1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: t; c-basic-offset: 8 -*- */
2 /****************************************************************
4 * The author of this software is David M. Gay.
6 * Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
8 * Permission to use, copy, modify, and distribute this software for any
9 * purpose without fee is hereby granted, provided that this entire notice
10 * is included in all copies of any software which is or includes a copy
11 * or modification of this software and in all copies of the supporting
12 * documentation for such software.
14 * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
15 * WARRANTY. IN PARTICULAR, NEITHER THE AUTHOR NOR LUCENT MAKES ANY
16 * REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
17 * OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
19 ***************************************************************/
21 /* Please send bug reports to David M. Gay (dmg at acm dot org,
22 * with " at " changed at "@" and " dot " changed to "."). */
24 /* On a machine with IEEE extended-precision registers, it is
25 * necessary to specify double-precision (53-bit) rounding precision
26 * before invoking strtod or dtoa. If the machine uses (the equivalent
27 * of) Intel 80x87 arithmetic, the call
28 * _control87(PC_53, MCW_PC);
29 * does this with many compilers. Whether this or another call is
30 * appropriate depends on the compiler; for this to work, it may be
31 * necessary to #include "float.h" or another system-dependent header
35 /* strtod for IEEE-, VAX-, and IBM-arithmetic machines.
37 * This strtod returns a nearest machine number to the input decimal
38 * string (or sets errno to ERANGE). With IEEE arithmetic, ties are
39 * broken by the IEEE round-even rule. Otherwise ties are broken by
40 * biased rounding (add half and chop).
42 * Inspired loosely by William D. Clinger's paper "How to Read Floating
43 * Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101].
47 * 1. We only require IEEE, IBM, or VAX double-precision
48 * arithmetic (not IEEE double-extended).
49 * 2. We get by with floating-point arithmetic in a case that
50 * Clinger missed -- when we're computing d * 10^n
51 * for a small integer d and the integer n is not too
52 * much larger than 22 (the maximum integer k for which
53 * we can represent 10^k exactly), we may be able to
54 * compute (d*10^k) * 10^(e-k) with just one roundoff.
55 * 3. Rather than a bit-at-a-time adjustment of the binary
56 * result in the hard case, we use floating-point
57 * arithmetic to determine the adjustment to within
58 * one bit; only in really hard cases do we need to
59 * compute a second residual.
60 * 4. Because of 3., we don't need a large table of powers of 10
61 * for ten-to-e (just some small tables, e.g. of 10^k
66 * #define IEEE_8087 for IEEE-arithmetic machines where the least
67 * significant byte has the lowest address.
68 * #define IEEE_MC68k for IEEE-arithmetic machines where the most
69 * significant byte has the lowest address.
70 * #define Long int on machines with 32-bit ints and 64-bit longs.
71 * #define IBM for IBM mainframe-style floating-point arithmetic.
72 * #define VAX for VAX-style floating-point arithmetic (D_floating).
73 * #define No_leftright to omit left-right logic in fast floating-point
74 * computation of dtoa.
75 * #define Honor_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
76 * and strtod and dtoa should round accordingly.
77 * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3
78 * and Honor_FLT_ROUNDS is not #defined.
79 * #define RND_PRODQUOT to use rnd_prod and rnd_quot (assembly routines
80 * that use extended-precision instructions to compute rounded
81 * products and quotients) with IBM.
82 * #define ROUND_BIASED for IEEE-format with biased rounding.
83 * #define Inaccurate_Divide for IEEE-format with correctly rounded
84 * products but inaccurate quotients, e.g., for Intel i860.
85 * #define NO_LONG_LONG on machines that do not have a "long long"
86 * integer type (of >= 64 bits). On such machines, you can
87 * #define Just_16 to store 16 bits per 32-bit Long when doing
88 * high-precision integer arithmetic. Whether this speeds things
89 * up or slows things down depends on the machine and the number
90 * being converted. If long long is available and the name is
91 * something other than "long long", #define Llong to be the name,
92 * and if "unsigned Llong" does not work as an unsigned version of
93 * Llong, #define #ULLong to be the corresponding unsigned type.
94 * #define KR_headers for old-style C function headers.
95 * #define Bad_float_h if your system lacks a float.h or if it does not
96 * define some or all of DBL_DIG, DBL_MAX_10_EXP, DBL_MAX_EXP,
97 * FLT_RADIX, FLT_ROUNDS, and DBL_MAX.
98 * #define MALLOC your_malloc, where your_malloc(n) acts like malloc(n)
99 * if memory is available and otherwise does something you deem
100 * appropriate. If MALLOC is undefined, malloc will be invoked
101 * directly -- and assumed always to succeed. Similarly, if you
102 * want something other than the system's free() to be called to
103 * recycle memory acquired from MALLOC, #define FREE to be the
104 * name of the alternate routine. (Unless you #define
105 * NO_GLOBAL_STATE and call destroydtoa, FREE or free is only
106 * called in pathological cases, e.g., in a dtoa call after a dtoa
107 * return in mode 3 with thousands of digits requested.)
108 * #define Omit_Private_Memory to omit logic (added Jan. 1998) for making
109 * memory allocations from a private pool of memory when possible.
110 * When used, the private pool is PRIVATE_MEM bytes long: 2304 bytes,
111 * unless #defined to be a different length. This default length
112 * suffices to get rid of MALLOC calls except for unusual cases,
113 * such as decimal-to-binary conversion of a very long string of
114 * digits. The longest string dtoa can return is about 751 bytes
115 * long. For conversions by strtod of strings of 800 digits and
116 * all dtoa conversions in single-threaded executions with 8-byte
117 * pointers, PRIVATE_MEM >= 7400 appears to suffice; with 4-byte
118 * pointers, PRIVATE_MEM >= 7112 appears adequate.
119 * #define MULTIPLE_THREADS if the system offers preemptively scheduled
120 * multiple threads. In this case, you must provide (or suitably
121 * #define) two locks, acquired by ACQUIRE_DTOA_LOCK(n) and freed
122 * by FREE_DTOA_LOCK(n) for n = 0 or 1. (The second lock, accessed
123 * in pow5mult, ensures lazy evaluation of only one copy of high
124 * powers of 5; omitting this lock would introduce a small
125 * probability of wasting memory, but would otherwise be harmless.)
126 * You must also invoke freedtoa(s) to free the value s returned by
127 * dtoa. You may do so whether or not MULTIPLE_THREADS is #defined.
128 * #define NO_IEEE_Scale to disable new (Feb. 1997) logic in strtod that
129 * avoids underflows on inputs whose result does not underflow.
130 * If you #define NO_IEEE_Scale on a machine that uses IEEE-format
131 * floating-point numbers and flushes underflows to zero rather
132 * than implementing gradual underflow, then you must also #define
134 * #define USE_LOCALE to use the current locale's decimal_point value.
135 * #define SET_INEXACT if IEEE arithmetic is being used and extra
136 * computation should be done to set the inexact flag when the
137 * result is inexact and avoid setting inexact when the result
138 * is exact. In this case, dtoa.c must be compiled in
139 * an environment, perhaps provided by #include "dtoa.c" in a
140 * suitable wrapper, that defines two functions,
141 * int get_inexact(void);
142 * void clear_inexact(void);
143 * such that get_inexact() returns a nonzero value if the
144 * inexact bit is already set, and clear_inexact() sets the
145 * inexact bit to 0. When SET_INEXACT is #defined, strtod
146 * also does extra computations to set the underflow and overflow
147 * flags when appropriate (i.e., when the result is tiny and
148 * inexact or when it is a numeric value rounded to +-infinity).
149 * #define NO_ERRNO if strtod should not assign errno = ERANGE when
150 * the result overflows to +-Infinity or underflows to 0.
