << 2) + k;
1146                 *bits = 4*P + 8 - k - hi0bits(word0(d) & Frac_mask);
1147 #else
1148                 *e = de - Bias - (P-1) + k;
1149                 *bits = P - k;
1150 #endif
1151 #ifndef Sudden_Underflow
1152                 }
1153         else {
1154                 *e = de - Bias - (P-1) + 1 + k;
1155 #ifdef Pack_32
1156                 *bits = 32*i - hi0bits(x[i-1]);
1157 #else
1158                 *bits = (i+2)*16 - hi0bits(x[i]);
1159 #endif
1160                 }
1161 #endif
1162         return b;
1163         }
1164 #undef d0
1165 #undef d1
1166
1167  static double
1168 ratio
1169 #ifdef KR_headers
1170         (a, b) Bigint *a, *b;
1171 #else
1172         (Bigint *a, Bigint *b)
1173 #endif
1174 {
1175         double da, db;
1176         int k, ka, kb;
1177
1178         da = b2d(a, &ka);
1179         db = b2d(b, &kb);
1180 #ifdef Pack_32
1181         k = ka - kb + 32*(a->wds - b->wds);
1182 #else
1183         k = ka - kb + 16*(a->wds - b->wds);
1184 #endif
1185 #ifdef IBM
1186         if (k > 0) {
1187                 word0(da) += (k >> 2)*Exp_msk1;
1188                 if (k &= 3)
1189                         da *= 1 << k;
1190                 }
1191         else {
1192                 k = -k;
1193                 word0(db) += (k >> 2)*Exp_msk1;
1194                 if (k &= 3)
1195                         db *= 1 << k;
1196                 }
1197 #else
1198         if (k > 0)
1199                 word0(da) += k*Exp_msk1;
1200         else {
1201                 k = -k;
1202                 word0(db) += k*Exp_msk1;
1203                 }
1204 #endif
1205         return da / db;
1206         }
1207
1208 static CONST double
1209 tens[] = {
1210                 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
1211                 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
1212                 1e20, 1e21, 1e22
1213 #ifdef VAX
1214                 , 1e23, 1e24
1215 #endif
1216                 };
1217
1218 #ifdef IEEE_Arith
1219 static CONST double bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 };
1220 static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128, 1e-256 };
1221 #define n_bigtens 5
1222 #else
1223 #ifdef IBM
1224 static CONST double bigtens[] = { 1e16, 1e32, 1e64 };
1225 static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64 };
1226 #define n_bigtens 3
1227 #else
1228 static CONST double bigtens[] = { 1e16, 1e32 };
1229 static CONST double tinytens[] = { 1e-16, 1e-32 };
1230 #define n_bigtens 2
1231 #endif
1232 #endif
1233
1234  double
1235 strtod
1236 #ifdef KR_headers
1237         (s00, se) CONST char *s00; char **se;
1238 #else
1239         (CONST char *s00, char **se)
1240 #endif
1241 {
1242         int bb2, bb5, bbe, bd2, bd5, bbbits, bs2, c, dsign,
1243                  e, e1, esign, i, j, k, nd, nd0, nf, nz, nz0, sign;
1244         CONST char *s, *s0, *s1;
1245         double aadj, aadj1, adj, rv, rv0;
1246         Long L;
1247         ULong y, z;
1248         Bigint *bb, *bb1, *bd, *bd0, *bs, *delta;
1249
1250 #ifndef KR_headers
1251         CONST char decimal_point = localeconv()->decimal_point[0];
1252 #else
1253         CONST char decimal_point = '.';
1254 #endif
1255
1256         sign = nz0 = nz = 0;
1257         rv = 0.;
1258
1259
1260         for(s = s00; isspace((unsigned char) *s); s++)
1261                 ;
1262
1263         if (*s == '-') {
1264                 sign = 1;
1265                 s++;
1266         } else if (*s == '+') {
1267                 s++;
1268         }
1269
1270         if (*s == '\0') {
1271                 s = s00;
1272                 goto ret;
1273         }
1274
1275         if (*s == '0') {
1276                 nz0 = 1;
1277                 while(*++s == '0') ;
1278                 if (!*s)
1279                         goto ret;
1280                 }
1281         s0 = s;
1282         y = z = 0;
1283         for(nd = nf = 0; (c = *s) >= '0' && c <= '9'; nd++, s++)
1284                 if (nd < 9)
1285                         y = 10*y + c - '0';
1286                 else if (nd < 16)
1287                         z = 10*z + c - '0';
1288         nd0 = nd;
1289         if (c == decimal_point) {
1290                 c = *++s;
1291                 if (!nd) {
1292                         for(; c == '0'; c = *++s)
1293                                 nz++;
1294                         if (c > '0' && c <= '9') {
1295                                 s0 = s;
1296                                 nf += nz;
1297                                 nz = 0;
1298                                 goto have_dig;
1299                                 }
1300                         goto dig_done;
1301                         }
1302                 for(; c >= '0' && c <= '9'; c = *++s) {
1303  have_dig:
1304                         nz++;
1305                         if (c -= '0') {
1306                                 nf += nz;
1307                                 for(i = 1; i < nz; i++)
1308                                         if (nd++ < 9)
1309                                                 y *= 10;
1310                                         else if (nd <= DBL_DIG + 1)
1311                                                 z *= 10;
1312                                 if (nd++ < 9)
1313                                         y = 10*y + c;
1314                                 else if (nd <= DBL_DIG + 1)
1315                                         z = 10*z + c;
1316                                 nz = 0;
1317                                 }
1318                         }
1319                 }
1320  dig_done:
1321         e = 0;
1322         if (c == 'e' || c == 'E') {
1323                 if (!nd && !nz && !nz0) {
1324                         s = s00;
1325                         goto ret;
1326                         }
1327                 s00 = s;
1328                 esign = 0;
1329                 switch(c = *++s) {
1330                         case '-':
1331                                 esign = 1;
1332                         case '+':
1333                                 c = *++s;
1334                         }
1335                 if (c >= '0' && c <= '9') {
1336                         while(c == '0')
1337                                 c = *++s;
1338                         if (c > '0' && c <= '9') {
1339                                 L = c - '0';
1340                                 s1 = s;
1341                                 while((c = *++s) >= '0' && c <= '9')
1342                                         L = 10*L + c - '0';
1343                                 if (s - s1 > 8 || L > 19999)
1344                                         /* Avoid confusion from exponents
1345                                          * so large that e might overflow.
