1 /****************************************************************
3 * The author of this software is David M. Gay.
5 * Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
7 * Permission to use, copy, modify, and distribute this software for any
8 * purpose without fee is hereby granted, provided that this entire notice
9 * is included in all copies of any software which is or includes a copy
10 * or modification of this software and in all copies of the supporting
11 * documentation for such software.
13 * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
14 * WARRANTY. IN PARTICULAR, NEITHER THE AUTHOR NOR LUCENT MAKES ANY
15 * REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
16 * OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
18 ***************************************************************/
21 #define freedtoa __freedtoa
24 G_LOCK_DEFINE_STATIC(str_mutex0);
25 G_LOCK_DEFINE_STATIC(str_mutex1);
26 #define MULTIPLE_THREADS 1
27 #define ACQUIRE_DTOA_LOCK(n) G_LOCK (str_mutex##n)
28 #define FREE_DTOA_LOCK(n) G_UNLOCK (str_mutex##n)
30 /* Please send bug reports to David M. Gay (dmg at acm dot org,
31 * with " at " changed at "@" and " dot " changed to "."). */
33 /* On a machine with IEEE extended-precision registers, it is
34 * necessary to specify double-precision (53-bit) rounding precision
35 * before invoking strtod or dtoa. If the machine uses (the equivalent
36 * of) Intel 80x87 arithmetic, the call
37 * _control87(PC_53, MCW_PC);
38 * does this with many compilers. Whether this or another call is
39 * appropriate depends on the compiler; for this to work, it may be
40 * necessary to #include "float.h" or another system-dependent header
44 /* strtod for IEEE-, VAX-, and IBM-arithmetic machines.
46 * This strtod returns a nearest machine number to the input decimal
47 * string (or sets errno to ERANGE). With IEEE arithmetic, ties are
48 * broken by the IEEE round-even rule. Otherwise ties are broken by
49 * biased rounding (add half and chop).
51 * Inspired loosely by William D. Clinger's paper "How to Read Floating
52 * Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101].
56 * 1. We only require IEEE, IBM, or VAX double-precision
57 * arithmetic (not IEEE double-extended).
58 * 2. We get by with floating-point arithmetic in a case that
59 * Clinger missed -- when we're computing d * 10^n
60 * for a small integer d and the integer n is not too
61 * much larger than 22 (the maximum integer k for which
62 * we can represent 10^k exactly), we may be able to
63 * compute (d*10^k) * 10^(e-k) with just one roundoff.
64 * 3. Rather than a bit-at-a-time adjustment of the binary
65 * result in the hard case, we use floating-point
66 * arithmetic to determine the adjustment to within
67 * one bit; only in really hard cases do we need to
68 * compute a second residual.
69 * 4. Because of 3., we don't need a large table of powers of 10
70 * for ten-to-e (just some small tables, e.g. of 10^k
75 * #define IEEE_8087 for IEEE-arithmetic machines where the least
76 * significant byte has the lowest address.
77 * #define IEEE_MC68k for IEEE-arithmetic machines where the most
78 * significant byte has the lowest address.
79 * #define Long int on machines with 32-bit ints and 64-bit longs.
80 * #define IBM for IBM mainframe-style floating-point arithmetic.
81 * #define VAX for VAX-style floating-point arithmetic (D_floating).
82 * #define No_leftright to omit left-right logic in fast floating-point
83 * computation of dtoa.
84 * #define Honor_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
85 * and strtod and dtoa should round accordingly.
86 * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
87 * and Honor_FLT_ROUNDS is not #defined.
88 * #define RND_PRODQUOT to use rnd_prod and rnd_quot (assembly routines
89 * that use extended-precision instructions to compute rounded
90 * products and quotients) with IBM.
91 * #define ROUND_BIASED for IEEE-format with biased rounding.
92 * #define Inaccurate_Divide for IEEE-format with correctly rounded
93 * products but inaccurate quotients, e.g., for Intel i860.
94 * #define NO_LONG_LONG on machines that do not have a "long long"
95 * integer type (of >= 64 bits). On such machines, you can
96 * #define Just_16 to store 16 bits per 32-bit Long when doing
97 * high-precision integer arithmetic. Whether this speeds things
98 * up or slows things down depends on the machine and the number
99 * being converted. If long long is available and the name is
100 * something other than "long long", #define Llong to be the name,
101 * and if "unsigned Llong" does not work as an unsigned version of
102 * Llong, #define #ULLong to be the corresponding unsigned type.
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 * #define Omit_Private_Memory to omit logic (added Jan. 1998) for making
112 * memory allocations from a private pool of memory when possible.
113 * When used, the private pool is PRIVATE_MEM bytes long: 2304 bytes,
114 * unless #defined to be a different length. This default length
115 * suffices to get rid of MALLOC calls except for unusual cases,
116 * such as decimal-to-binary conversion of a very long string of
117 * digits. The longest string dtoa can return is about 751 bytes
118 * long. For conversions by strtod of strings of 800 digits and
119 * all dtoa conversions in single-threaded executions with 8-byte
120 * pointers, PRIVATE_MEM >= 7400 appears to suffice; with 4-byte
121 * pointers, PRIVATE_MEM >= 7112 appears adequate.
122 * #define INFNAN_CHECK on IEEE systems to cause strtod to check for
123 * Infinity and NaN (case insensitively). On some systems (e.g.,
124 * some HP systems), it may be necessary to #define NAN_WORD0
125 * appropriately -- to the most significant word of a quiet NaN.
126 * (On HP Series 700/800 machines, -DNAN_WORD0=0x7ff40000 works.)
127 * When INFNAN_CHECK is #defined and No_Hex_NaN is not #defined,
128 * strtod also accepts (case insensitively) strings of the form
129 * NaN(x), where x is a string of hexadecimal digits and spaces;
130 * if there is only one string of hexadecimal digits, it is taken
131 * for the 52 fraction bits of the resulting NaN; if there are two
132 * or more strings of hex digits, the first is for the high 20 bits,
133 * the second and subsequent for the low 32 bits, with intervening
134 * white space ignored; but if this results in none of the 52
135 * fraction bits being on (an IEEE Infinity symbol), then NAN_WORD0
136 * and NAN_WORD1 are used instead.
137 * #define MULTIPLE_THREADS if the system offers preemptively scheduled
138 * multiple threads. In this case, you must provide (or suitably
139 * #define) two locks, acquired by ACQUIRE_DTOA_LOCK(n) and freed
140 * by FREE_DTOA_LOCK(n) for n = 0 or 1. (The second lock, accessed
141 * in pow5mult, ensures lazy evaluation of only one copy of high
142 * powers of 5; omitting this lock would introduce a small
143 * probability of wasting memory, but would otherwise be harmless.)
144 * You must also invoke freedtoa(s) to free the value s returned by
145 * dtoa. You may do so whether or not MULTIPLE_THREADS is #defined.
146 * #define NO_IEEE_Scale to disable new (Feb. 1997) logic in strtod that
147 * avoids underflows on inputs whose result does not underflow.
148 * If you #define NO_IEEE_Scale on a machine that uses IEEE-format
149 * floating-point numbers and flushes underflows to zero rather
150 * than implementing gradual underflow, then you must also #define
152 * #define YES_ALIAS to permit aliasing certain double values with
153 * arrays of ULongs. This leads to slightly better code with
154 * some compilers and was always used prior to 19990916, but it
155 * is not strictly legal and can cause trouble with aggressively
156 * optimizing compilers (e.g., gcc 2.95.1 under -O2).