151 * #define NO_GLOBAL_STATE to avoid defining any non-const global or
152 * static variables. Instead the necessary state is stored in an
153 * opaque struct, DtoaState, a pointer to which must be passed to
154 * every entry point. Two new functions are added to the API:
155 * DtoaState *newdtoa(void);
156 * void destroydtoa(DtoaState *);
163 typedef unsigned Long ULong
;
168 #define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
180 extern char *MALLOC();
182 extern void *MALLOC(size_t);
185 #define MALLOC malloc
192 #ifndef Omit_Private_Memory
194 #define PRIVATE_MEM 2304
196 #define PRIVATE_mem ((PRIVATE_MEM+sizeof(double)-1)/sizeof(double))
200 #undef Avoid_Underflow
214 #define DBL_MAX_10_EXP 308
215 #define DBL_MAX_EXP 1024
217 #endif /*IEEE_Arith*/
221 #define DBL_MAX_10_EXP 75
222 #define DBL_MAX_EXP 63
224 #define DBL_MAX 7.2370055773322621e+75
229 #define DBL_MAX_10_EXP 38
230 #define DBL_MAX_EXP 127
232 #define DBL_MAX 1.7014118346046923e+38
236 #define LONG_MAX 2147483647
239 #else /* ifndef Bad_float_h */
241 #endif /* Bad_float_h */
249 #define CONST /* blank */
255 #if defined(IEEE_8087) + defined(IEEE_MC68k) + defined(VAX) + defined(IBM) != 1
256 Exactly one of IEEE_8087
, IEEE_MC68k
, VAX
, or IBM should be defined
.
259 typedef union { double d
; ULong L
[2]; } U
;
261 #define dval(x) ((x).d)
263 #define word0(x) ((x).L[1])
264 #define word1(x) ((x).L[0])
266 #define word0(x) ((x).L[0])
267 #define word1(x) ((x).L[1])
270 /* The following definition of Storeinc is appropriate for MIPS processors.
271 * An alternative that might be better on some machines is
272 * #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
274 #if defined(IEEE_8087) + defined(VAX)
275 #define Storeinc(a,b,c) (((unsigned short *)a)[1] = (unsigned short)b, \
276 ((unsigned short *)a)[0] = (unsigned short)c, a++)
278 #define Storeinc(a,b,c) (((unsigned short *)a)[0] = (unsigned short)b, \
279 ((unsigned short *)a)[1] = (unsigned short)c, a++)
282 /* #define P DBL_MANT_DIG */
283 /* Ten_pmax = floor(P*log(2)/log(5)) */
284 /* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
285 /* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
286 /* Int_max = floor(P*log(FLT_RADIX)/log(10) - 1) */
290 #define Exp_shift1 20
291 #define Exp_msk1 0x100000
292 #define Exp_msk11 0x100000
293 #define Exp_mask 0x7ff00000
297 #define Exp_1 0x3ff00000
298 #define Exp_11 0x3ff00000
300 #define Frac_mask 0xfffff
301 #define Frac_mask1 0xfffff
304 #define Bndry_mask 0xfffff
305 #define Bndry_mask1 0xfffff
307 #define Sign_bit 0x80000000
313 #ifndef NO_IEEE_Scale
314 #define Avoid_Underflow
315 #ifdef Flush_Denorm /* debugging option */
316 #undef Sudden_Underflow
322 #define Flt_Rounds FLT_ROUNDS
326 #endif /*Flt_Rounds*/
328 #ifdef Honor_FLT_ROUNDS
329 #define Rounding rounding
330 #undef Check_FLT_ROUNDS
331 #define Check_FLT_ROUNDS
333 #define Rounding Flt_Rounds
336 #else /* ifndef IEEE_Arith */
337 #undef Check_FLT_ROUNDS
338 #undef Honor_FLT_ROUNDS
340 #undef Sudden_Underflow
341 #define Sudden_Underflow
346 #define Exp_shift1 24
347 #define Exp_msk1 0x1000000
348 #define Exp_msk11 0x1000000
349 #define Exp_mask 0x7f000000
352 #define Exp_1 0x41000000
353 #define Exp_11 0x41000000
354 #define Ebits 8 /* exponent has 7 bits, but 8 is the right value in b2d */
355 #define Frac_mask 0xffffff
356 #define Frac_mask1 0xffffff
359 #define Bndry_mask 0xefffff
360 #define Bndry_mask1 0xffffff
362 #define Sign_bit 0x80000000
364 #define Tiny0 0x100000
373 #define Exp_msk1 0x80
374 #define Exp_msk11 0x800000
375 #define Exp_mask 0x7f80
378 #define Exp_1 0x40800000
379 #define Exp_11 0x4080
381 #define Frac_mask 0x7fffff
382 #define Frac_mask1 0xffff007f
385 #define Bndry_mask 0xffff007f
386 #define Bndry_mask1 0xffff007f
388 #define Sign_bit 0x8000
394 #endif /* IBM, VAX */
395 #endif /* IEEE_Arith */
402 #define rounded_product(a,b) a = rnd_prod(a, b)
403 #define rounded_quotient(a,b) a = rnd_quot(a, b)
405 extern double rnd_prod(), rnd_quot();
407 extern double rnd_prod(double, double), rnd_quot(double, double);
410 #define rounded_product(a,b) a *= b
411 #define rounded_quotient(a,b) a /= b
414 #define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
415 #define Big1 0xffffffff
422 #define FFFFFFFF ((((unsigned long)0xffff)<<16)|(unsigned long)0xffff)
424 #define FFFFFFFF 0xffffffffUL
431 /* When Pack_32 is not defined, we store 16 bits per 32-bit Long.
432 * This makes some inner loops simpler and sometimes saves work
433 * during multiplications, but it often seems to make things slightly
434 * slower. Hence the default is now to store 32 bits per Long.
437 #else /* long long available */
439 #define Llong long long
442 #define ULLong unsigned Llong
444 #endif /* NO_LONG_LONG */
446 #ifndef MULTIPLE_THREADS
447 #define ACQUIRE_DTOA_LOCK(n) /*nothing*/
448 #define FREE_DTOA_LOCK(n) /*nothing*/
456 int k
, maxwds
, sign
, wds
;
460 typedef struct Bigint Bigint
;
462 #ifdef NO_GLOBAL_STATE
463 #ifdef MULTIPLE_THREADS
464 #error "cannot have both NO_GLOBAL_STATE and MULTIPLE_THREADS"
468 #define DECLARE_GLOBAL_STATE /* nothing */
470 #define DECLARE_GLOBAL_STATE static
473 DECLARE_GLOBAL_STATE Bigint
*freelist
[Kmax
+1];
474 DECLARE_GLOBAL_STATE Bigint
*p5s
;
475 #ifndef Omit_Private_Memory
476 DECLARE_GLOBAL_STATE
double private_mem
[PRIVATE_mem
];
477 DECLARE_GLOBAL_STATE
double *pmem_next
478 #ifndef NO_GLOBAL_STATE
483 #ifdef NO_GLOBAL_STATE
485 typedef struct DtoaState DtoaState
;
487 #define STATE_PARAM state,