1346                                          */
1347                                         e = 19999; /* safe for 16 bit ints */
1348                                 else
1349                                         e = (int)L;
1350                                 if (esign)
1351                                         e = -e;
1352                                 }
1353                         else
1354                                 e = 0;
1355                         }
1356                 else
1357                         s = s00;
1358                 }
1359         if (!nd) {
1360                 if (!nz && !nz0)
1361                         s = s00;
1362                 goto ret;
1363                 }
1364         e1 = e -= nf;
1365
1366         /* Now we have nd0 digits, starting at s0, followed by a
1367          * decimal point, followed by nd-nd0 digits.  The number we're
1368          * after is the integer represented by those digits times
1369          * 10**e */
1370
1371         if (!nd0)
1372                 nd0 = nd;
1373         k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1;
1374         rv = y;
1375         if (k > 9)
1376                 rv = tens[k - 9] * rv + z;
1377         bd0 = 0;
1378         if (nd <= DBL_DIG
1379 #ifndef RND_PRODQUOT
1380                 && FLT_ROUNDS == 1
1381 #endif
1382                         ) {
1383                 if (!e)
1384                         goto ret;
1385                 if (e > 0) {
1386                         if (e <= Ten_pmax) {
1387 #ifdef VAX
1388                                 goto vax_ovfl_check;
1389 #else
1390                                 /* rv = */ rounded_product(rv, tens[e]);
1391                                 goto ret;
1392 #endif
1393                                 }
1394                         i = DBL_DIG - nd;
1395                         if (e <= Ten_pmax + i) {
1396                                 /* A fancier test would sometimes let us do
1397                                  * this for larger i values.
1398                                  */
1399                                 e -= i;
1400                                 rv *= tens[i];
1401 #ifdef VAX
1402                                 /* VAX exponent range is so narrow we must
1403                                  * worry about overflow here...
1404                                  */
1405  vax_ovfl_check:
1406                                 word0(rv) -= P*Exp_msk1;
1407                                 /* rv = */ rounded_product(rv, tens[e]);
1408                                 if ((word0(rv) & Exp_mask)
1409                                  > Exp_msk1*(DBL_MAX_EXP+Bias-1-P))
1410                                         goto ovfl;
1411                                 word0(rv) += P*Exp_msk1;
1412 #else
1413                                 /* rv = */ rounded_product(rv, tens[e]);
1414 #endif
1415                                 goto ret;
1416                                 }
1417                         }
1418 #ifndef Inaccurate_Divide
1419                 else if (e >= -Ten_pmax) {
1420                         /* rv = */ rounded_quotient(rv, tens[-e]);
1421                         goto ret;
1422                         }
1423 #endif
1424                 }
1425         e1 += nd - k;
1426
1427         /* Get starting approximation = rv * 10**e1 */
1428
1429         if (e1 > 0) {
1430                 if (i = e1 & 15)
1431                         rv *= tens[i];
1432                 if (e1 &= ~15) {
1433                         if (e1 > DBL_MAX_10_EXP) {
1434  ovfl:
1435                                 errno = ERANGE;
1436 #ifdef __STDC__
1437                                 rv = HUGE_VAL;
1438 #else
1439                                 /* Can't trust HUGE_VAL */
1440 #ifdef IEEE_Arith
1441                                 word0(rv) = Exp_mask;
1442                                 word1(rv) = 0;
1443 #else
1444                                 word0(rv) = Big0;
1445                                 word1(rv) = Big1;
1446 #endif
1447 #endif
1448                                 if (bd0)
1449                                         goto retfree;
1450                                 goto ret;
1451                                 }
1452                         if (e1 >>= 4) {
1453                                 for(j = 0; e1 > 1; j++, e1 >>= 1)
1454                                         if (e1 & 1)
1455                                                 rv *= bigtens[j];
1456                         /* The last multiplication could overflow. */
1457                                 word0(rv) -= P*Exp_msk1;
1458                                 rv *= bigtens[j];
1459                                 if ((z = word0(rv) & Exp_mask)
1460                                  > Exp_msk1*(DBL_MAX_EXP+Bias-P))
1461                                         goto ovfl;
1462                                 if (z > Exp_msk1*(DBL_MAX_EXP+Bias-1-P)) {
1463                                         /* set to largest number */
1464                                         /* (Can't trust DBL_MAX) */
1465                                         word0(rv) = Big0;
1466                                         word1(rv) = Big1;
1467                                         }
1468                                 else
1469                                         word0(rv) += P*Exp_msk1;
1470                                 }
1471
1472                         }
1473                 }
1474         else if (e1 < 0) {
1475                 e1 = -e1;
1476                 if (i = e1 & 15)
1477                         rv /= tens[i];
1478                 if (e1 &= ~15) {
1479                         e1 >>= 4;
1480                         if (e1 >= 1 << n_bigtens)
1481                                 goto undfl;
1482                         for(j = 0; e1 > 1; j++, e1 >>= 1)
1483                                 if (e1 & 1)
1484                                         rv *= tinytens[j];
1485                         /* The last multiplication could underflow. */
1486                         rv0 = rv;
1487                         rv *= tinytens[j];
1488                         if (!rv) {
1489                                 rv = 2.*rv0;
1490                                 rv *= tinytens[j];
1491                                 if (!rv) {
1492  undfl:
1493                                         rv = 0.;
1494                                         errno = ERANGE;
1495                                         if (bd0)
1496                                                 goto retfree;
1497                                         goto ret;
1498                                         }
1499                                 word0(rv) = Tiny0;
1500                                 word1(rv) = Tiny1;
1501                                 /* The refinement below will clean
1502                                  * this approximation up.