157 * #define USE_LOCALE to use the current locale's decimal_point value.
158 * #define SET_INEXACT if IEEE arithmetic is being used and extra
159 * computation should be done to set the inexact flag when the
160 * result is inexact and avoid setting inexact when the result
161 * is exact. In this case, dtoa.c must be compiled in
162 * an environment, perhaps provided by #include "dtoa.c" in a
163 * suitable wrapper, that defines two functions,
164 * int get_inexact(void);
165 * void clear_inexact(void);
166 * such that get_inexact() returns a nonzero value if the
167 * inexact bit is already set, and clear_inexact() sets the
168 * inexact bit to 0. When SET_INEXACT is #defined, strtod
169 * also does extra computations to set the underflow and overflow
170 * flags when appropriate (i.e., when the result is tiny and
171 * inexact or when it is a numeric value rounded to +-infinity).
172 * #define NO_ERRNO if strtod should not assign errno = ERANGE when
173 * the result overflows to +-Infinity or underflows to 0.
175 #if defined(i386) || defined(mips) && defined(MIPSEL) || defined (__arm__)
179 #elif defined(__x86_64__) || defined(__alpha__)
183 #elif defined(__ia64)
191 #elif defined(__hppa)
200 #define ULong guint32
204 #define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
217 extern char *MALLOC();
219 extern void *MALLOC(size_t);
222 #define MALLOC malloc
225 #define Omit_Private_Memory
226 #ifndef Omit_Private_Memory
228 #define PRIVATE_MEM 2304
230 #define PRIVATE_mem ((PRIVATE_MEM+sizeof(double)-1)/sizeof(double))
231 static double private_mem[PRIVATE_mem], *pmem_next = private_mem;
235 #undef Avoid_Underflow
249 #define DBL_MAX_10_EXP 308
250 #define DBL_MAX_EXP 1024
252 #endif /*IEEE_Arith*/
256 #define DBL_MAX_10_EXP 75
257 #define DBL_MAX_EXP 63
259 #define DBL_MAX 7.2370055773322621e+75
264 #define DBL_MAX_10_EXP 38
265 #define DBL_MAX_EXP 127
267 #define DBL_MAX 1.7014118346046923e+38
271 #define LONG_MAX 2147483647
274 #else /* ifndef Bad_float_h */
276 #endif /* Bad_float_h */
288 #define CONST /* blank */
294 #if defined(IEEE_8087) + defined(IEEE_MC68k) + defined(VAX) + defined(IBM) != 1
295 Exactly one of IEEE_8087, IEEE_MC68k, VAX, or IBM should be defined.
298 typedef union { double d; ULong L[2]; } U;
303 #define word0(x) ((ULong *)&x)[1]
304 #define word1(x) ((ULong *)&x)[0]
306 #define word0(x) ((ULong *)&x)[0]
307 #define word1(x) ((ULong *)&x)[1]
311 #define word0(x) ((U*)&x)->L[1]
312 #define word1(x) ((U*)&x)->L[0]
314 #define word0(x) ((U*)&x)->L[0]
315 #define word1(x) ((U*)&x)->L[1]
317 #define dval(x) ((U*)&x)->d
320 /* The following definition of Storeinc is appropriate for MIPS processors.
321 * An alternative that might be better on some machines is
322 * #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
324 #if defined(IEEE_8087) + defined(VAX)
325 #define Storeinc(a,b,c) do { (((unsigned short *)a)[1] = (unsigned short)b, \
326 ((unsigned short *)a)[0] = (unsigned short)c, a++) } while (0)
328 #define Storeinc(a,b,c) do { (((unsigned short *)a)[0] = (unsigned short)b, \
329 ((unsigned short *)a)[1] = (unsigned short)c, a++) } while (0)
332 /* #define P DBL_MANT_DIG */
333 /* Ten_pmax = floor(P*log(2)/log(5)) */
334 /* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
335 /* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
336 /* Int_max = floor(P*log(FLT_RADIX)/log(10) - 1) */
340 #define Exp_shift1 20
341 #define Exp_msk1 0x100000
342 #define Exp_msk11 0x100000
343 #define Exp_mask 0x7ff00000
347 #define Exp_1 0x3ff00000
348 #define Exp_11 0x3ff00000
350 #define Frac_mask 0xfffff
351 #define Frac_mask1 0xfffff
354 #define Bndry_mask 0xfffff
355 #define Bndry_mask1 0xfffff
357 #define Sign_bit 0x80000000
363 #ifndef NO_IEEE_Scale
364 #define Avoid_Underflow
365 #ifdef Flush_Denorm /* debugging option */
366 #undef Sudden_Underflow
372 #define Flt_Rounds FLT_ROUNDS
376 #endif /*Flt_Rounds*/
378 #ifdef Honor_FLT_ROUNDS
379 #define Rounding rounding
380 #undef Check_FLT_ROUNDS
381 #define Check_FLT_ROUNDS
383 #define Rounding Flt_Rounds
386 #else /* ifndef IEEE_Arith */
387 #undef Check_FLT_ROUNDS
388 #undef Honor_FLT_ROUNDS
390 #undef Sudden_Underflow
391 #define Sudden_Underflow
396 #define Exp_shift1 24
397 #define Exp_msk1 0x1000000
398 #define Exp_msk11 0x1000000
399 #define Exp_mask 0x7f000000
402 #define Exp_1 0x41000000
403 #define Exp_11 0x41000000
404 #define Ebits 8 /* exponent has 7 bits, but 8 is the right value in b2d */
405 #define Frac_mask 0xffffff
406 #define Frac_mask1 0xffffff
409 #define Bndry_mask 0xefffff
410 #define Bndry_mask1 0xffffff
412 #define Sign_bit 0x80000000
414 #define Tiny0 0x100000
423 #define Exp_msk1 0x80
424 #define Exp_msk11 0x800000
425 #define Exp_mask 0x7f80
428 #define Exp_1 0x40800000
429 #define Exp_11 0x4080
431 #define Frac_mask 0x7fffff
432 #define Frac_mask1 0xffff007f
435 #define Bndry_mask 0xffff007f
436 #define Bndry_mask1 0xffff007f
438 #define Sign_bit 0x8000
444 #endif /* IBM, VAX */
445 #endif /* IEEE_Arith */
452 #define rounded_product(a,b) a = rnd_prod(a, b)
453 #define rounded_quotient(a,b) a = rnd_quot(a, b)
455 extern double rnd_prod(), rnd_quot();
457 extern double rnd_prod(double, double), rnd_quot(double, double);
460 #define rounded_product(a,b) a *= b
461 #define rounded_quotient(a,b) a /= b
464 #define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
465 #define Big1 0xffffffff
472 #define FFFFFFFF ((((unsigned long)0xffff)<<16)|(unsigned long)0xffff)
474 #define FFFFFFFF 0xffffffffUL
481 /* When Pack_32 is not defined, we store 16 bits per 32-bit Long.
482 * This makes some inner loops simpler and sometimes saves work
483 * during multiplications, but it often seems to make things slightly
484 * slower. Hence the default is now to store 32 bits per Long.