488 #define STATE_PARAM_DECL DtoaState *state;
490 #define STATE_PARAM DtoaState *state,
492 #define PASS_STATE state,
493 #define GET_STATE(field) (state->field)
498 DtoaState
*state
= (DtoaState
*) MALLOC(sizeof(DtoaState
));
500 memset(state
, 0, sizeof(DtoaState
));
501 #ifndef Omit_Private_Memory
502 state
->pmem_next
= state
->private_mem
;
511 (state
) STATE_PARAM_DECL
519 for (i
= 0; i
<= Kmax
; i
++) {
520 for (v
= GET_STATE(freelist
)[i
]; v
; v
= next
) {
522 #ifndef Omit_Private_Memory
523 if ((double*)v
< GET_STATE(private_mem
) ||
524 (double*)v
>= GET_STATE(private_mem
) + PRIVATE_mem
)
533 #define STATE_PARAM /* nothing */
534 #define STATE_PARAM_DECL /* nothing */
535 #define PASS_STATE /* nothing */
536 #define GET_STATE(name) name
542 (STATE_PARAM k
) STATE_PARAM_DECL
int k
;
549 #ifndef Omit_Private_Memory
553 ACQUIRE_DTOA_LOCK(0);
554 /* The k > Kmax case does not need ACQUIRE_DTOA_LOCK(0), */
555 /* but this case seems very unlikely. */
556 if (k
<= Kmax
&& (rv
= GET_STATE(freelist
)[k
]))
557 GET_STATE(freelist
)[k
] = rv
->next
;
560 #ifdef Omit_Private_Memory
561 rv
= (Bigint
*)MALLOC(sizeof(Bigint
) + (x
-1)*sizeof(ULong
));
563 len
= (sizeof(Bigint
) + (x
-1)*sizeof(ULong
) + sizeof(double) - 1)
565 if (k
<= Kmax
&& GET_STATE(pmem_next
) - GET_STATE(private_mem
) + len
<= PRIVATE_mem
) {
566 rv
= (Bigint
*)GET_STATE(pmem_next
);
567 GET_STATE(pmem_next
) += len
;
570 rv
= (Bigint
*)MALLOC(len
*sizeof(double));
576 rv
->sign
= rv
->wds
= 0;
583 (STATE_PARAM v
) STATE_PARAM_DECL Bigint
*v
;
585 (STATE_PARAM Bigint
*v
)
592 ACQUIRE_DTOA_LOCK(0);
593 v
->next
= GET_STATE(freelist
)[v
->k
];
594 GET_STATE(freelist
)[v
->k
] = v
;
600 #define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
601 y->wds*sizeof(Long) + 2*sizeof(int))
606 (STATE_PARAM b
, m
, a
) STATE_PARAM_DECL Bigint
*b
; int m
, a
;
608 (STATE_PARAM Bigint
*b
, int m
, int a
) /* multiply by m and add a */
629 y
= *x
* (ULLong
)m
+ carry
;
631 *x
++ = (ULong
) y
& FFFFFFFF
;
635 y
= (xi
& 0xffff) * m
+ carry
;
636 z
= (xi
>> 16) * m
+ (y
>> 16);
638 *x
++ = (z
<< 16) + (y
& 0xffff);
648 if (wds
>= b
->maxwds
) {
649 b1
= Balloc(PASS_STATE b
->k
+1);
654 b
->x
[wds
++] = (ULong
) carry
;
663 (STATE_PARAM s
, nd0
, nd
, y9
) STATE_PARAM_DECL CONST
char *s
; int nd0
, nd
; ULong y9
;
665 (STATE_PARAM CONST
char *s
, int nd0
, int nd
, ULong y9
)
673 for(k
= 0, y
= 1; x
> y
; y
<<= 1, k
++) ;
675 b
= Balloc(PASS_STATE k
);
679 b
= Balloc(PASS_STATE k
+1);
680 b
->x
[0] = y9
& 0xffff;
681 b
->wds
= (b
->x
[1] = y9
>> 16) ? 2 : 1;
687 do b
= multadd(PASS_STATE b
, 10, *s
++ - '0');
694 b
= multadd(PASS_STATE b
, 10, *s
++ - '0');
708 if (!(x
& 0xffff0000)) {
712 if (!(x
& 0xff000000)) {
716 if (!(x
& 0xf0000000)) {
720 if (!(x
& 0xc0000000)) {
724 if (!(x
& 0x80000000)) {
726 if (!(x
& 0x40000000))
783 (STATE_PARAM i
) STATE_PARAM_DECL
int i
;
790 b
= Balloc(PASS_STATE
1);
799 (STATE_PARAM a
, b
) STATE_PARAM_DECL Bigint
*a
, *b
;
801 (STATE_PARAM Bigint
*a
, Bigint
*b
)
806 ULong
*x
, *xa
, *xae
, *xb
, *xbe
, *xc
, *xc0
;
817 if (a
->wds
< b
->wds
) {
828 c
= Balloc(PASS_STATE k
);
829 for(x
= c
->x
, xa
= x
+ wc
; x
< xa
; x
++)
837 for(; xb
< xbe
; xc0
++) {
843 z
= *x
++ * (ULLong
)y
+ *xc
+ carry
;
845 *xc
++ = (ULong
) z
& FFFFFFFF
;
853 for(; xb
< xbe
; xb
++, xc0
++) {
854 if (y
= *xb
& 0xffff) {
859 z
= (*x
& 0xffff) * y
+ (*xc
& 0xffff) + carry
;
861 z2
= (*x
++ >> 16) * y
+ (*xc
>> 16) + carry
;
874 z
= (*x
& 0xffff) * y
+ (*xc
>> 16) + carry
;
877 z2
= (*x
++ >> 16) * y
+ (*xc
& 0xffff) + carry
;
885 for(; xb
< xbe
; xc0
++) {
891 z
= *x
++ * y
+ *xc
+ carry
;
901 for(xc0
= c
->x
, xc
= xc0
+ wc
; wc
> 0 && !*--xc
; --wc
) ;
909 (STATE_PARAM b
, k
) STATE_PARAM_DECL Bigint
*b
; int k
;
911 (STATE_PARAM Bigint
*b
, int k
)
914 Bigint
*b1
, *p5
, *p51
;
916 static CONST
int p05
[3] = { 5, 25, 125 };
919 b
= multadd(PASS_STATE b
, p05
[i
-1], 0);
923 if (!(p5
= GET_STATE(p5s
))) {
925 #ifdef MULTIPLE_THREADS
926 ACQUIRE_DTOA_LOCK(1);
933 p5
= GET_STATE(p5s
) = i2b(PASS_STATE
625);
939 b1
= mult(PASS_STATE b
, p5
);
945 if (!(p51
= p5
->next
)) {
946 #ifdef MULTIPLE_THREADS
947 ACQUIRE_DTOA_LOCK(1);
948 if (!(p51
= p5
->next
)) {
949 p51
= p5
->next
= mult(p5
,p5
);
954 p51
= p5
->next
= mult(PASS_STATE p5
,p5
);
966 (STATE_PARAM b
, k
) STATE_PARAM_DECL Bigint
*b
; int k
;
968 (STATE_PARAM Bigint
*b
, int k
)
973 ULong
*x
, *x1
, *xe
, z
;
982 for(i
= b
->maxwds
; n1
> i
; i
<<= 1)
984 b1
= Balloc(PASS_STATE k1
);
986 for(i
= 0; i
< n
; i
++)
1007 *x1
++ = *x
<< k
& 0xffff | z
;
1019 Bfree(PASS_STATE b
);
1026 (a
, b
) Bigint
*a
, *b
;
1028 (Bigint
*a
, Bigint
*b
)
1031 ULong
*xa
, *xa0
, *xb
, *xb0
;
1037 if (i
> 1 && !a
->x
[i
-1])
1038 Bug("cmp called with a->x[a->wds-1] == 0");
1039 if (j
> 1 && !b
->x
[j
-1])
1040 Bug("cmp called with b->x[b->wds-1] == 0");
1050 return *xa
< *xb
? -1 : 1;
1060 (STATE_PARAM a
, b
) STATE_PARAM_DECL Bigint
*a
, *b
;
1062 (STATE_PARAM Bigint
*a
, Bigint
*b
)
1067 ULong
*xa
, *xae
, *xb
, *xbe
, *xc
;
1079 c
= Balloc(PASS_STATE
0);
1092 c
= Balloc(PASS_STATE a
->k
);
1104 y
= (ULLong
)*xa
++ - *xb
++ - borrow
;
1105 borrow
= y
>> 32 & (ULong
)1;
1106 *xc
++ = (ULong
) y
& FFFFFFFF
;
1111 borrow
= y
>> 32 & (ULong
)1;
1112 *xc
++ = (ULong
) y
& FFFFFFFF
;
1117 y
= (*xa
& 0xffff) - (*xb
& 0xffff) - borrow
;
1118 borrow
= (y
& 0x10000) >> 16;
1119 z
= (*xa
++ >> 16) - (*xb
++ >> 16) - borrow
;
1120 borrow
= (z
& 0x10000) >> 16;
1125 y
= (*xa
& 0xffff) - borrow
;
1126 borrow
= (y
& 0x10000) >> 16;
1127 z