1503                                  */
1504                                 }
1505                         }
1506                 }
1507
1508         /* Now the hard part -- adjusting rv to the correct value.*/
1509
1510         /* Put digits into bd: true value = bd * 10^e */
1511
1512         bd0 = s2b(s0, nd0, nd, y);
1513
1514         for(;;) {
1515                 bd = Balloc(bd0->k);
1516                 Bcopy(bd, bd0);
1517                 bb = d2b(rv, &bbe, &bbbits);    /* rv = bb * 2^bbe */
1518                 bs = i2b(1);
1519
1520                 if (e >= 0) {
1521                         bb2 = bb5 = 0;
1522                         bd2 = bd5 = e;
1523                         }
1524                 else {
1525                         bb2 = bb5 = -e;
1526                         bd2 = bd5 = 0;
1527                         }
1528                 if (bbe >= 0)
1529                         bb2 += bbe;
1530                 else
1531                         bd2 -= bbe;
1532                 bs2 = bb2;
1533 #ifdef Sudden_Underflow
1534 #ifdef IBM
1535                 j = 1 + 4*P - 3 - bbbits + ((bbe + bbbits - 1) & 3);
1536 #else
1537                 j = P + 1 - bbbits;
1538 #endif
1539 #else
1540                 i = bbe + bbbits - 1;   /* logb(rv) */
1541                 if (i < Emin)   /* denormal */
1542                         j = bbe + (P-Emin);
1543                 else
1544                         j = P + 1 - bbbits;
1545 #endif
1546                 bb2 += j;
1547                 bd2 += j;
1548                 i = bb2 < bd2 ? bb2 : bd2;
1549                 if (i > bs2)
1550                         i = bs2;
1551                 if (i > 0) {
1552                         bb2 -= i;
1553                         bd2 -= i;
1554                         bs2 -= i;
1555                         }
1556                 if (bb5 > 0) {
1557                         bs = pow5mult(bs, bb5);
1558                         bb1 = mult(bs, bb);
1559                         Bfree(bb);
1560                         bb = bb1;
1561                         }
1562                 if (bb2 > 0)
1563                         bb = lshift(bb, bb2);
1564                 if (bd5 > 0)
1565                         bd = pow5mult(bd, bd5);
1566                 if (bd2 > 0)
1567                         bd = lshift(bd, bd2);
1568                 if (bs2 > 0)
1569                         bs = lshift(bs, bs2);
1570                 delta = diff(bb, bd);
1571                 dsign = delta->sign;
1572                 delta->sign = 0;
1573                 i = cmp(delta, bs);
1574                 if (i < 0) {
1575                         /* Error is less than half an ulp -- check for
1576                          * special case of mantissa a power of two.
1577                          */
1578                         if (dsign || word1(rv) || word0(rv) & Bndry_mask)
1579                                 break;
1580                         delta = lshift(delta,Log2P);
1581                         if (cmp(delta, bs) > 0)
1582                                 goto drop_down;
1583                         break;
1584                         }
1585                 if (i == 0) {
1586                         /* exactly half-way between */
1587                         if (dsign) {
1588                                 if ((word0(rv) & Bndry_mask1) == Bndry_mask1
1589                                  &&  word1(rv) == 0xffffffff) {
1590                                         /*boundary case -- increment exponent*/
1591                                         word0(rv) = (word0(rv) & Exp_mask)
1592                                                 + Exp_msk1
1593 #ifdef IBM
1594                                                 | Exp_msk1 >> 4
1595 #endif
1596                                                 ;
1597                                         word1(rv) = 0;
1598                                         break;
1599                                         }
1600                                 }
1601                         else if (!(word0(rv) & Bndry_mask) && !word1(rv)) {
1602  drop_down:
1603                                 /* boundary case -- decrement exponent */
1604 #ifdef Sudden_Underflow
1605                                 L = word0(rv) & Exp_mask;
1606 #ifdef IBM
1607                                 if (L <  Exp_msk1)
1608 #else
1609                                 if (L <= Exp_msk1)
1610 #endif
1611                                         goto undfl;
1612                                 L -= Exp_msk1;
1613 #else
1614                                 L = (word0(rv) & Exp_mask) - Exp_msk1;
1615 #endif
1616                                 word0(rv) = L | Bndry_mask1;
1617                                 word1(rv) = 0xffffffff;
1618 #ifdef IBM
1619                                 goto cont;
1620 #else
1621                                 break;
1622 #endif
1623                                 }
1624 #ifndef ROUND_BIASED
1625                         if (!(word1(rv) & LSB))
1626                                 break;
1627 #endif
1628                         if (dsign)
1629                                 rv += ulp(rv);
1630 #ifndef ROUND_BIASED
1631                         else {
1632                                 rv -= ulp(rv);
1633 #ifndef Sudden_Underflow
1634                                 if (!rv)
1635                                         goto undfl;
1636 #endif
1637                                 }
1638 #endif
1639                         break;
1640                         }
1641                 if ((aadj = ratio(delta, bs)) <= 2.) {
1642                         if (dsign)
1643                                 aadj = aadj1 = 1.;
1644                         else if (word1(rv) || word0(rv) & Bndry_mask) {
1645 #ifndef Sudden_Underflow
1646                                 if (word1(rv) == Tiny1 && !word0(rv))
1647                                         goto undfl;
1648 #endif
1649                                 aadj = 1.;
1650                                 aadj1 = -1.;
1651                                 }
1652                         else {
1653                                 /* special case -- power of FLT_RADIX to be */
1654                                 /* rounded down... */
1655
1656                                 if (aadj < 2./FLT_RADIX)
1657                                         aadj = 1./FLT_RADIX;
1658                                 else
1659                                         aadj *= 0.5;
1660                                 aadj1 = -aadj;
1661                                 }