487 #else /* long long available */
489 #define Llong long long
492 #define ULLong unsigned Llong
494 #endif /* NO_LONG_LONG */
496 #ifndef MULTIPLE_THREADS
497 #define ACQUIRE_DTOA_LOCK(n) /*nothing*/
498 #define FREE_DTOA_LOCK(n) /*nothing*/
504 extern "C" double strtod(const char *s00, char **se);
505 extern "C" char *dtoa(double d, int mode, int ndigits,
506 int *decpt, int *sign, char **rve);
512 int k, maxwds, sign, wds;
516 typedef struct Bigint Bigint;
518 static Bigint *freelist[Kmax+1];
530 #ifndef Omit_Private_Memory
534 ACQUIRE_DTOA_LOCK(0);
535 if ((rv = freelist[k])) {
536 freelist[k] = rv->next;
540 #ifdef Omit_Private_Memory
541 rv = (Bigint *)MALLOC(sizeof(Bigint) + (x-1)*sizeof(ULong));
543 len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
545 if (pmem_next - private_mem + len <= PRIVATE_mem) {
546 rv = (Bigint*)pmem_next;
550 rv = (Bigint*)MALLOC(len*sizeof(double));
556 rv->sign = rv->wds = 0;
569 ACQUIRE_DTOA_LOCK(0);
570 v->next = freelist[v->k];
576 #define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
577 y->wds*sizeof(Long) + 2*sizeof(int))
582 (b, m, a) Bigint *b; int m, a;
584 (Bigint *b, int m, int a) /* multiply by m and add a */
605 y = *x * (ULLong)m + carry;
611 y = (xi & 0xffff) * m + carry;
612 z = (xi >> 16) * m + (y >> 16);
614 *x++ = (z << 16) + (y & 0xffff);
624 if (wds >= b->maxwds) {
639 (s, nd0, nd, y9) CONST char *s; int nd0, nd; ULong y9;
641 (CONST char *s, int nd0, int nd, ULong y9)
649 for(k = 0, y = 1; x > y; y <<= 1, k++) ;
656 b->x[0] = y9 & 0xffff;
657 b->wds = (b->x[1] = y9 >> 16) ? 2 : 1;
663 do b = multadd(b, 10, *s++ - '0');
670 b = multadd(b, 10, *s++ - '0');
677 (x) register ULong x;
684 if (!(x & 0xffff0000)) {
688 if (!(x & 0xff000000)) {
692 if (!(x & 0xf0000000)) {
696 if (!(x & 0xc0000000)) {
700 if (!(x & 0x80000000)) {
702 if (!(x & 0x40000000))
717 register ULong x = *y;
775 (a, b) Bigint *a, *b;
777 (Bigint *a, Bigint *b)
782 ULong *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
793 if (a->wds < b->wds) {
805 for(x = c->x, xa = x + wc; x < xa; x++)
813 for(; xb < xbe; xc0++) {
819 z = *x++ * (ULLong)y + *xc + carry;
821 *xc++ = z & FFFFFFFF;
829 for(; xb < xbe; xb++, xc0++) {
830 if (y = *xb & 0xffff) {
835 z = (*x & 0xffff) * y + (*xc & 0xffff) + carry;
837 z2 = (*x++ >> 16) * y + (*xc >> 16) + carry;
850 z = (*x & 0xffff) * y + (*xc >> 16) + carry;
853 z2 = (*x++ >> 16) * y + (*xc & 0xffff) + carry;
861 for(; xb < xbe; xc0++) {
867 z = *x++ * y + *xc + carry;
877 for(xc0 = c->x, xc = xc0 + wc; wc > 0 && !*--xc; --wc) ;
887 (b, k) Bigint *b; int k;
892 Bigint *b1, *p5, *p51;
894 static int p05[3] = { 5, 25, 125 };
897 b = multadd(b, p05[i-1], 0);
903 #ifdef MULTIPLE_THREADS
904 ACQUIRE_DTOA_LOCK(1);
923 if (!(p51 = p5->next)) {
924 #ifdef MULTIPLE_THREADS
925 ACQUIRE_DTOA_LOCK(1);
926 if (!(p51 = p5->next)) {
927 p51 = p5->next = mult(p5,p5);
932 p51 = p5->next = mult(p5,p5);
944 (b, k) Bigint *b; int k;
951 ULong *x, *x1, *xe, z;
960 for(i = b->maxwds; n1 > i; i <<= 1)
964 for(i = 0; i < n; i++)
985 *x1++ = *x << k & 0xffff | z;
1004 (a, b) Bigint *a, *b;
1006 (Bigint *a, Bigint *b)
1009 ULong *xa, *xa0, *xb, *xb0;
1015 if (i > 1 && !a->x[i-1])
1016 Bug("cmp called with a->x[a->wds-1] == 0");
1017 if (j > 1 && !b->x[j-1])
1018 Bug("cmp called with b->x[b->wds-1] == 0");
1028 return *xa < *xb ? -1 : 1;
1038 (a, b) Bigint *a, *b;
1040 (Bigint *a, Bigint *b)
1045 ULong *xa, *xae, *xb, *xbe, *xc;
1082 y = (ULLong)*xa++ - *xb++ - borrow;
1083 borrow = y >> 32 & (ULong)1;
1084 *xc++ = y & FFFFFFFF;
1089 borrow = y >> 32 & (ULong)1;
1090 *xc++ = y & FFFFFFFF;
1095 y = (*xa & 0xffff) - (*xb & 0xffff) - borrow;
1096 borrow = (y & 0x10000) >> 16;
1097 z = (*xa++ >> 16) - (*xb++ >> 16) - borrow;
1098 borrow = (z & 0x10000) >> 16;
1103 y = (*xa & 0xffff) - borrow;
1104 borrow = (y & 0x10000) >> 16;
1105 z = (*xa++ >> 16) - borrow;
1106 borrow = (z & 0x10000) >> 16;
1111 y = *xa++ - *xb++ - borrow;
1112 borrow = (y & 0x10000) >> 16;
1118 borrow = (y & 0x10000) >> 16;
1140 L = (word0(x) & Exp_mask) - (P-1)*Exp_msk1;
1141 #ifndef Avoid_Underflow
1142 #ifndef Sudden_Underflow
1151 #ifndef Avoid_Underflow
1152 #ifndef Sudden_Underflow
1155 L = -L >> Exp_shift;
1156 if (L < Exp_shift) {
1157 word0(a) = 0x80000 >> L;
1163 word1(a) = L >= 31 ? 1 : 1 << 31 - L;
1174 (a, e) Bigint *a; int *e;
1179 ULong *xa, *xa0, w, y, z;
1193 if (!y) Bug("zero y in b2d");
1199 d0 = Exp_1 | (y >> (Ebits - k));
1200 w = xa > xa0 ? *--xa : 0;
1201 d1 = y << ((32-Ebits) + k) | (w >> (Ebits - k));
1204 z = xa > xa0 ? *--xa : 0;
1206 d0 = Exp_1 | y << k | (z >> (32 - k));
1207 y = xa > xa0 ? *--xa : 0;
1208 d1 = z << k | (y >> (32 - k));
1215 if (k < Ebits + 16) {
1216 z = xa > xa0 ? *--xa : 0;
1217 d0 = Exp_1 | y << k - Ebits | z >> Ebits + 16 - k;
1218 w = xa > xa0 ? *--xa : 0;
1219 y = xa > xa0 ? *--xa : 0;
1220 d1 = z << k + 16 - Ebits | w << k - Ebits | y >> 16 + Ebits - k;
1223 z = xa > xa0 ? *--xa : 0;
1224 w = xa > xa0 ? *--xa : 0;
1226 d0 = Exp_1 | y << k + 16 | z << k | w >> 16 - k;
1227 y = xa > xa0 ? *--xa : 0;
1228 d1 = w << k + 16 | y << k;