= (*xa
++ >> 16) - borrow
;
1128 borrow
= (z
& 0x10000) >> 16;
1133 y
= *xa
++ - *xb
++ - borrow
;
1134 borrow
= (y
& 0x10000) >> 16;
1140 borrow
= (y
& 0x10000) >> 16;
1162 L
= (word0(x
) & Exp_mask
) - (P
-1)*Exp_msk1
;
1163 #ifndef Avoid_Underflow
1164 #ifndef Sudden_Underflow
1173 #ifndef Avoid_Underflow
1174 #ifndef Sudden_Underflow
1177 L
= -L
>> Exp_shift
;
1178 if (L
< Exp_shift
) {
1179 word0(a
) = 0x80000 >> L
;
1185 word1(a
) = L
>= 31 ? 1 : 1 << 31 - L
;
1196 (a
, e
) Bigint
*a
; int *e
;
1201 ULong
*xa
, *xa0
, w
, y
, z
;
1215 if (!y
) Bug("zero y in b2d");
1221 d0
= Exp_1
| y
>> (Ebits
- k
);
1222 w
= xa
> xa0
? *--xa
: 0;
1223 d1
= y
<< ((32-Ebits
) + k
) | w
>> (Ebits
- k
);
1226 z
= xa
> xa0
? *--xa
: 0;
1228 d0
= Exp_1
| y
<< k
| z
>> (32 - k
);
1229 y
= xa
> xa0
? *--xa
: 0;
1230 d1
= z
<< k
| y
>> (32 - k
);
1237 if (k
< Ebits
+ 16) {
1238 z
= xa
> xa0
? *--xa
: 0;
1239 d0
= Exp_1
| y
<< k
- Ebits
| z
>> Ebits
+ 16 - k
;
1240 w
= xa
> xa0
? *--xa
: 0;
1241 y
= xa
> xa0
? *--xa
: 0;
1242 d1
= z
<< k
+ 16 - Ebits
| w
<< k
- Ebits
| y
>> 16 + Ebits
- k
;
1245 z
= xa
> xa0
? *--xa
: 0;
1246 w
= xa
> xa0
? *--xa
: 0;
1248 d0
= Exp_1
| y
<< k
+ 16 | z
<< k
| w
>> 16 - k
;
1249 y
= xa
> xa0
? *--xa
: 0;
1250 d1
= w
<< k
+ 16 | y
<< k
;
1254 word0(d
) = d0
>> 16 | d0
<< 16;
1255 word1(d
) = d1
>> 16 | d1
<< 16;
1266 (STATE_PARAM d
, e
, bits
) STATE_PARAM_DECL U d
; int *e
, *bits
;
1268 (STATE_PARAM U d
, int *e
, int *bits
)
1274 #ifndef Sudden_Underflow
1279 d0
= word0(d
) >> 16 | word0(d
) << 16;
1280 d1
= word1(d
) >> 16 | word1(d
) << 16;
1287 b
= Balloc(PASS_STATE
1);
1289 b
= Balloc(PASS_STATE
2);
1294 d0
&= 0x7fffffff; /* clear sign bit, which we ignore */
1295 #ifdef Sudden_Underflow
1296 de
= (int)(d0
>> Exp_shift
);
1301 if ((de
= (int)(d0
>> Exp_shift
)))
1306 if ((k
= lo0bits(&y
))) {
1307 x
[0] = y
| z
<< (32 - k
);
1312 #ifndef Sudden_Underflow
1315 b
->wds
= (x
[1] = z
) ? 2 : 1;
1320 #ifndef Sudden_Underflow
1328 if (k
= lo0bits(&y
))
1330 x
[0] = y
| z
<< 32 - k
& 0xffff;
1331 x
[1] = z
>> k
- 16 & 0xffff;
1337 x
[1] = y
>> 16 | z
<< 16 - k
& 0xffff;
1338 x
[2] = z
>> k
& 0xffff;
1353 Bug("Zero passed to d2b");
1371 #ifndef Sudden_Underflow
1375 *e
= (de
- Bias
- (P
-1) << 2) + k
;
1376 *bits
= 4*P
+ 8 - k
- hi0bits(word0(d
) & Frac_mask
);
1378 *e
= de
- Bias
- (P
-1) + k
;
1381 #ifndef Sudden_Underflow
1384 *e
= de
- Bias
- (P
-1) + 1 + k
;
1386 *bits
= 32*i
- hi0bits(x
[i
-1]);
1388 *bits
= (i
+2)*16 - hi0bits(x
[i
]);
1400 (a
, b
) Bigint
*a
, *b
;
1402 (Bigint
*a
, Bigint
*b
)
1408 dval(da
) = b2d(a
, &ka
);
1409 dval(db
) = b2d(b
, &kb
);
1411 k
= ka
- kb
+ 32*(a
->wds
- b
->wds
);
1413 k
= ka
- kb
+ 16*(a
->wds
- b
->wds
);
1417 word0(da
) += (k
>> 2)*Exp_msk1
;
1423 word0(db
) += (k
>> 2)*Exp_msk1
;
1429 word0(da
) += k
*Exp_msk1
;
1432 word0(db
) += k
*Exp_msk1
;
1435 return dval(da
) / dval(db
);
1440 1e0
, 1e1
, 1e2
, 1e3
, 1e4
, 1e5
, 1e6
, 1e7
, 1e8
, 1e9
,
1441 1e10
, 1e11
, 1e12
, 1e13
, 1e14
, 1e15
, 1e16
, 1e17
, 1e18
, 1e19
,
1450 bigtens
[] = { 1e16
, 1e32
, 1e64
, 1e128
, 1e256
};
1451 static CONST
double tinytens
[] = { 1e-16, 1e-32, 1e-64, 1e-128,
1452 #ifdef Avoid_Underflow
1453 9007199254740992.*9007199254740992.e
-256
1454 /* = 2^106 * 1e-53 */
1459 /* The factor of 2^53 in tinytens[4] helps us avoid setting the underflow */
1460 /* flag unnecessarily. It leads to a song and dance at the end of strtod. */
1461 #define Scale_Bit 0x10
1465 bigtens
[] = { 1e16
, 1e32
, 1e64
};
1466 static CONST
double tinytens
[] = { 1e-16, 1e-32, 1e-64 };
1469 bigtens
[] = { 1e16
, 1e32
};
1470 static CONST
double tinytens
[] = { 1e-16, 1e-32 };
1478 (STATE_PARAM s00
, se
) STATE_PARAM_DECL CONST
char *s00
; char **se
;
1480 (STATE_PARAM CONST
char *s00
, char **se
)
1483 #ifdef Avoid_Underflow
1486 int bb2
, bb5
, bbe
, bd2
, bd5
, bbbits
, bs2
, c
, dsign
,
1487 e
, e1
, esign
, i
, j
, k
, nd
, nd0
, nf
, nz
, nz0
, sign
;
1488 CONST
char *s
, *s0
, *s1
;
1493 Bigint
*bb
, *bb1
, *bd
, *bd0
, *bs
, *delta
;
1495 int inexact
, oldinexact
;
1497 #ifdef Honor_FLT_ROUNDS
1505 delta
= bb
= bd
= bs
= 0;
1508 sign
= nz0
= nz
= 0;
1510 for(s
= s00
;;s
++) switch(*s
) {
1533 while(*++s
== '0') ;
1539 for(nd
= nf
= 0; (c
= *s
) >= '0' && c
<= '9'; nd
++, s
++)
1546 s1
= localeconv()->decimal_point
;
1567 for(; c
== '0'; c
= *++s
)
1569 if (c
> '0' && c
<= '9') {
1577 for(; c
>= '0' && c
<= '9'; c
= *++s
) {
1582 for(i
= 1; i
< nz
; i
++)
1585 else if (nd
<= DBL_DIG
+ 1)
1589 else if (nd
<= DBL_DIG
+ 1)
1597 if (c
== 'e' || c
== 'E') {
1598 if (!nd
&& !nz
&& !nz0
) {
1609 if (c
>= '0' && c
<= '9') {
1612 if (c
> '0' && c
<= '9') {
1615 while((c
= *++s
) >= '0' && c
<= '9')
1617 if (s
- s1
> 8 || L
> 19999)
1618 /* Avoid confusion from exponents
1619 * so large that e might overflow.
1621 e
= 19999; /* safe for 16 bit ints */
1643 /* Now we have nd0 digits, starting at s0, followed by a
1644 * decimal point, followed by nd-nd0 digits. The number we're
1645 * after is the integer represented by those digits times
1650 k
= nd
< DBL_DIG
+ 1 ? nd
: DBL_DIG
+ 1;
1655 oldinexact
= get_inexact();
1657 dval(rv
) = tens
[k
- 9] * dval(rv
) + z
;
1661 #ifndef RND_PRODQUOT
1662 #ifndef Honor_FLT_ROUNDS
1670 if (e
<= Ten_pmax
) {
1672 goto vax_ovfl_check
;
1674 #ifdef Honor_FLT_ROUNDS
1675 /* round correctly FLT_ROUNDS = 2 or 3 */
1681 /* rv = */ rounded_product(dval(rv
), tens
[e
]);
1686 if (e
<= Ten_pmax
+ i
) {
1687 /* A fancier test would sometimes let us do
1688 * this for larger i values.