1662                         }
1663                 else {
1664                         aadj *= 0.5;
1665                         aadj1 = dsign ? aadj : -aadj;
1666 #ifdef Check_FLT_ROUNDS
1667                         switch(FLT_ROUNDS) {
1668                                 case 2: /* towards +infinity */
1669                                         aadj1 -= 0.5;
1670                                         break;
1671                                 case 0: /* towards 0 */
1672                                 case 3: /* towards -infinity */
1673                                         aadj1 += 0.5;
1674                                 }
1675 #else
1676                         if (FLT_ROUNDS == 0)
1677                                 aadj1 += 0.5;
1678 #endif
1679                         }
1680                 y = word0(rv) & Exp_mask;
1681
1682                 /* Check for overflow */
1683
1684                 if (y == Exp_msk1*(DBL_MAX_EXP+Bias-1)) {
1685                         rv0 = rv;
1686                         word0(rv) -= P*Exp_msk1;
1687                         adj = aadj1 * ulp(rv);
1688                         rv += adj;
1689                         if ((word0(rv) & Exp_mask) >=
1690                                         Exp_msk1*(DBL_MAX_EXP+Bias-P)) {
1691                                 if (word0(rv0) == Big0 && word1(rv0) == Big1)
1692                                         goto ovfl;
1693                                 word0(rv) = Big0;
1694                                 word1(rv) = Big1;
1695                                 goto cont;
1696                                 }
1697                         else
1698                                 word0(rv) += P*Exp_msk1;
1699                         }
1700                 else {
1701 #ifdef Sudden_Underflow
1702                         if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
1703                                 rv0 = rv;
1704                                 word0(rv) += P*Exp_msk1;
1705                                 adj = aadj1 * ulp(rv);
1706                                 rv += adj;
1707 #ifdef IBM
1708                                 if ((word0(rv) & Exp_mask) <  P*Exp_msk1)
1709 #else
1710                                 if ((word0(rv) & Exp_mask) <= P*Exp_msk1)
1711 #endif
1712                                         {
1713                                         if (word0(rv0) == Tiny0
1714                                          && word1(rv0) == Tiny1)
1715                                                 goto undfl;
1716                                         word0(rv) = Tiny0;
1717                                         word1(rv) = Tiny1;
1718                                         goto cont;
1719                                         }
1720                                 else
1721                                         word0(rv) -= P*Exp_msk1;
1722                                 }
1723                         else {
1724                                 adj = aadj1 * ulp(rv);
1725                                 rv += adj;
1726                                 }
1727 #else
1728                         /* Compute adj so that the IEEE rounding rules will
1729                          * correctly round rv + adj in some half-way cases.
1730                          * If rv * ulp(rv) is denormalized (i.e.,
1731                          * y <= (P-1)*Exp_msk1), we must adjust aadj to avoid
1732                          * trouble from bits lost to denormalization;
1733                          * example: 1.2e-307 .
1734                          */
1735                         if (y <= (P-1)*Exp_msk1 && aadj >= 1.) {
1736                                 aadj1 = (double)(int)(aadj + 0.5);
1737                                 if (!dsign)
1738                                         aadj1 = -aadj1;
1739                                 }
1740                         adj = aadj1 * ulp(rv);
1741                         rv += adj;
1742 #endif
1743                         }
1744                 z = word0(rv) & Exp_mask;
1745                 if (y == z) {
1746                         /* Can we stop now? */
1747                         L = aadj;
1748                         aadj -= L;
1749                         /* The tolerances below are conservative. */
1750                         if (dsign || word1(rv) || word0(rv) & Bndry_mask) {
1751                                 if (aadj < .4999999 || aadj > .5000001)
1752                                         break;
1753                                 }
1754                         else if (aadj < .4999999/FLT_RADIX)
1755                                 break;
1756                         }
1757  cont:
1758                 Bfree(bb);
1759                 Bfree(bd);
1760                 Bfree(bs);
1761                 Bfree(delta);
1762                 }
1763  retfree:
1764         Bfree(bb);
1765         Bfree(bd);
1766         Bfree(bs);
1767         Bfree(bd0);
1768         Bfree(delta);
1769  ret:
1770         if (se)
1771                 *se = (char *)s;
1772         return sign ? -rv : rv;
1773         }
1774
1775  static int
1776 quorem
1777 #ifdef KR_headers
1778         (b, S) Bigint *b, *S;
1779 #else
1780         (Bigint *b, Bigint *S)
1781 #endif
1782 {
1783         int n;
1784         Long borrow, y;
1785         ULong carry, q, ys;
1786         ULong *bx, *bxe, *sx, *sxe;
1787 #ifdef Pack_32
1788         Long z;
1789         ULong si, zs;
1790 #endif
1791
1792         n = S->wds;
1793 #ifdef DEBUG
1794         /*debug*/ if (b->wds > n)
1795         /*debug*/       Bug("oversize b in quorem");
1796 #endif
1797         if (b->wds < n)
1798                 return 0;
1799         sx = S->x;
1800         sxe = sx + --n;
1801         bx = b->x;
1802         bxe = bx + n;
1803         q = *bxe / (*sxe + 1);  /* ensure q <= true quotient */
1804 #ifdef DEBUG
1805         /*debug*/ if (q > 9)
1806         /*debug*/       Bug("oversized quotient in quorem");
1807 #endif
1808         if (q) {
1809                 borrow = 0;
1810                 carry = 0;
1811                 do {
1812 #ifdef Pack_32
1813                         si = *sx++;
1814                         ys = (si & 0xffff) * q + carry;
1815                         zs = (si >> 16) * q + (ys >> 16);
1816                         carry = zs >> 16;
1817                         y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
1818                         borrow = y >> 16;
1819                         Sign_Extend(borrow, y);
1820                         z = (*bx >> 16) - (zs & 0xffff) + borrow;