1232 word0(d) = d0 >> 16 | d0 << 16;
1233 word1(d) = d1 >> 16 | d1 << 16;
1244 (d, e, bits) double d; int *e, *bits;
1246 (double d, int *e, int *bits)
1252 #ifndef Sudden_Underflow
1257 d0 = word0(d) >> 16 | word0(d) << 16;
1258 d1 = word1(d) >> 16 | word1(d) << 16;
1272 d0 &= 0x7fffffff; /* clear sign bit, which we ignore */
1273 #ifdef Sudden_Underflow
1274 de = (int)(d0 >> Exp_shift);
1279 if ((de = (int)(d0 >> Exp_shift)))
1284 if ((k = lo0bits(&y))) {
1285 x[0] = y | (z << (32 - k));
1290 #ifndef Sudden_Underflow
1293 b->wds = (x[1] = z) ? 2 : 1;
1298 Bug("Zero passed to d2b");
1302 #ifndef Sudden_Underflow
1310 if (k = lo0bits(&y))
1312 x[0] = y | z << 32 - k & 0xffff;
1313 x[1] = z >> k - 16 & 0xffff;
1319 x[1] = y >> 16 | z << 16 - k & 0xffff;
1320 x[2] = z >> k & 0xffff;
1335 Bug("Zero passed to d2b");
1353 #ifndef Sudden_Underflow
1357 *e = (de - Bias - (P-1) << 2) + k;
1358 *bits = 4*P + 8 - k - hi0bits(word0(d) & Frac_mask);
1360 *e = de - Bias - (P-1) + k;
1363 #ifndef Sudden_Underflow
1366 *e = de - Bias - (P-1) + 1 + k;
1368 *bits = 32*i - hi0bits(x[i-1]);
1370 *bits = (i+2)*16 - hi0bits(x[i]);
1382 (a, b) Bigint *a, *b;
1384 (Bigint *a, Bigint *b)
1390 dval(da) = b2d(a, &ka);
1391 dval(db) = b2d(b, &kb);
1393 k = ka - kb + 32*(a->wds - b->wds);
1395 k = ka - kb + 16*(a->wds - b->wds);
1399 word0(da) += (k >> 2)*Exp_msk1;
1405 word0(db) += (k >> 2)*Exp_msk1;
1411 word0(da) += k*Exp_msk1;
1414 word0(db) += k*Exp_msk1;
1417 return dval(da) / dval(db);
1422 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
1423 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
1432 bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 };
1433 static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128,
1434 #ifdef Avoid_Underflow
1435 9007199254740992.*9007199254740992.e-256
1436 /* = 2^106 * 1e-53 */
1441 /* The factor of 2^53 in tinytens[4] helps us avoid setting the underflow */
1442 /* flag unnecessarily. It leads to a song and dance at the end of strtod. */
1443 #define Scale_Bit 0x10
1447 bigtens[] = { 1e16, 1e32, 1e64 };
1448 static CONST double tinytens[] = { 1e-16, 1e-32, 1e-64 };
1451 bigtens[] = { 1e16, 1e32 };
1452 static CONST double tinytens[] = { 1e-16, 1e-32 };
1464 #define NAN_WORD0 0x7ff80000
1474 (sp, t) char **sp, *t;
1476 (CONST char **sp, char *t)
1480 CONST char *s = *sp;
1483 if ((c = *++s) >= 'A' && c <= 'Z')
1496 (rvp, sp) double *rvp; CONST char **sp;
1498 (double *rvp, CONST char **sp)
1503 int havedig, udx0, xshift;
1506 havedig = xshift = 0;
1509 while(c = *(CONST unsigned char*)++s) {
1510 if (c >= '0' && c <= '9')
1512 else if (c >= 'a' && c <= 'f')
1514 else if (c >= 'A' && c <= 'F')
1516 else if (c <= ' ') {
1517 if (udx0 && havedig) {
1523 else if (/*(*/ c == ')' && havedig) {
1528 return; /* invalid form: don't change *sp */
1536 x[0] = (x[0] << 4) | (x[1] >> 28);
1537 x[1] = (x[1] << 4) | c;
1539 if ((x[0] &= 0xfffff) || x[1]) {
1540 word0(*rvp) = Exp_mask | x[0];
1544 #endif /*No_Hex_NaN*/
1545 #endif /* INFNAN_CHECK */
1550 (s00, se) CONST char *s00; char **se;
1552 (CONST char *s00, char **se)
1555 #ifdef Avoid_Underflow
1558 int bb2, bb5, bbe, bd2, bd5, bbbits, bs2, c, dsign,
1559 e, e1, esign, i, j, k, nd, nd0, nf, nz, nz0, sign;
1560 CONST char *s, *s0, *s1;
1561 double aadj, aadj1, adj, rv, rv0;
1564 Bigint *bb, *bb1, *bd, *bd0, *bs, *delta;
1566 int inexact, oldinexact;
1568 #ifdef Honor_FLT_ROUNDS
1575 sign = nz0 = nz = 0;
1577 for(s = s00;;s++) switch(*s) {
1600 while(*++s == '0') ;
1606 for(nd = nf = 0; (c = *s) >= '0' && c <= '9'; nd++, s++)
1613 s1 = localeconv()->decimal_point;
1634 for(; c == '0'; c = *++s)
1636 if (c > '0' && c <= '9') {
1644 for(; c >= '0' && c <= '9'; c = *++s) {
1649 for(i = 1; i < nz; i++)
1652 else if (nd <= DBL_DIG + 1)
1656 else if (nd <= DBL_DIG + 1)
1664 if (c == 'e' || c == 'E') {
1665 if (!nd && !nz && !nz0) {
1676 if (c >= '0' && c <= '9') {
1679 if (c > '0' && c <= '9') {
1682 while((c = *++s) >= '0' && c <= '9')
1684 if (s - s1 > 8 || L > 19999)
1685 /* Avoid confusion from exponents
1686 * so large that e might overflow.
1688 e = 19999; /* safe for 16 bit ints */
1703 /* Check for Nan and Infinity */
1707 if (match(&s,"nf")) {
1709 if (!match(&s,"inity"))
1711 word0(rv) = 0x7ff00000;
1718 if (match(&s, "an")) {
1719 word0(rv) = NAN_WORD0;
1720 word1(rv) = NAN_WORD1;
1722 if (*s == '(') /*)*/
1728 #endif /* INFNAN_CHECK */
1737 /* Now we have nd0 digits, starting at s0, followed by a
1738 * decimal point, followed by nd-nd0 digits. The number we're
1739 * after is the integer represented by those digits times
1744 k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1;
1749 oldinexact = get_inexact();
1751 dval(rv) = tens[k - 9] * dval(rv) + z;
1755 #ifndef RND_PRODQUOT
1756 #ifndef Honor_FLT_ROUNDS
1764 if (e <= Ten_pmax) {
1766 goto vax_ovfl_check;
1768 #ifdef Honor_FLT_ROUNDS
1769 /* round correctly FLT_ROUNDS = 2 or 3 */
1775 /* rv = */ rounded_product(dval(rv), tens[e]);
1780 if (e <= Ten_pmax + i) {
1781 /* A fancier test would sometimes let us do
1782 * this for larger i values.