1690 #ifdef Honor_FLT_ROUNDS
1691 /* round correctly FLT_ROUNDS = 2 or 3 */
1698 dval(rv
) *= tens
[i
];
1700 /* VAX exponent range is so narrow we must
1701 * worry about overflow here...
1704 word0(rv
) -= P
*Exp_msk1
;
1705 /* rv = */ rounded_product(dval(rv
), tens
[e
]);
1706 if ((word0(rv
) & Exp_mask
)
1707 > Exp_msk1
*(DBL_MAX_EXP
+Bias
-1-P
))
1709 word0(rv
) += P
*Exp_msk1
;
1711 /* rv = */ rounded_product(dval(rv
), tens
[e
]);
1716 #ifndef Inaccurate_Divide
1717 else if (e
>= -Ten_pmax
) {
1718 #ifdef Honor_FLT_ROUNDS
1719 /* round correctly FLT_ROUNDS = 2 or 3 */
1725 /* rv = */ rounded_quotient(dval(rv
), tens
[-e
]);
1736 oldinexact
= get_inexact();
1738 #ifdef Avoid_Underflow
1741 #ifdef Honor_FLT_ROUNDS
1742 if ((rounding
= Flt_Rounds
) >= 2) {
1744 rounding
= rounding
== 2 ? 0 : 2;
1750 #endif /*IEEE_Arith*/
1752 /* Get starting approximation = rv * 10**e1 */
1756 dval(rv
) *= tens
[i
];
1758 if (e1
> DBL_MAX_10_EXP
) {
1763 /* Can't trust HUGE_VAL */
1765 #ifdef Honor_FLT_ROUNDS
1767 case 0: /* toward 0 */
1768 case 3: /* toward -infinity */
1773 word0(rv
) = Exp_mask
;
1776 #else /*Honor_FLT_ROUNDS*/
1777 word0(rv
) = Exp_mask
;
1779 #endif /*Honor_FLT_ROUNDS*/
1781 /* set overflow bit */
1783 dval(rv0
) *= dval(rv0
);
1785 #else /*IEEE_Arith*/
1788 #endif /*IEEE_Arith*/
1794 for(j
= 0; e1
> 1; j
++, e1
>>= 1)
1796 dval(rv
) *= bigtens
[j
];
1797 /* The last multiplication could overflow. */
1798 word0(rv
) -= P
*Exp_msk1
;
1799 dval(rv
) *= bigtens
[j
];
1800 if ((z
= word0(rv
) & Exp_mask
)
1801 > Exp_msk1
*(DBL_MAX_EXP
+Bias
-P
))
1803 if (z
> Exp_msk1
*(DBL_MAX_EXP
+Bias
-1-P
)) {
1804 /* set to largest number */
1805 /* (Can't trust DBL_MAX) */
1810 word0(rv
) += P
*Exp_msk1
;
1816 dval(rv
) /= tens
[i
];
1818 if (e1
>= 1 << n_bigtens
)
1820 #ifdef Avoid_Underflow
1823 for(j
= 0; e1
> 0; j
++, e1
>>= 1)
1825 dval(rv
) *= tinytens
[j
];
1826 if (scale
&& (j
= 2*P
+ 1 - ((word0(rv
) & Exp_mask
)
1827 >> Exp_shift
)) > 0) {
1828 /* scaled rv is denormal; zap j low bits */
1832 word0(rv
) = (P
+2)*Exp_msk1
;
1834 word0(rv
) &= 0xffffffff << (j
-32);
1837 word1(rv
) &= 0xffffffff << j
;
1840 for(j
= 0; e1
> 1; j
++, e1
>>= 1)
1842 dval(rv
) *= tinytens
[j
];
1843 /* The last multiplication could underflow. */
1844 dval(rv0
) = dval(rv
);
1845 dval(rv
) *= tinytens
[j
];
1847 dval(rv
) = 2.*dval(rv0
);
1848 dval(rv
) *= tinytens
[j
];
1860 #ifndef Avoid_Underflow
1863 /* The refinement below will clean
1864 * this approximation up.
1871 /* Now the hard part -- adjusting rv to the correct value.*/
1873 /* Put digits into bd: true value = bd * 10^e */
1875 bd0
= s2b(PASS_STATE s0
, nd0
, nd
, y
);
1878 bd
= Balloc(PASS_STATE bd0
->k
);
1880 bb
= d2b(PASS_STATE rv
, &bbe
, &bbbits
); /* rv = bb * 2^bbe */
1881 bs
= i2b(PASS_STATE
1);
1896 #ifdef Honor_FLT_ROUNDS
1900 #ifdef Avoid_Underflow
1902 i
= j
+ bbbits
- 1; /* logb(rv) */
1903 if (i
< Emin
) /* denormal */
1907 #else /*Avoid_Underflow*/
1908 #ifdef Sudden_Underflow
1910 j
= 1 + 4*P
- 3 - bbbits
+ ((bbe
+ bbbits
- 1) & 3);
1914 #else /*Sudden_Underflow*/
1916 i
= j
+ bbbits
- 1; /* logb(rv) */
1917 if (i
< Emin
) /* denormal */
1921 #endif /*Sudden_Underflow*/
1922 #endif /*Avoid_Underflow*/
1925 #ifdef Avoid_Underflow
1928 i
= bb2
< bd2
? bb2
: bd2
;
1937 bs
= pow5mult(PASS_STATE bs
, bb5
);
1938 bb1
= mult(PASS_STATE bs
, bb
);
1939 Bfree(PASS_STATE bb
);
1943 bb
= lshift(PASS_STATE bb
, bb2
);
1945 bd
= pow5mult(PASS_STATE bd
, bd5
);
1947 bd
= lshift(PASS_STATE bd
, bd2
);
1949 bs
= lshift(PASS_STATE bs
, bs2
);
1950 delta
= diff(PASS_STATE bb
, bd
);
1951 dsign
= delta
->sign
;
1954 #ifdef Honor_FLT_ROUNDS
1955 if (rounding
!= 1) {
1957 /* Error is less than an ulp */
1958 if (!delta
->x
[0] && delta
->wds
<= 1) {
1974 && !(word0(rv
) & Frac_mask
)) {
1975 y
= word0(rv
) & Exp_mask
;
1976 #ifdef Avoid_Underflow
1977 if (!scale
|| y
> 2*P
*Exp_msk1
)
1982 delta
= lshift(PASS_STATE delta
,Log2P
);
1983 if (cmp(delta
, bs
) <= 0)
1988 #ifdef Avoid_Underflow
1989 if (scale
&& (y
= word0(rv
) & Exp_mask
)
1991 word0(adj
) += (2*P
+1)*Exp_msk1
- y
;
1993 #ifdef Sudden_Underflow
1994 if ((word0(rv
) & Exp_mask
) <=
1996 word0(rv
) += P
*Exp_msk1
;
1997 dval(rv
) += adj
*ulp(rv
);
1998 word0(rv
) -= P
*Exp_msk1
;
2001 #endif /*Sudden_Underflow*/
2002 #endif /*Avoid_Underflow*/
2003 dval(rv
) += adj
*ulp(rv
);
2007 adj
= ratio(delta
, bs
);
2010 if (adj
<= 0x7ffffffe) {
2011 /* adj = rounding ? ceil(adj) : floor(adj); */
2014 if (!((rounding
>>1) ^ dsign
))
2019 #ifdef Avoid_Underflow
2020 if (scale
&& (y
= word0(rv
) & Exp_mask
) <= 2*P
*Exp_msk1
)
2021 word0(adj
) += (2*P
+1)*Exp_msk1
- y
;
2023 #ifdef Sudden_Underflow
2024 if ((word0(rv
) & Exp_mask
) <= P
*Exp_msk1
) {
2025 word0(rv
) += P
*Exp_msk1
;
2031 word0(rv
) -= P
*Exp_msk1
;
2034 #endif /*Sudden_Underflow*/
2035 #endif /*Avoid_Underflow*/
2043 #endif /*Honor_FLT_ROUNDS*/
2046 /* Error is less than half an ulp -- check for
2047 * special case of mantissa a power of two.