1821                         borrow = z >> 16;
1822                         Sign_Extend(borrow, z);
1823                         Storeinc(bx, z, y);
1824 #else
1825                         ys = *sx++ * q + carry;
1826                         carry = ys >> 16;
1827                         y = *bx - (ys & 0xffff) + borrow;
1828                         borrow = y >> 16;
1829                         Sign_Extend(borrow, y);
1830                         *bx++ = y & 0xffff;
1831 #endif
1832                         }
1833                         while(sx <= sxe);
1834                 if (!*bxe) {
1835                         bx = b->x;
1836                         while(--bxe > bx && !*bxe)
1837                                 --n;
1838                         b->wds = n;
1839                         }
1840                 }
1841         if (cmp(b, S) >= 0) {
1842                 q++;
1843                 borrow = 0;
1844                 carry = 0;
1845                 bx = b->x;
1846                 sx = S->x;
1847                 do {
1848 #ifdef Pack_32
1849                         si = *sx++;
1850                         ys = (si & 0xffff) + carry;
1851                         zs = (si >> 16) + (ys >> 16);
1852                         carry = zs >> 16;
1853                         y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
1854                         borrow = y >> 16;
1855                         Sign_Extend(borrow, y);
1856                         z = (*bx >> 16) - (zs & 0xffff) + borrow;
1857                         borrow = z >> 16;
1858                         Sign_Extend(borrow, z);
1859                         Storeinc(bx, z, y);
1860 #else
1861                         ys = *sx++ + carry;
1862                         carry = ys >> 16;
1863                         y = *bx - (ys & 0xffff) + borrow;
1864                         borrow = y >> 16;
1865                         Sign_Extend(borrow, y);
1866                         *bx++ = y & 0xffff;
1867 #endif
1868                         }
1869                         while(sx <= sxe);
1870                 bx = b->x;
1871                 bxe = bx + n;
1872                 if (!*bxe) {
1873                         while(--bxe > bx && !*bxe)
1874                                 --n;
1875                         b->wds = n;
1876                         }
1877                 }
1878         return q;
1879         }
1880
1881 /* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
1882  *
1883  * Inspired by "How to Print Floating-Point Numbers Accurately" by
1884  * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 92-101].
1885  *
1886  * Modifications:
1887  *      1. Rather than iterating, we use a simple numeric overestimate
1888  *         to determine k = floor(log10(d)).  We scale relevant
1889  *         quantities using O(log2(k)) rather than O(k) multiplications.
1890  *      2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
1891  *         try to generate digits strictly left to right.  Instead, we
1892  *         compute with fewer bits and propagate the carry if necessary
1893  *         when rounding the final digit up.  This is often faster.
1894  *      3. Under the assumption that input will be rounded nearest,
1895  *         mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
1896  *         That is, we allow equality in stopping tests when the
1897  *         round-nearest rule will give the same floating-point value
1898  *         as would satisfaction of the stopping test with strict
1899  *         inequality.
1900  *      4. We remove common factors of powers of 2 from relevant
1901  *         quantities.
1902  *      5. When converting floating-point integers less than 1e16,
1903  *         we use floating-point arithmetic rather than resorting
1904  *         to multiple-precision integers.
1905  *      6. When asked to produce fewer than 15 digits, we first try
1906  *         to get by with floating-point arithmetic; we resort to
1907  *         multiple-precision integer arithmetic only if we cannot
1908  *         guarantee that the floating-point calculation has given
1909  *         the correctly rounded result.  For k requested digits and
1910  *         "uniformly" distributed input, the probability is
1911  *         something like 10^(k-15) that we must resort to the Long
1912  *         calculation.
1913  */
1914
1915  char *
1916 __dtoa
1917 #ifdef KR_headers
1918         (d, mode, ndigits, decpt, sign, rve)
1919         double d; int mode, ndigits, *decpt, *sign; char **rve;
1920 #else
1921         (double d, int mode, int ndigits, int *decpt, int *sign, char **rve)
1922 #endif
1923 {
1924  /*     Arguments ndigits, decpt, sign are similar to those
1925         of ecvt and fcvt; trailing zeros are suppressed from
1926         the returned string.  If not null, *rve is set to point
1927         to the end of the return value.  If d is +-Infinity or NaN,
1928         then *decpt is set to 9999.
1929
1930         mode:
1931                 0 ==> shortest string that yields d when read in
1932                         and rounded to nearest.
1933                 1 ==> like 0, but with Steele & White stopping rule;
1934                         e.g. with IEEE P754 arithmetic , mode 0 gives
1935                         1e23 whereas mode 1 gives 9.999999999999999e22.
1936                 2 ==> max(1,ndigits) significant digits.  This gives a
1937                         return value similar to that of ecvt, except
1938                         that trailing zeros are suppressed.
1939                 3 ==> through ndigits past the decimal point.  This
1940                         gives a return value similar to that from fcvt,
1941                         except that trailing zeros are suppressed, and
1942                         ndigits can be negative.
1943                 4-9 should give the same return values as 2-3, i.e.,
1944                         4 <= mode <= 9 ==> same return as mode
1945                         2 + (mode & 1).  These modes are mainly for
1946                         debugging; often they run slower but sometimes
1947                         faster than modes 2-3.
1948                 4,5,8,9 ==> left-to-right digit generation.
1949                 6-9 ==> don't try fast floating-point estimate
1950                         (if applicable).
1951
1952                 Values of mode other than 0-9 are treated as mode 0.
1953
1954                 Sufficient space is allocated to the return value
1955                 to hold the suppressed trailing zeros.