1784 #ifdef Honor_FLT_ROUNDS
1785 /* round correctly FLT_ROUNDS = 2 or 3 */
1792 dval(rv) *= tens[i];
1794 /* VAX exponent range is so narrow we must
1795 * worry about overflow here...
1798 word0(rv) -= P*Exp_msk1;
1799 /* rv = */ rounded_product(dval(rv), tens[e]);
1800 if ((word0(rv) & Exp_mask)
1801 > Exp_msk1*(DBL_MAX_EXP+Bias-1-P))
1803 word0(rv) += P*Exp_msk1;
1805 /* rv = */ rounded_product(dval(rv), tens[e]);
1810 #ifndef Inaccurate_Divide
1811 else if (e >= -Ten_pmax) {
1812 #ifdef Honor_FLT_ROUNDS
1813 /* round correctly FLT_ROUNDS = 2 or 3 */
1819 /* rv = */ rounded_quotient(dval(rv), tens[-e]);
1830 oldinexact = get_inexact();
1832 #ifdef Avoid_Underflow
1835 #ifdef Honor_FLT_ROUNDS
1836 if ((rounding = Flt_Rounds) >= 2) {
1838 rounding = rounding == 2 ? 0 : 2;
1844 #endif /*IEEE_Arith*/
1846 /* Get starting approximation = rv * 10**e1 */
1850 dval(rv) *= tens[i];
1852 if (e1 > DBL_MAX_10_EXP) {
1857 /* Can't trust HUGE_VAL */
1859 #ifdef Honor_FLT_ROUNDS
1861 case 0: /* toward 0 */
1862 case 3: /* toward -infinity */
1867 word0(rv) = Exp_mask;
1870 #else /*Honor_FLT_ROUNDS*/
1871 word0(rv) = Exp_mask;
1873 #endif /*Honor_FLT_ROUNDS*/
1875 /* set overflow bit */
1877 dval(rv0) *= dval(rv0);
1879 #else /*IEEE_Arith*/
1882 #endif /*IEEE_Arith*/
1888 for(j = 0; e1 > 1; j++, e1 >>= 1)
1890 dval(rv) *= bigtens[j];
1891 /* The last multiplication could overflow. */
1892 word0(rv) -= P*Exp_msk1;
1893 dval(rv) *= bigtens[j];
1894 if ((z = word0(rv) & Exp_mask)
1895 > Exp_msk1*(DBL_MAX_EXP+Bias-P))
1897 if (z > Exp_msk1*(DBL_MAX_EXP+Bias-1-P)) {
1898 /* set to largest number */
1899 /* (Can't trust DBL_MAX) */
1904 word0(rv) += P*Exp_msk1;
1910 dval(rv) /= tens[i];
1912 if (e1 >= 1 << n_bigtens)
1914 #ifdef Avoid_Underflow
1917 for(j = 0; e1 > 0; j++, e1 >>= 1)
1919 dval(rv) *= tinytens[j];
1920 if (scale && (j = 2*P + 1 - ((word0(rv) & Exp_mask)
1921 >> Exp_shift)) > 0) {
1922 /* scaled rv is denormal; zap j low bits */
1926 word0(rv) = (P+2)*Exp_msk1;
1928 word0(rv) &= 0xffffffff << (j-32);
1931 word1(rv) &= 0xffffffff << j;
1934 for(j = 0; e1 > 1; j++, e1 >>= 1)
1936 dval(rv) *= tinytens[j];
1937 /* The last multiplication could underflow. */
1938 dval(rv0) = dval(rv);
1939 dval(rv) *= tinytens[j];
1941 dval(rv) = 2.*dval(rv0);
1942 dval(rv) *= tinytens[j];
1954 #ifndef Avoid_Underflow
1957 /* The refinement below will clean
1958 * this approximation up.
1965 /* Now the hard part -- adjusting rv to the correct value.*/
1967 /* Put digits into bd: true value = bd * 10^e */
1969 bd0 = s2b(s0, nd0, nd, y);
1972 bd = Balloc(bd0->k);
1974 bb = d2b(dval(rv), &bbe, &bbbits); /* rv = bb * 2^bbe */
1990 #ifdef Honor_FLT_ROUNDS
1994 #ifdef Avoid_Underflow
1996 i = j + bbbits - 1; /* logb(rv) */
1997 if (i < Emin) /* denormal */
2001 #else /*Avoid_Underflow*/
2002 #ifdef Sudden_Underflow
2004 j = 1 + 4*P - 3 - bbbits + ((bbe + bbbits - 1) & 3);
2008 #else /*Sudden_Underflow*/
2010 i = j + bbbits - 1; /* logb(rv) */
2011 if (i < Emin) /* denormal */
2015 #endif /*Sudden_Underflow*/
2016 #endif /*Avoid_Underflow*/
2019 #ifdef Avoid_Underflow
2022 i = bb2 < bd2 ? bb2 : bd2;
2031 bs = pow5mult(bs, bb5);
2037 bb = lshift(bb, bb2);
2039 bd = pow5mult(bd, bd5);
2041 bd = lshift(bd, bd2);
2043 bs = lshift(bs, bs2);
2044 delta = diff(bb, bd);
2045 dsign = delta->sign;
2048 #ifdef Honor_FLT_ROUNDS
2049 if (rounding != 1) {
2051 /* Error is less than an ulp */
2052 if (!delta->x[0] && delta->wds <= 1) {
2068 && !(word0(rv) & Frac_mask)) {
2069 y = word0(rv) & Exp_mask;
2070 #ifdef Avoid_Underflow
2071 if (!scale || y > 2*P*Exp_msk1)
2076 delta = lshift(delta,Log2P);
2077 if (cmp(delta, bs) <= 0)
2082 #ifdef Avoid_Underflow
2083 if (scale && (y = word0(rv) & Exp_mask)
2085 word0(adj) += (2*P+1)*Exp_msk1 - y;
2087 #ifdef Sudden_Underflow
2088 if ((word0(rv) & Exp_mask) <=
2090 word0(rv) += P*Exp_msk1;
2091 dval(rv) += adj*ulp(dval(rv));
2092 word0(rv) -= P*Exp_msk1;
2095 #endif /*Sudden_Underflow*/
2096 #endif /*Avoid_Underflow*/
2097 dval(rv) += adj*ulp(dval(rv));
2101 adj = ratio(delta, bs);
2104 if (adj <= 0x7ffffffe) {
2105 /* adj = rounding ? ceil(adj) : floor(adj); */
2108 if (!((rounding>>1) ^ dsign))
2113 #ifdef Avoid_Underflow
2114 if (scale && (y = word0(rv) & Exp_mask) <= 2*P*Exp_msk1)
2115 word0(adj) += (2*P+1)*Exp_msk1 - y;
2117 #ifdef Sudden_Underflow
2118 if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
2119 word0(rv) += P*Exp_msk1;
2120 adj *= ulp(dval(rv));
2125 word0(rv) -= P*Exp_msk1;
2128 #endif /*Sudden_Underflow*/
2129 #endif /*Avoid_Underflow*/
2130 adj *= ulp(dval(rv));
2137 #endif /*Honor_FLT_ROUNDS*/
2140 /* Error is less than half an ulp -- check for
2141 * special case of mantissa a power of two.