2049 if (dsign
|| word1(rv
) || word0(rv
) & Bndry_mask
2051 #ifdef Avoid_Underflow
2052 || (word0(rv
) & Exp_mask
) <= (2*P
+1)*Exp_msk1
2054 || (word0(rv
) & Exp_mask
) <= Exp_msk1
2059 if (!delta
->x
[0] && delta
->wds
<= 1)
2064 if (!delta
->x
[0] && delta
->wds
<= 1) {
2071 delta
= lshift(PASS_STATE delta
,Log2P
);
2072 if (cmp(delta
, bs
) > 0)
2077 /* exactly half-way between */
2079 if ((word0(rv
) & Bndry_mask1
) == Bndry_mask1
2081 #ifdef Avoid_Underflow
2082 (scale
&& (y
= word0(rv
) & Exp_mask
) <= 2*P
*Exp_msk1
)
2083 ? (0xffffffff & (0xffffffff << (2*P
+1-(y
>>Exp_shift
)))) :
2086 /*boundary case -- increment exponent*/
2087 word0(rv
) = (word0(rv
) & Exp_mask
)
2094 #ifdef Avoid_Underflow
2100 else if (!(word0(rv
) & Bndry_mask
) && !word1(rv
)) {
2102 /* boundary case -- decrement exponent */
2103 #ifdef Sudden_Underflow /*{{*/
2104 L
= word0(rv
) & Exp_mask
;
2108 #ifdef Avoid_Underflow
2109 if (L
<= (scale
? (2*P
+1)*Exp_msk1
: Exp_msk1
))
2112 #endif /*Avoid_Underflow*/
2116 #else /*Sudden_Underflow}{*/
2117 #ifdef Avoid_Underflow
2119 L
= word0(rv
) & Exp_mask
;
2120 if (L
<= (2*P
+1)*Exp_msk1
) {
2121 if (L
> (P
+2)*Exp_msk1
)
2122 /* round even ==> */
2125 /* rv = smallest denormal */
2129 #endif /*Avoid_Underflow*/
2130 L
= (word0(rv
) & Exp_mask
) - Exp_msk1
;
2131 #endif /*Sudden_Underflow}}*/
2132 word0(rv
) = L
| Bndry_mask1
;
2133 word1(rv
) = 0xffffffff;
2140 #ifndef ROUND_BIASED
2141 if (!(word1(rv
) & LSB
))
2145 dval(rv
) += ulp(rv
);
2146 #ifndef ROUND_BIASED
2148 dval(rv
) -= ulp(rv
);
2149 #ifndef Sudden_Underflow
2154 #ifdef Avoid_Underflow
2160 if ((aadj
= ratio(delta
, bs
)) <= 2.) {
2162 aadj
= dval(aadj1
) = 1.;
2163 else if (word1(rv
) || word0(rv
) & Bndry_mask
) {
2164 #ifndef Sudden_Underflow
2165 if (word1(rv
) == Tiny1
&& !word0(rv
))
2172 /* special case -- power of FLT_RADIX to be */
2173 /* rounded down... */
2175 if (aadj
< 2./FLT_RADIX
)
2176 aadj
= 1./FLT_RADIX
;
2179 dval(aadj1
) = -aadj
;
2184 dval(aadj1
) = dsign
? aadj
: -aadj
;
2185 #ifdef Check_FLT_ROUNDS
2187 case 2: /* towards +infinity */
2190 case 0: /* towards 0 */
2191 case 3: /* towards -infinity */
2195 if (Flt_Rounds
== 0)
2197 #endif /*Check_FLT_ROUNDS*/
2199 y
= word0(rv
) & Exp_mask
;
2201 /* Check for overflow */
2203 if (y
== Exp_msk1
*(DBL_MAX_EXP
+Bias
-1)) {
2204 dval(rv0
) = dval(rv
);
2205 word0(rv
) -= P
*Exp_msk1
;
2206 adj
= dval(aadj1
) * ulp(rv
);
2208 if ((word0(rv
) & Exp_mask
) >=
2209 Exp_msk1
*(DBL_MAX_EXP
+Bias
-P
)) {
2210 if (word0(rv0
) == Big0
&& word1(rv0
) == Big1
)
2217 word0(rv
) += P
*Exp_msk1
;
2220 #ifdef Avoid_Underflow
2221 if (scale
&& y
<= 2*P
*Exp_msk1
) {
2222 if (aadj
<= 0x7fffffff) {
2223 if ((z
= (ULong
) aadj
) <= 0)
2226 dval(aadj1
) = dsign
? aadj
: -aadj
;
2228 word0(aadj1
) += (2*P
+1)*Exp_msk1
- y
;
2230 adj
= dval(aadj1
) * ulp(rv
);
2233 #ifdef Sudden_Underflow
2234 if ((word0(rv
) & Exp_mask
) <= P
*Exp_msk1
) {
2235 dval(rv0
) = dval(rv
);
2236 word0(rv
) += P
*Exp_msk1
;
2237 adj
= dval(aadj1
) * ulp(rv
);
2240 if ((word0(rv
) & Exp_mask
) < P
*Exp_msk1
)
2242 if ((word0(rv
) & Exp_mask
) <= P
*Exp_msk1
)
2245 if (word0(rv0
) == Tiny0
2246 && word1(rv0
) == Tiny1
)
2253 word0(rv
) -= P
*Exp_msk1
;
2256 adj
= dval(aadj1
) * ulp(rv
);
2259 #else /*Sudden_Underflow*/
2260 /* Compute adj so that the IEEE rounding rules will
2261 * correctly round rv + adj in some half-way cases.
2262 * If rv * ulp(rv) is denormalized (i.e.,
2263 * y <= (P-1)*Exp_msk1), we must adjust aadj to avoid
2264 * trouble from bits lost to denormalization;
2265 * example: 1.2e-307 .