1956         */
1957
1958         int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
1959                 j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
1960                 spec_case, try_quick;
1961         Long L;
1962 #ifndef Sudden_Underflow
1963         int denorm;
1964         ULong x;
1965 #endif
1966         Bigint *b, *b1, *delta, *mlo, *mhi, *S;
1967         double d2, ds, eps;
1968         char *s, *s0;
1969         static Bigint *result;
1970         static int result_k;
1971
1972         if (result) {
1973                 result->k = result_k;
1974                 result->maxwds = 1 << result_k;
1975                 Bfree(result);
1976                 result = 0;
1977                 }
1978
1979         if (word0(d) & Sign_bit) {
1980                 /* set sign for everything, including 0's and NaNs */
1981                 *sign = 1;
1982                 word0(d) &= ~Sign_bit;  /* clear sign bit */
1983                 }
1984         else
1985                 *sign = 0;
1986
1987 #if defined(IEEE_Arith) + defined(VAX)
1988 #ifdef IEEE_Arith
1989         if ((word0(d) & Exp_mask) == Exp_mask)
1990 #else
1991         if (word0(d)  == 0x8000)
1992 #endif
1993                 {
1994                 /* Infinity or NaN */
1995                 *decpt = 9999;
1996                 s =
1997 #ifdef IEEE_Arith
1998                         !word1(d) && !(word0(d) & 0xfffff) ? "Infinity" :
1999 #endif
2000                                 "NaN";
2001                 if (rve)
2002                         *rve =
2003 #ifdef IEEE_Arith
2004                                 s[3] ? s + 8 :
2005 #endif
2006                                                 s + 3;
2007                 return s;
2008                 }
2009 #endif
2010 #ifdef IBM
2011         d += 0; /* normalize */
2012 #endif
2013         if (!d) {
2014                 *decpt = 1;
2015                 s = "0";
2016                 if (rve)
2017                         *rve = s + 1;
2018                 return s;
2019                 }
2020
2021         b = d2b(d, &be, &bbits);
2022 #ifdef Sudden_Underflow
2023         i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1));
2024 #else
2025         if (i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1))) {
2026 #endif
2027                 d2 = d;
2028                 word0(d2) &= Frac_mask1;
2029                 word0(d2) |= Exp_11;
2030 #ifdef IBM
2031                 if (j = 11 - hi0bits(word0(d2) & Frac_mask))
2032                         d2 /= 1 << j;
2033 #endif
2034
2035                 /* log(x)       ~=~ log(1.5) + (x-1.5)/1.5
2036                  * log10(x)      =  log(x) / log(10)
2037                  *              ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
2038                  * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
2039                  *
2040                  * This suggests computing an approximation k to log10(d) by
2041                  *
2042                  * k = (i - Bias)*0.301029995663981
2043                  *      + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
2044                  *
2045                  * We want k to be too large rather than too small.
2046                  * The error in the first-order Taylor series approximation
2047                  * is in our favor, so we just round up the constant enough
2048                  * to compensate for any error in the multiplication of
2049                  * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
2050                  * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
2051                  * adding 1e-13 to the constant term more than suffices.
2052                  * Hence we adjust the constant term to 0.1760912590558.
2053                  * (We could get a more accurate k by invoking log10,
2054                  *  but this is probably not worthwhile.)
2055                  */
2056
2057                 i -= Bias;
2058 #ifdef IBM
2059                 i <<= 2;
2060                 i += j;
2061 #endif
2062 #ifndef Sudden_Underflow
2063                 denorm = 0;
2064                 }
2065         else {
2066                 /* d is denormalized */
2067
2068                 i = bbits + be + (Bias + (P-1) - 1);
2069                 x = i > 32  ? word0(d) << 64 - i | word1(d) >> i - 32
2070                             : word1(d) << 32 - i;
2071                 d2 = x;
2072                 word0(d2) -= 31*Exp_msk1; /* adjust exponent */
2073                 i -= (Bias + (P-1) - 1) + 1;
2074                 denorm = 1;
2075                 }
2076 #endif
2077         ds = (d2-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
2078         k = (int)ds;
2079         if (ds < 0. && ds != k)
2080                 k--;    /* want k = floor(ds) */
2081         k_check = 1;
2082         if (k >= 0 && k <= Ten_pmax) {
2083                 if (d < tens[k])
2084                         k--;
2085                 k_check = 0;
2086                 }
2087         j = bbits - i - 1;
2088         if (j >= 0) {
2089                 b2 = 0;
2090                 s2 = j;
2091                 }
2092         else {
2093                 b2 = -j;
2094                 s2 = 0;
2095                 }
2096         if (k >= 0) {
2097                 b5 = 0;
2098                 s5 = k;
2099                 s2 += k;
2100                 }
2101         else {
2102                 b2 -= k;
2103                 b5 = -k;
2104                 s5 = 0;
2105                 }
2106         if (mode < 0 || mode > 9)
2107                 mode = 0;
2108         try_quick = 1;
2109         if (mode > 5) {
2110                 mode -= 4;
2111                 try_quick = 0;
2112                 }
2113         leftright = 1;
2114         switch(mode) {
2115                 case 0:
2116                 case 1:
2117                         ilim = ilim1 = -1;
2118                         i = 18;
2119                         ndigits = 0;
2120                         break;
2121                 case 2:
2122                         leftright = 0;
2123                         /* no break */
2124                 case 4:
2125                         if (ndigits <= 0)
2126                                 ndigits = 1;
2127                         ilim = ilim1 = i = ndigits;
2128                         break;
2129                 case 3:
2130                         leftright = 0;
2131                         /* no break */
2132                 case 5:
2133                         i = ndigits + k + 1;
2134                         ilim = i;
2135                         ilim1 = i - 1;
2136                         if (i <= 0)
2137                                 i = 1;
2138                 }
2139         j = sizeof(ULong);
2140         for(result_k = 0; sizeof(Bigint) - sizeof(ULong) + j <= i;
2141                 j <<= 1) result_k++;
2142         result = Balloc(result_k);
2143         s = s0 = (char *)result;
2144
2145         if (ilim >= 0 && ilim <= Quick_max && try_quick) {
2146
2147                 /* Try to get by with floating-point arithmetic. */
2148
2149                 i = 0;
2150                 d2 = d;
2151                 k0 = k;
2152                 ilim0 = ilim;
2153                 ieps = 2; /* conservative */
2154                 if (k > 0) {
2155                         ds = tens[k&0xf];
2156                         j = k >> 4;
2157                         if (j & Bletch) {
2158                                 /* prevent overflows */
2159                                 j &= Bletch - 1;
2160                                 d /= bigtens[n_bigtens-1];
2161                                 ieps++;
2162                                 }
2163                         for(; j; j >>= 1, i++)
2164                                 if (j & 1) {
2165                                         ieps++;
2166                                         ds *= bigtens[i];
2167                                         }
2168                         d /= ds;
2169                         }
2170                 else if (j1 = -k) {
2171                         d *= tens[j1 & 0xf];
2172                         for(j = j1 >> 4; j; j >>= 1, i++)
2173                                 if (j & 1) {
2174                                         ieps++;
2175                                         d *= bigtens[i];
2176                                         }
2177                         }
2178                 if (k_check && d < 1. && ilim > 0) {
2179                         if (ilim1 <= 0)
2180                                 goto fast_failed;
2181                         ilim = ilim1;
2182                         k--;
2183                         d *= 10.;
2184                         ieps++;
2185                         }
2186                 eps = ieps*d + 7.;
2187                 word0(eps) -= (P-1)*Exp_msk1;
2188                 if (ilim == 0) {
2189                         S = mhi = 0;
2190                         d -= 5.;
2191                         if (d > eps)
2192                                 goto one_digit;
2193                         if (d < -eps)
2194                                 goto no_digits;
2195                         goto fast_failed;
2196                         }
2197 #ifndef No_leftright
2198                 if (leftright) {
2199                         /* Use Steele & White method of only
2200                          * generating digits needed.