2143 if (dsign || word1(rv) || word0(rv) & Bndry_mask
2145 #ifdef Avoid_Underflow
2146 || (word0(rv) & Exp_mask) <= (2*P+1)*Exp_msk1
2148 || (word0(rv) & Exp_mask) <= Exp_msk1
2153 if (!delta->x[0] && delta->wds <= 1)
2158 if (!delta->x[0] && delta->wds <= 1) {
2165 delta = lshift(delta,Log2P);
2166 if (cmp(delta, bs) > 0)
2171 /* exactly half-way between */
2173 if ((word0(rv) & Bndry_mask1) == Bndry_mask1
2175 #ifdef Avoid_Underflow
2176 (scale && (y = word0(rv) & Exp_mask) <= 2*P*Exp_msk1)
2177 ? (0xffffffff & (0xffffffff << (2*P+1-(y>>Exp_shift)))) :
2180 /*boundary case -- increment exponent*/
2181 word0(rv) = (word0(rv) & Exp_mask)
2188 #ifdef Avoid_Underflow
2194 else if (!(word0(rv) & Bndry_mask) && !word1(rv)) {
2196 /* boundary case -- decrement exponent */
2197 #ifdef Sudden_Underflow /*{{*/
2198 L = word0(rv) & Exp_mask;
2202 #ifdef Avoid_Underflow
2203 if (L <= (scale ? (2*P+1)*Exp_msk1 : Exp_msk1))
2206 #endif /*Avoid_Underflow*/
2210 #else /*Sudden_Underflow}{*/
2211 #ifdef Avoid_Underflow
2213 L = word0(rv) & Exp_mask;
2214 if (L <= (2*P+1)*Exp_msk1) {
2215 if (L > (P+2)*Exp_msk1)
2216 /* round even ==> */
2219 /* rv = smallest denormal */
2223 #endif /*Avoid_Underflow*/
2224 L = (word0(rv) & Exp_mask) - Exp_msk1;
2225 #endif /*Sudden_Underflow}}*/
2226 word0(rv) = L | Bndry_mask1;
2227 word1(rv) = 0xffffffff;
2234 #ifndef ROUND_BIASED
2235 if (!(word1(rv) & LSB))
2239 dval(rv) += ulp(dval(rv));
2240 #ifndef ROUND_BIASED
2242 dval(rv) -= ulp(dval(rv));
2243 #ifndef Sudden_Underflow
2248 #ifdef Avoid_Underflow
2254 if ((aadj = ratio(delta, bs)) <= 2.) {
2257 else if (word1(rv) || word0(rv) & Bndry_mask) {
2258 #ifndef Sudden_Underflow
2259 if (word1(rv) == Tiny1 && !word0(rv))
2266 /* special case -- power of FLT_RADIX to be */
2267 /* rounded down... */
2269 if (aadj < 2./FLT_RADIX)
2270 aadj = 1./FLT_RADIX;
2278 aadj1 = dsign ? aadj : -aadj;
2279 #ifdef Check_FLT_ROUNDS
2281 case 2: /* towards +infinity */
2284 case 0: /* towards 0 */
2285 case 3: /* towards -infinity */
2289 if (Flt_Rounds == 0)
2291 #endif /*Check_FLT_ROUNDS*/
2293 y = word0(rv) & Exp_mask;
2295 /* Check for overflow */
2297 if (y == Exp_msk1*(DBL_MAX_EXP+Bias-1)) {
2298 dval(rv0) = dval(rv);
2299 word0(rv) -= P*Exp_msk1;
2300 adj = aadj1 * ulp(dval(rv));
2302 if ((word0(rv) & Exp_mask) >=
2303 Exp_msk1*(DBL_MAX_EXP+Bias-P)) {
2304 if (word0(rv0) == Big0 && word1(rv0) == Big1)
2311 word0(rv) += P*Exp_msk1;
2314 #ifdef Avoid_Underflow
2315 if (scale && y <= 2*P*Exp_msk1) {
2316 if (aadj <= 0x7fffffff) {
2317 if ((z = aadj) <= 0)
2320 aadj1 = dsign ? aadj : -aadj;
2322 word0(aadj1) += (2*P+1)*Exp_msk1 - y;
2324 adj = aadj1 * ulp(dval(rv));
2327 #ifdef Sudden_Underflow
2328 if ((word0(rv) & Exp_mask) <= P*Exp_msk1) {
2329 dval(rv0) = dval(rv);
2330 word0(rv) += P*Exp_msk1;
2331 adj = aadj1 * ulp(dval(rv));
2334 if ((word0(rv) & Exp_mask) < P*Exp_msk1)
2336 if ((word0(rv) & Exp_mask) <= P*Exp_msk1)
2339 if (word0(rv0) == Tiny0
2340 && word1(rv0) == Tiny1)
2347 word0(rv) -= P*Exp_msk1;
2350 adj = aadj1 * ulp(dval(rv));
2353 #else /*Sudden_Underflow*/
2354 /* Compute adj so that the IEEE rounding rules will
2355 * correctly round rv + adj in some half-way cases.
2356 * If rv * ulp(rv) is denormalized (i.e.,
2357 * y <= (P-1)*Exp_msk1), we must adjust aadj to avoid
2358 * trouble from bits lost to denormalization;
2359 * example: 1.2e-307 .
2361 if (y <= (P-1)*Exp_msk1 && aadj > 1.) {
2362 aadj1 = (double)(int)(aadj + 0.5);
2366 adj = aadj1 * ulp(dval(rv));
2368 #endif /*Sudden_Underflow*/
2369 #endif /*Avoid_Underflow*/
2371 z = word0(rv) & Exp_mask;
2373 #ifdef Avoid_Underflow
2377 /* Can we stop now? */
2380 /* The tolerances below are conservative. */
2381 if (dsign || word1(rv) || word0(rv) & Bndry_mask) {
2382 if (aadj < .4999999 || aadj > .5000001)
2385 else if (aadj < .4999999/FLT_RADIX)
2398 word0(rv0) = Exp_1 + (70 << Exp_shift);
2403 else if (!oldinexact)
2406 #ifdef Avoid_Underflow
2408 word0(rv0) = Exp_1 - 2*P*Exp_msk1;
2410 dval(rv) *= dval(rv0);
2412 /* try to avoid the bug of testing an 8087 register value */
2413 if (word0(rv) == 0 && word1(rv) == 0)
2417 #endif /* Avoid_Underflow */
2419 if (inexact && !(word0(rv) & Exp_mask)) {
2420 /* set underflow bit */
2422 dval(rv0) *= dval(rv0);
2434 return sign ? -dval(rv) : dval(rv);
2440 (b, S) Bigint *b, *S;
2442 (Bigint *b, Bigint *S)
2446 ULong *bx, *bxe, q, *sx, *sxe;
2448 ULLong borrow, carry, y, ys;
2450 ULong borrow, carry, y, ys;
2458 /*debug*/ if (b->wds > n)
2459 /*debug*/ Bug("oversize b in quorem");
2467 q = *bxe / (*sxe + 1); /* ensure q <= true quotient */
2469 /*debug*/ if (q > 9)
2470 /*debug*/ Bug("oversized quotient in quorem");
2477 ys = *sx++ * (ULLong)q + carry;
2479 y = *bx - (ys & FFFFFFFF) - borrow;
2480 borrow = y >> 32 & (ULong)1;
2481 *bx++ = y & FFFFFFFF;
2485 ys = (si & 0xffff) * q + carry;
2486 zs = (si >> 16) * q + (ys >> 16);
2488 y = (*bx & 0xffff) - (ys & 0xffff) - borrow;
2489 borrow = (y & 0x10000) >> 16;
2490 z = (*bx >> 16) - (zs & 0xffff) - borrow;
2491 borrow = (z & 0x10000) >> 16;
2494 ys = *sx++ * q + carry;
2496 y = *bx - (ys & 0xffff) - borrow;
2497 borrow = (y & 0x10000) >> 16;
2505 while(--bxe > bx && !*bxe)
2510 if (cmp(b, S) >= 0) {
2520 y = *bx - (ys & FFFFFFFF) - borrow;
2521 borrow = y >> 32 & (ULong)1;
2522 *bx++ = y & FFFFFFFF;
2526 ys = (si & 0xffff) + carry;
2527 zs = (si >> 16) + (ys >> 16);
2529 y = (*bx & 0xffff) - (ys & 0xffff) - borrow;
2530 borrow = (y & 0x10000) >> 16;
2531 z = (*bx >> 16) - (zs & 0xffff) - borrow;
2532 borrow = (z & 0x10000) >> 16;
2537 y = *bx - (ys & 0xffff) - borrow;
2538 borrow = (y & 0x10000) >> 16;
2547 while(--bxe > bx && !*bxe)
2555 #ifndef MULTIPLE_THREADS
2556 static char *dtoa_result;
2570 sizeof(Bigint) - sizeof(ULong) - sizeof(int) + j <= i;
2573 r = (int*)Balloc(k);
2576 #ifndef MULTIPLE_THREADS
2584 nrv_alloc(s, rve, n) char *s, **rve; int n;
2586 nrv_alloc(char *s, char **rve, int n)
2591 t = rv = rv_alloc(n);
2592 while((*t = *s++)) t++;
2598 /* freedtoa(s) must be used to free values s returned by dtoa
2599 * when MULTIPLE_THREADS is #defined. It should be used in all cases,
2600 * but for consistency with earlier versions of dtoa, it is optional
2601 * when MULTIPLE_THREADS is not defined.