2267 if (y
<= (P
-1)*Exp_msk1
&& aadj
> 1.) {
2268 dval(aadj1
) = (double)(int)(aadj
+ 0.5);
2270 dval(aadj1
) = -dval(aadj1
);
2272 adj
= dval(aadj1
) * ulp(rv
);
2274 #endif /*Sudden_Underflow*/
2275 #endif /*Avoid_Underflow*/
2277 z
= word0(rv
) & Exp_mask
;
2279 #ifdef Avoid_Underflow
2283 /* Can we stop now? */
2286 /* The tolerances below are conservative. */
2287 if (dsign
|| word1(rv
) || word0(rv
) & Bndry_mask
) {
2288 if (aadj
< .4999999 || aadj
> .5000001)
2291 else if (aadj
< .4999999/FLT_RADIX
)
2296 Bfree(PASS_STATE bb
);
2297 Bfree(PASS_STATE bd
);
2298 Bfree(PASS_STATE bs
);
2299 Bfree(PASS_STATE delta
);
2304 word0(rv0
) = Exp_1
+ (70 << Exp_shift
);
2309 else if (!oldinexact
)
2312 #ifdef Avoid_Underflow
2314 word0(rv0
) = Exp_1
- 2*P
*Exp_msk1
;
2316 dval(rv
) *= dval(rv0
);
2318 /* try to avoid the bug of testing an 8087 register value */
2319 if (word0(rv
) == 0 && word1(rv
) == 0)
2323 #endif /* Avoid_Underflow */
2325 if (inexact
&& !(word0(rv
) & Exp_mask
)) {
2326 /* set underflow bit */
2328 dval(rv0
) *= dval(rv0
);
2332 Bfree(PASS_STATE bb
);
2333 Bfree(PASS_STATE bd
);
2334 Bfree(PASS_STATE bs
);
2335 Bfree(PASS_STATE bd0
);
2336 Bfree(PASS_STATE delta
);
2340 return sign
? -dval(rv
) : dval(rv
);
2346 (b
, S
) Bigint
*b
, *S
;
2348 (Bigint
*b
, Bigint
*S
)
2352 ULong
*bx
, *bxe
, q
, *sx
, *sxe
;
2354 ULLong borrow
, carry
, y
, ys
;
2356 ULong borrow
, carry
, y
, ys
;
2364 /*debug*/ if (b
->wds
> n
)
2365 /*debug*/ Bug("oversize b in quorem");
2373 q
= *bxe
/ (*sxe
+ 1); /* ensure q <= true quotient */
2375 /*debug*/ if (q
> 9)
2376 /*debug*/ Bug("oversized quotient in quorem");
2383 ys
= *sx
++ * (ULLong
)q
+ carry
;
2385 y
= *bx
- (ys
& FFFFFFFF
) - borrow
;
2386 borrow
= y
>> 32 & (ULong
)1;
2387 *bx
++ = (ULong
) y
& FFFFFFFF
;
2391 ys
= (si
& 0xffff) * q
+ carry
;
2392 zs
= (si
>> 16) * q
+ (ys
>> 16);
2394 y
= (*bx
& 0xffff) - (ys
& 0xffff) - borrow
;
2395 borrow
= (y
& 0x10000) >> 16;
2396 z
= (*bx
>> 16) - (zs
& 0xffff) - borrow
;
2397 borrow
= (z
& 0x10000) >> 16;
2400 ys
= *sx
++ * q
+ carry
;
2402 y
= *bx
- (ys
& 0xffff) - borrow
;
2403 borrow
= (y
& 0x10000) >> 16;
2411 while(--bxe
> bx
&& !*bxe
)
2416 if (cmp(b
, S
) >= 0) {
2426 y
= *bx
- (ys
& FFFFFFFF
) - borrow
;
2427 borrow
= y
>> 32 & (ULong
)1;
2428 *bx
++ = (ULong
) y
& FFFFFFFF
;
2432 ys
= (si
& 0xffff) + carry
;
2433 zs
= (si
>> 16) + (ys
>> 16);
2435 y
= (*bx
& 0xffff) - (ys
& 0xffff) - borrow
;
2436 borrow
= (y
& 0x10000) >> 16;
2437 z
= (*bx
>> 16) - (zs
& 0xffff) - borrow
;
2438 borrow
= (z
& 0x10000) >> 16;
2443 y
= *bx
- (ys
& 0xffff) - borrow
;
2444 borrow
= (y
& 0x10000) >> 16;
2453 while(--bxe
> bx
&& !*bxe
)
2461 #if !defined(MULTIPLE_THREADS) && !defined(NO_GLOBAL_STATE)
2462 #define USE_DTOA_RESULT 1
2463 static char *dtoa_result
;
2468 rv_alloc(STATE_PARAM i
) STATE_PARAM_DECL
int i
;
2470 rv_alloc(STATE_PARAM
int i
)
2477 sizeof(Bigint
) - sizeof(ULong
) - sizeof(int) + j
<= (unsigned) i
;
2480 r
= (int*)Balloc(PASS_STATE k
);
2483 #ifdef USE_DTOA_RESULT
2491 nrv_alloc(STATE_PARAM s
, rve
, n
) STATE_PARAM_DECL
char *s
, **rve
; int n
;
2493 nrv_alloc(STATE_PARAM CONST
char *s
, char **rve
, int n
)
2498 t
= rv
= rv_alloc(PASS_STATE n
);
2499 while((*t
= *s
++)) t
++;
2505 /* freedtoa(s) must be used to free values s returned by dtoa
2506 * when MULTIPLE_THREADS is #defined. It should be used in all cases,
2507 * but for consistency with earlier versions of dtoa, it is optional
2508 * when MULTIPLE_THREADS is not defined.
2513 freedtoa(STATE_PARAM s
) STATE_PARAM_DECL
char *s
;
2515 freedtoa(STATE_PARAM
char *s
)
2518 Bigint
*b
= (Bigint
*)((int *)s
- 1);
2519 b
->maxwds
= 1 << (b
->k
= *(int*)b
);
2520 Bfree(PASS_STATE b
);
2521 #ifdef USE_DTOA_RESULT
2522 if (s
== dtoa_result
)
2527 /* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
2529 * Inspired by "How to Print Floating-Point Numbers Accurately" by
2530 * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 112-126].
2533 * 1. Rather than iterating, we use a simple numeric overestimate
2534 * to determine k = floor(log10(d)). We scale relevant
2535 * quantities using O(log2(k)) rather than O(k) multiplications.
2536 * 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
2537 * try to generate digits strictly left to right. Instead, we
2538 * compute with fewer bits and propagate the carry if necessary
2539 * when rounding the final digit up. This is often faster.
2540 * 3. Under the assumption that input will be rounded nearest,
2541 * mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
2542 * That is, we allow equality in stopping tests when the
2543 * round-nearest rule will give the same floating-point value
2544 * as would satisfaction of the stopping test with strict
2546 * 4. We remove common factors of powers of 2 from relevant
2548 * 5. When converting floating-point integers less than 1e16,
2549 * we use floating-point arithmetic rather than resorting
2550 * to multiple-precision integers.
2551 * 6. When asked to produce fewer than 15 digits, we first try
2552 * to get by with floating-point arithmetic; we resort to
2553 * multiple-precision integer arithmetic only if we cannot
2554 * guarantee that the floating-point calculation has given
2555 * the correctly rounded result. For k requested digits and
2556 * "uniformly" distributed input, the probability is
2557 * something like 10^(k-15) that we must resort to the Long
2564 (STATE_PARAM d
, mode
, ndigits
, decpt
, sign
, rve
)
2565 STATE_PARAM_DECL U d
; int mode
, ndigits
, *decpt
, *sign
; char **rve
;
2567 (STATE_PARAM U d
, int mode
, int ndigits
, int *decpt
, int *sign
, char **rve
)
2570 /* Arguments ndigits, decpt, sign are similar to those
2571 of ecvt and fcvt; trailing zeros are suppressed from
2572 the returned string. If not null, *rve is set to point
2573 to the end of the return value. If d is +-Infinity or NaN,
2574 then *decpt is set to 9999.
2577 0 ==> shortest string that yields d when read in
2578 and rounded to nearest.
2579 1 ==> like 0, but with Steele & White stopping rule;
2580 e.g. with IEEE P754 arithmetic , mode 0 gives
2581 1e23 whereas mode 1 gives 9.999999999999999e22.
2582 2 ==> max(1,ndigits) significant digits. This gives a
2583 return value similar to that of ecvt, except
2584 that trailing zeros are suppressed.
2585 3 ==> through ndigits past the decimal point. This
2586 gives a return value similar to that from fcvt,
2587 except that trailing zeros are suppressed, and
2588 ndigits can be negative.
2589 4,5 ==> similar to 2 and 3, respectively, but (in
2590 round-nearest mode) with the tests of mode 0 to
2591 possibly return a shorter string that rounds to d.
2592 With IEEE arithmetic and compilation with
2593 -DHonor_FLT_ROUNDS, modes 4 and 5 behave the same
2594 as modes 2 and 3 when FLT_ROUNDS != 1.
2595 6-9 ==> Debugging modes similar to mode - 4: don't try
2596 fast floating-point estimate (if applicable).
2598 Values of mode other than 0-9 are treated as mode 0.
2600 Sufficient space is allocated to the return value
2601 to hold the suppressed trailing zeros.