2201                          */
2202                         eps = 0.5/tens[ilim-1] - eps;
2203                         for(i = 0;;) {
2204                                 L = d;
2205                                 d -= L;
2206                                 *s++ = '0' + (int)L;
2207                                 if (d < eps)
2208                                         goto ret1;
2209                                 if (1. - d < eps)
2210                                         goto bump_up;
2211                                 if (++i >= ilim)
2212                                         break;
2213                                 eps *= 10.;
2214                                 d *= 10.;
2215                                 }
2216                         }
2217                 else {
2218 #endif
2219                         /* Generate ilim digits, then fix them up. */
2220                         eps *= tens[ilim-1];
2221                         for(i = 1;; i++, d *= 10.) {
2222                                 L = d;
2223                                 d -= L;
2224                                 *s++ = '0' + (int)L;
2225                                 if (i == ilim) {
2226                                         if (d > 0.5 + eps)
2227                                                 goto bump_up;
2228                                         else if (d < 0.5 - eps) {
2229                                                 while(*--s == '0');
2230                                                 s++;
2231                                                 goto ret1;
2232                                                 }
2233                                         break;
2234                                         }
2235                                 }
2236 #ifndef No_leftright
2237                         }
2238 #endif
2239  fast_failed:
2240                 s = s0;
2241                 d = d2;
2242                 k = k0;
2243                 ilim = ilim0;
2244                 }
2245
2246         /* Do we have a "small" integer? */
2247
2248         if (be >= 0 && k <= Int_max) {
2249                 /* Yes. */
2250                 ds = tens[k];
2251                 if (ndigits < 0 && ilim <= 0) {
2252                         S = mhi = 0;
2253                         if (ilim < 0 || d <= 5*ds)
2254                                 goto no_digits;
2255                         goto one_digit;
2256                         }
2257                 for(i = 1;; i++) {
2258                         L = d / ds;
2259                         d -= L*ds;
2260 #ifdef Check_FLT_ROUNDS
2261                         /* If FLT_ROUNDS == 2, L will usually be high by 1 */
2262                         if (d < 0) {
2263                                 L--;
2264                                 d += ds;
2265                                 }
2266 #endif
2267                         *s++ = '0' + (int)L;
2268                         if (i == ilim) {
2269                                 d += d;
2270                                 if (d > ds || d == ds && L & 1) {
2271  bump_up:
2272                                         while(*--s == '9')
2273                                                 if (s == s0) {
2274                                                         k++;
2275                                                         *s = '0';
2276                                                         break;
2277                                                         }
2278                                         ++*s++;
2279                                         }
2280                                 break;
2281                                 }
2282                         if (!(d *= 10.))
2283                                 break;
2284                         }
2285                 goto ret1;
2286                 }
2287
2288         m2 = b2;
2289         m5 = b5;
2290         mhi = mlo = 0;
2291         if (leftright) {
2292                 if (mode < 2) {
2293                         i =
2294 #ifndef Sudden_Underflow
2295                                 denorm ? be + (Bias + (P-1) - 1 + 1) :
2296 #endif
2297 #ifdef IBM
2298                                 1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
2299 #else
2300                                 1 + P - bbits;
2301 #endif
2302                         }
2303                 else {
2304                         j = ilim - 1;
2305                         if (m5 >= j)
2306                                 m5 -= j;
2307                         else {
2308                                 s5 += j -= m5;
2309                                 b5 += j;
2310                                 m5 = 0;
2311                                 }
2312                         if ((i = ilim) < 0) {
2313                                 m2 -= i;
2314                                 i = 0;
2315                                 }
2316                         }
2317                 b2 += i;
2318                 s2 += i;
2319                 mhi = i2b(1);
2320                 }
2321         if (m2 > 0 && s2 > 0) {
2322                 i = m2 < s2 ? m2 : s2;
2323                 b2 -= i;
2324                 m2 -= i;
2325                 s2 -= i;
2326                 }
2327         if (b5 > 0) {
2328                 if (leftright) {
2329                         if (m5 > 0) {
2330                                 mhi = pow5mult(mhi, m5);
2331                                 b1 = mult(mhi, b);
2332                                 Bfree(b);
2333                                 b = b1;
2334                                 }
2335                         if (j = b5 - m5)
2336                                 b = pow5mult(b, j);
2337                         }
2338                 else
2339                         b = pow5mult(b, b5);
2340                 }
2341         S = i2b(1);
2342         if (s5 > 0)
2343                 S = pow5mult(S, s5);
2344
2345         /* Check for special case that d is a normalized power of 2. */
2346
2347         if (mode < 2) {
2348                 if (!word1(d) && !(word0(d) & Bndry_mask)
2349 #ifndef Sudden_Underflow
2350                  && word0(d) & Exp_mask
2351 #endif
2352                                 ) {
2353                         /* The special case */
2354                         b2 += Log2P;
2355                         s2 += Log2P;
2356                         spec_case = 1;
2357                         }
2358                 else
2359                         spec_case = 0;
2360                 }
2361
2362         /* Arrange for convenient computation of quotients:
2363          * shift left if necessary so divisor has 4 leading 0 bits.