2604 static void freedtoa (char *s);
2608 freedtoa(s) char *s;
2613 Bigint *b = (Bigint *)((int *)s - 1);
2614 b->maxwds = 1 << (b->k = *(int*)b);
2616 #ifndef MULTIPLE_THREADS
2617 if (s == dtoa_result)
2623 /* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
2625 * Inspired by "How to Print Floating-Point Numbers Accurately" by
2626 * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 112-126].
2629 * 1. Rather than iterating, we use a simple numeric overestimate
2630 * to determine k = floor(log10(d)). We scale relevant
2631 * quantities using O(log2(k)) rather than O(k) multiplications.
2632 * 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
2633 * try to generate digits strictly left to right. Instead, we
2634 * compute with fewer bits and propagate the carry if necessary
2635 * when rounding the final digit up. This is often faster.
2636 * 3. Under the assumption that input will be rounded nearest,
2637 * mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
2638 * That is, we allow equality in stopping tests when the
2639 * round-nearest rule will give the same floating-point value
2640 * as would satisfaction of the stopping test with strict
2642 * 4. We remove common factors of powers of 2 from relevant
2644 * 5. When converting floating-point integers less than 1e16,
2645 * we use floating-point arithmetic rather than resorting
2646 * to multiple-precision integers.
2647 * 6. When asked to produce fewer than 15 digits, we first try
2648 * to get by with floating-point arithmetic; we resort to
2649 * multiple-precision integer arithmetic only if we cannot
2650 * guarantee that the floating-point calculation has given
2651 * the correctly rounded result. For k requested digits and
2652 * "uniformly" distributed input, the probability is
2653 * something like 10^(k-15) that we must resort to the Long
2660 (d, mode, ndigits, decpt, sign, rve)
2661 double d; int mode, ndigits, *decpt, *sign; char **rve;
2663 (double d, int mode, int ndigits, int *decpt, int *sign, char **rve)
2666 /* Arguments ndigits, decpt, sign are similar to those
2667 of ecvt and fcvt; trailing zeros are suppressed from
2668 the returned string. If not null, *rve is set to point
2669 to the end of the return value. If d is +-Infinity or NaN,
2670 then *decpt is set to 9999.
2673 0 ==> shortest string that yields d when read in
2674 and rounded to nearest.
2675 1 ==> like 0, but with Steele & White stopping rule;
2676 e.g. with IEEE P754 arithmetic , mode 0 gives
2677 1e23 whereas mode 1 gives 9.999999999999999e22.
2678 2 ==> max(1,ndigits) significant digits. This gives a
2679 return value similar to that of ecvt, except
2680 that trailing zeros are suppressed.
2681 3 ==> through ndigits past the decimal point. This
2682 gives a return value similar to that from fcvt,
2683 except that trailing zeros are suppressed, and
2684 ndigits can be negative.
2685 4,5 ==> similar to 2 and 3, respectively, but (in
2686 round-nearest mode) with the tests of mode 0 to
2687 possibly return a shorter string that rounds to d.
2688 With IEEE arithmetic and compilation with
2689 -DHonor_FLT_ROUNDS, modes 4 and 5 behave the same
2690 as modes 2 and 3 when FLT_ROUNDS != 1.
2691 6-9 ==> Debugging modes similar to mode - 4: don't try
2692 fast floating-point estimate (if applicable).
2694 Values of mode other than 0-9 are treated as mode 0.
2696 Sufficient space is allocated to the return value
2697 to hold the suppressed trailing zeros.
2700 int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
2701 j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
2702 spec_case, try_quick;
2704 #ifndef Sudden_Underflow
2708 Bigint *b, *b1, *delta, *mlo, *mhi, *S;
2711 #ifdef Honor_FLT_ROUNDS
2715 int inexact, oldinexact;
2718 #ifndef MULTIPLE_THREADS
2720 freedtoa(dtoa_result);
2725 if (word0(d) & Sign_bit) {
2726 /* set sign for everything, including 0's and NaNs */
2728 word0(d) &= ~Sign_bit; /* clear sign bit */
2733 #if defined(IEEE_Arith) + defined(VAX)
2735 if ((word0(d) & Exp_mask) == Exp_mask)
2737 if (word0(d) == 0x8000)
2740 /* Infinity or NaN */
2743 if (!word1(d) && !(word0(d) & 0xfffff))
2744 return nrv_alloc("Infinity", rve, 8);
2746 return nrv_alloc("NaN", rve, 3);
2750 dval(d) += 0; /* normalize */
2754 return nrv_alloc("0", rve, 1);
2758 try_quick = oldinexact = get_inexact();
2761 #ifdef Honor_FLT_ROUNDS
2762 if ((rounding = Flt_Rounds) >= 2) {
2764 rounding = rounding == 2 ? 0 : 2;
2771 b = d2b(dval(d), &be, &bbits);
2772 #ifdef Sudden_Underflow
2773 i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1));
2775 if (i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1))) {
2778 word0(d2) &= Frac_mask1;
2779 word0(d2) |= Exp_11;
2781 if (j = 11 - hi0bits(word0(d2) & Frac_mask))
2785 /* log(x) ~=~ log(1.5) + (x-1.5)/1.5
2786 * log10(x) = log(x) / log(10)
2787 * ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
2788 * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
2790 * This suggests computing an approximation k to log10(d) by
2792 * k = (i - Bias)*0.301029995663981
2793 * + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
2795 * We want k to be too large rather than too small.
2796 * The error in the first-order Taylor series approximation
2797 * is in our favor, so we just round up the constant enough
2798 * to compensate for any error in the multiplication of
2799 * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
2800 * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
2801 * adding 1e-13 to the constant term more than suffices.
2802 * Hence we adjust the constant term to 0.1760912590558.
2803 * (We could get a more accurate k by invoking log10,
2804 * but this is probably not worthwhile.)
2812 #ifndef Sudden_Underflow
2816 /* d is denormalized */
2818 i = bbits + be + (Bias + (P-1) - 1);
2819 x = i > 32 ? word0(d) << 64 - i | word1(d) >> i - 32
2820 : word1(d) << 32 - i;
2822 word0(d2) -= 31*Exp_msk1; /* adjust exponent */
2823 i -= (Bias + (P-1) - 1) + 1;
2827 ds = (dval(d2)-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
2829 if (ds < 0. && ds != k)
2830 k--; /* want k = floor(ds) */
2832 if (k >= 0 && k <= Ten_pmax) {
2833 if (dval(d) < tens[k])
2856 if (mode < 0 || mode > 9)
2860 #ifdef Check_FLT_ROUNDS
2861 try_quick = Rounding == 1;
2865 #endif /*SET_INEXACT*/
2885 ilim = ilim1 = i = ndigits;
2891 i = ndigits + k + 1;
2897 s = s0 = rv_alloc(i);
2899 #ifdef Honor_FLT_ROUNDS
2900 if (mode > 1 && rounding != 1)
2904 if (ilim >= 0 && ilim <= Quick_max && try_quick) {
2906 /* Try to get by with floating-point arithmetic. */
2912 ieps = 2; /* conservative */
2917 /* prevent overflows */
2919 dval(d) /= bigtens[n_bigtens-1];
2922 for(; j; j >>= 1, i++)
2930 dval(d) *= tens[j1 & 0xf];
2931 for(j = j1 >> 4; j; j >>= 1, i++)
2934 dval(d) *= bigtens[i];
2937 if (k_check && dval(d) < 1. && ilim > 0) {
2945 dval(eps) = ieps*dval(d) + 7.;
2946 word0(eps) -= (P-1)*Exp_msk1;
2950 if (dval(d) > dval(eps))
2952 if (dval(d) < -dval(eps))
2956 #ifndef No_leftright
2958 /* Use Steele & White method of only
2959 * generating digits needed.
2961 dval(eps) = 0.5/tens[ilim-1] - dval(eps);
2965 *s++ = '0' + (int)L;
2966 if (dval(d) < dval(eps))
2968 if (1. - dval(d) < dval(eps))
2978 /* Generate ilim digits, then fix them up. */
2979 dval(eps) *= tens[ilim-1];
2980 for(i = 1;; i++, dval(d) *= 10.) {
2981 L = (Long)(dval(d));
2982 if (!(dval(d) -= L))
2984 *s++ = '0' + (int)L;
2986 if (dval(d) > 0.5 + dval(eps))
2988 else if (dval(d) < 0.5 - dval(eps)) {
2996 #ifndef No_leftright
3006 /* Do we have a "small" integer? */
3008 if (be >= 0 && k <= Int_max) {
3011 if (ndigits < 0 && ilim <= 0) {
3013 if (ilim < 0 || dval(d) <= 5*ds)
3017 for(i = 1;; i++, dval(d) *= 10.) {
3018 L = (Long)(dval(d) / ds);
3020 #ifdef Check_FLT_ROUNDS
3021 /* If FLT_ROUNDS == 2, L will usually be high by 1 */
3027 *s++ = '0' + (int)L;
3035 #ifdef Honor_FLT_ROUNDS
3039 case 2: goto bump_up;
3043 if (dval(d) > ds || dval(d) == ds && L & 1) {
3064 #ifndef Sudden_Underflow
3065 denorm ? be + (Bias + (P-1) - 1 + 1) :
3068 1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
3076 if (m2 > 0 && s2 > 0) {
3077 i = m2 < s2 ? m2 : s2;
3085 mhi = pow5mult(mhi, m5);
3094 b = pow5mult(b, b5);
3098 S = pow5mult(S, s5);
3100 /* Check for special case that d is a normalized power of 2. */
3103 if ((mode < 2 || leftright)
3104 #ifdef Honor_FLT_ROUNDS
3108 if (!word1(d) && !(word0(d) & Bndry_mask)
3109 #ifndef Sudden_Underflow
3110 && word0(d) & (Exp_mask & ~Exp_msk1)
3113 /* The special case */
3120 /* Arrange for convenient computation of quotients:
3121 * shift left if necessary so divisor has 4 leading 0 bits.
3123 * Perhaps we should just compute leading 28 bits of S once
3124 * and for all and pass them and a shift to quorem, so it
3125 * can do shifts and ors to compute the numerator for q.
3128 if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f)
3131 if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf)
3153 b = multadd(b, 10, 0); /* we botched the k estimate */
3155 mhi = multadd(mhi, 10, 0);
3159 if (ilim <= 0 && (mode == 3 || mode == 5)) {
3160 if (ilim < 0 || cmp(b,S = multadd(S,5,0)) <= 0) {
3161 /* no digits, fcvt style */
3173 mhi = lshift(mhi, m2);
3175 /* Compute mlo -- check for special case
3176 * that d is a normalized power of 2.
3181 mhi = Balloc(mhi->k);
3183 mhi = lshift(mhi, Log2P);
3187 dig = quorem(b,S) + '0';
3188 /* Do we yet have the shortest decimal string
3189 * that will round to d?
3192 delta = diff(S, mhi);
3193 j1 = delta->sign ? 1 : cmp(b, delta);
3195 #ifndef ROUND_BIASED
3196 if (j1 == 0 && mode != 1 && !(word1(d) & 1)
3197 #ifdef Honor_FLT_ROUNDS
3206 else if (!b->x[0] && b->wds <= 1)
3213 if (j < 0 || j == 0 && mode != 1
3214 #ifndef ROUND_BIASED
3218 if (!b->x[0] && b->wds <= 1) {
3224 #ifdef Honor_FLT_ROUNDS
3227 case 0: goto accept_dig;
3228 case 2: goto keep_dig;
3230 #endif /*Honor_FLT_ROUNDS*/
3234 if ((j1 > 0 || j1 == 0 && dig & 1)
3243 #ifdef Honor_FLT_ROUNDS
3247 if (dig == '9') { /* possible if i == 1 */
3255 #ifdef Honor_FLT_ROUNDS
3261 b = multadd(b, 10, 0);
3263 mlo = mhi = multadd(mhi, 10, 0);
3265 mlo = multadd(mlo, 10, 0);
3266 mhi = multadd(mhi, 10, 0);
3272 *s++ = dig = quorem(b,S) + '0';
3273 if (!b->x[0] && b->wds <= 1) {
3281 b = multadd(b, 10, 0);
3284 /* Round off last digit */
3286 #ifdef Honor_FLT_ROUNDS
3288 case 0: goto trimzeros;
3289 case 2: goto roundoff;
3294 if (j > 0 || j == 0 && dig & 1) {
3312 if (mlo && mlo != mhi)
3320 word0(d) = Exp_1 + (70 << Exp_shift);
3325 else if (!oldinexact)