2364          *
2365          * Perhaps we should just compute leading 28 bits of S once
2366          * and for all and pass them and a shift to quorem, so it
2367          * can do shifts and ors to compute the numerator for q.
2368          */
2369 #ifdef Pack_32
2370         if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f)
2371                 i = 32 - i;
2372 #else
2373         if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf)
2374                 i = 16 - i;
2375 #endif
2376         if (i > 4) {
2377                 i -= 4;
2378                 b2 += i;
2379                 m2 += i;
2380                 s2 += i;
2381                 }
2382         else if (i < 4) {
2383                 i += 28;
2384                 b2 += i;
2385                 m2 += i;
2386                 s2 += i;
2387                 }
2388         if (b2 > 0)
2389                 b = lshift(b, b2);
2390         if (s2 > 0)
2391                 S = lshift(S, s2);
2392         if (k_check) {
2393                 if (cmp(b,S) < 0) {
2394                         k--;
2395                         b = multadd(b, 10, 0);  /* we botched the k estimate */
2396                         if (leftright)
2397                                 mhi = multadd(mhi, 10, 0);
2398                         ilim = ilim1;
2399                         }
2400                 }
2401         if (ilim <= 0 && mode > 2) {
2402                 if (ilim < 0 || cmp(b,S = multadd(S,5,0)) <= 0) {
2403                         /* no digits, fcvt style */
2404  no_digits:
2405                         k = -1 - ndigits;
2406                         goto ret;
2407                         }
2408  one_digit:
2409                 *s++ = '1';
2410                 k++;
2411                 goto ret;
2412                 }
2413         if (leftright) {
2414                 if (m2 > 0)
2415                         mhi = lshift(mhi, m2);
2416
2417                 /* Compute mlo -- check for special case
2418                  * that d is a normalized power of 2.
2419                  */
2420
2421                 mlo = mhi;
2422                 if (spec_case) {
2423                         mhi = Balloc(mhi->k);
2424                         Bcopy(mhi, mlo);
2425                         mhi = lshift(mhi, Log2P);
2426                         }
2427
2428                 for(i = 1;;i++) {
2429                         dig = quorem(b,S) + '0';
2430                         /* Do we yet have the shortest decimal string
2431                          * that will round to d?
2432                          */
2433                         j = cmp(b, mlo);
2434                         delta = diff(S, mhi);
2435                         j1 = delta->sign ? 1 : cmp(b, delta);
2436                         Bfree(delta);
2437 #ifndef ROUND_BIASED
2438                         if (j1 == 0 && !mode && !(word1(d) & 1)) {
2439                                 if (dig == '9')
2440                                         goto round_9_up;
2441                                 if (j > 0)
2442                                         dig++;
2443                                 *s++ = dig;
2444                                 goto ret;
2445                                 }
2446 #endif
2447                         if (j < 0 || j == 0 && !mode
2448 #ifndef ROUND_BIASED
2449                                                         && !(word1(d) & 1)
2450 #endif
2451                                         ) {
2452                                 if (j1 > 0) {
2453                                         b = lshift(b, 1);
2454                                         j1 = cmp(b, S);
2455                                         if ((j1 > 0 || j1 == 0 && dig & 1)
2456                                         && dig++ == '9')
2457                                                 goto round_9_up;
2458                                         }
2459                                 *s++ = dig;
2460                                 goto ret;
2461                                 }
2462                         if (j1 > 0) {
2463                                 if (dig == '9') { /* possible if i == 1 */
2464  round_9_up:
2465                                         *s++ = '9';
2466                                         goto roundoff;
2467                                         }
2468                                 *s++ = dig + 1;
2469                                 goto ret;
2470                                 }
2471                         *s++ = dig;
2472                         if (i == ilim)
2473                                 break;
2474                         b = multadd(b, 10, 0);
2475                         if (mlo == mhi)
2476                                 mlo = mhi = multadd(mhi, 10, 0);
2477                         else {
2478                                 mlo = multadd(mlo, 10, 0);
2479                                 mhi = multadd(mhi, 10, 0);
2480                                 }
2481                         }
2482                 }
2483         else
2484                 for(i = 1;; i++) {
2485                         *s++ = dig = quorem(b,S) + '0';
2486                         if (i >= ilim)
2487                                 break;
2488                         b = multadd(b, 10, 0);
2489                         }
2490
2491         /* Round off last digit */
2492
2493         b = lshift(b, 1);
2494         j = cmp(b, S);
2495         if (j > 0 || j == 0 && dig & 1) {
2496  roundoff:
2497                 while(*--s == '9')
2498                         if (s == s0) {
2499                                 k++;
2500                                 *s++ = '1';
2501                                 goto ret;
2502                                 }
2503                 ++*s++;
2504                 }
2505         else {
2506                 while(*--s == '0');
2507                 s++;
2508                 }
2509  ret:
2510         Bfree(S);
2511         if (mhi) {
2512                 if (mlo && mlo != mhi)
2513                         Bfree(mlo);
2514                 Bfree(mhi);
2515                 }
2516  ret1:
2517         Bfree(b);
2518         if (s == s0) {                          /* don't return empty string */
2519                 *s++ = '0';
2520                 k = 0;
2521         }
2522         *s = 0;
2523         *decpt = k + 1;
2524         if (rve)
2525                 *rve = s;
2526         return s0;
2527         }
2528 #ifdef __cplusplus
2529 }
2530 #endif