2604 int bbits
, b2
, b5
, be
, dig
, i
, ieps
, ilim
, ilim0
, ilim1
,
2605 j
, j1
, k
, k0
, k_check
, leftright
, m2
, m5
, s2
, s5
,
2606 spec_case
, try_quick
;
2608 #ifndef Sudden_Underflow
2612 Bigint
*b
, *b1
, *delta
, *mlo
, *mhi
, *S
;
2616 #ifdef Honor_FLT_ROUNDS
2620 int inexact
, oldinexact
;
2628 #ifdef USE_DTOA_RESULT
2630 freedtoa(PASS_STATE dtoa_result
);
2635 if (word0(d
) & Sign_bit
) {
2636 /* set sign for everything, including 0's and NaNs */
2638 word0(d
) &= ~Sign_bit
; /* clear sign bit */
2643 #if defined(IEEE_Arith) + defined(VAX)
2645 if ((word0(d
) & Exp_mask
) == Exp_mask
)
2647 if (word0(d
) == 0x8000)
2650 /* Infinity or NaN */
2653 if (!word1(d
) && !(word0(d
) & 0xfffff))
2654 return nrv_alloc(PASS_STATE
"Infinity", rve
, 8);
2656 return nrv_alloc(PASS_STATE
"NaN", rve
, 3);
2660 dval(d
) += 0; /* normalize */
2664 return nrv_alloc(PASS_STATE
"0", rve
, 1);
2668 try_quick
= oldinexact
= get_inexact();
2671 #ifdef Honor_FLT_ROUNDS
2672 if ((rounding
= Flt_Rounds
) >= 2) {
2674 rounding
= rounding
== 2 ? 0 : 2;
2681 b
= d2b(PASS_STATE d
, &be
, &bbits
);
2682 #ifdef Sudden_Underflow
2683 i
= (int)(word0(d
) >> Exp_shift1
& (Exp_mask
>>Exp_shift1
));
2685 if ((i
= (int)(word0(d
) >> Exp_shift1
& (Exp_mask
>>Exp_shift1
)))) {
2688 word0(d2
) &= Frac_mask1
;
2689 word0(d2
) |= Exp_11
;
2691 if (j
= 11 - hi0bits(word0(d2
) & Frac_mask
))
2695 /* log(x) ~=~ log(1.5) + (x-1.5)/1.5
2696 * log10(x) = log(x) / log(10)
2697 * ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
2698 * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
2700 * This suggests computing an approximation k to log10(d) by
2702 * k = (i - Bias)*0.301029995663981
2703 * + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
2705 * We want k to be too large rather than too small.
2706 * The error in the first-order Taylor series approximation
2707 * is in our favor, so we just round up the constant enough
2708 * to compensate for any error in the multiplication of
2709 * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
2710 * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
2711 * adding 1e-13 to the constant term more than suffices.
2712 * Hence we adjust the constant term to 0.1760912590558.
2713 * (We could get a more accurate k by invoking log10,
2714 * but this is probably not worthwhile.)
2722 #ifndef Sudden_Underflow
2726 /* d is denormalized */
2728 i
= bbits
+ be
+ (Bias
+ (P
-1) - 1);
2729 x
= i
> 32 ? word0(d
) << (64 - i
) | word1(d
) >> (i
- 32)
2730 : word1(d
) << (32 - i
);
2732 word0(d2
) -= 31*Exp_msk1
; /* adjust exponent */
2733 i
-= (Bias
+ (P
-1) - 1) + 1;
2737 ds
= (dval(d2
)-1.5)*0.289529654602168 + 0.1760912590558 + i
*0.301029995663981;
2739 if (ds
< 0. && ds
!= k
)
2740 k
--; /* want k = floor(ds) */
2742 if (k
>= 0 && k
<= Ten_pmax
) {
2743 if (dval(d
) < tens
[k
])
2766 if (mode
< 0 || mode
> 9)
2770 #ifdef Check_FLT_ROUNDS
2771 try_quick
= Rounding
== 1;
2775 #endif /*SET_INEXACT*/
2795 ilim
= ilim1
= i
= ndigits
;
2801 i
= ndigits
+ k
+ 1;
2807 s
= s0
= rv_alloc(PASS_STATE i
);
2809 #ifdef Honor_FLT_ROUNDS
2810 if (mode
> 1 && rounding
!= 1)
2814 if (ilim
>= 0 && ilim
<= Quick_max
&& try_quick
) {
2816 /* Try to get by with floating-point arithmetic. */
2822 ieps
= 2; /* conservative */
2827 /* prevent overflows */
2829 dval(d
) /= bigtens
[n_bigtens
-1];
2832 for(; j
; j
>>= 1, i
++)
2839 else if ((j1
= -k
)) {
2840 dval(d
) *= tens
[j1
& 0xf];
2841 for(j
= j1
>> 4; j
; j
>>= 1, i
++)
2844 dval(d
) *= bigtens
[i
];
2847 if (k_check
&& dval(d
) < 1. && ilim
> 0) {
2855 dval(eps
) = ieps
*dval(d
) + 7.;
2856 word0(eps
) -= (P
-1)*Exp_msk1
;
2860 if (dval(d
) > dval(eps
))
2862 if (dval(d
) < -dval(eps
))
2866 #ifndef No_leftright
2868 /* Use Steele & White method of only
2869 * generating digits needed.
2871 dval(eps
) = 0.5/tens
[ilim
-1] - dval(eps
);
2873 L
= (ULong
) dval(d
);
2875 *s
++ = '0' + (int)L
;
2876 if (dval(d
) < dval(eps
))
2878 if (1. - dval(d
) < dval(eps
))
2888 /* Generate ilim digits, then fix them up. */
2889 dval(eps
) *= tens
[ilim
-1];
2890 for(i
= 1;; i
++, dval(d
) *= 10.) {
2891 L
= (Long
)(dval(d
));
2892 if (!(dval(d
) -= L
))
2894 *s
++ = '0' + (int)L
;
2896 if (dval(d
) > 0.5 + dval(eps
))
2898 else if (dval(d
) < 0.5 - dval(eps
)) {
2906 #ifndef No_leftright
2916 /* Do we have a "small" integer? */
2918 if (be
>= 0 && k
<= Int_max
) {
2921 if (ndigits
< 0 && ilim
<= 0) {
2923 if (ilim
< 0 || dval(d
) < 5*ds
)
2927 for(i
= 1;; i
++, dval(d
) *= 10.) {
2928 L
= (Long
)(dval(d
) / ds
);
2930 #ifdef Check_FLT_ROUNDS
2931 /* If FLT_ROUNDS == 2, L will usually be high by 1 */
2937 *s
++ = '0' + (int)L
;
2945 #ifdef Honor_FLT_ROUNDS
2949 case 2: goto bump_up
;
2953 if (dval(d
) > ds
|| (dval(d
) == ds
&& L
& 1)) {
2974 #ifndef Sudden_Underflow
2975 denorm
? be
+ (Bias
+ (P
-1) - 1 + 1) :
2978 1 + 4*P
- 3 - bbits
+ ((bbits
+ be
- 1) & 3);
2984 mhi
= i2b(PASS_STATE
1);
2986 if (m2
> 0 && s2
> 0) {
2987 i
= m2
< s2
? m2
: s2
;
2995 mhi
= pow5mult(PASS_STATE mhi
, m5
);
2996 b1
= mult(PASS_STATE mhi
, b
);
2997 Bfree(PASS_STATE b
);
3001 b
= pow5mult(PASS_STATE b
, j
);
3004 b
= pow5mult(PASS_STATE b
, b5
);
3006 S
= i2b(PASS_STATE
1);
3008 S
= pow5mult(PASS_STATE S
, s5
);
3010 /* Check for special case that d is a normalized power of 2. */
3013 if ((mode
< 2 || leftright
)
3014 #ifdef Honor_FLT_ROUNDS
3018 if (!word1(d
) && !(word0(d
) & Bndry_mask
)
3019 #ifndef Sudden_Underflow
3020 && word0(d
) & (Exp_mask
& ~Exp_msk1
)
3023 /* The special case */
3030 /* Arrange for convenient computation of quotients:
3031 * shift left if necessary so divisor has 4 leading 0 bits.
3033 * Perhaps we should just compute leading 28 bits of S once
3034 * and for all and pass them and a shift to quorem, so it
3035 * can do shifts and ors to compute the numerator for q.
3038 if ((i
= ((s5
? 32 - hi0bits(S
->x
[S
->wds
-1]) : 1) + s2
) & 0x1f))
3041 if (i
= ((s5
? 32 - hi0bits(S
->x
[S
->wds
-1]) : 1) + s2
) & 0xf)
3057 b
= lshift(PASS_STATE b
, b2
);
3059 S
= lshift(PASS_STATE S
, s2
);
3063 b
= multadd(PASS_STATE b
, 10, 0); /* we botched the k estimate */
3065 mhi
= multadd(PASS_STATE mhi
, 10, 0);
3069 if (ilim
<= 0 && (mode
== 3 || mode
== 5)) {
3070 if (ilim
< 0 || cmp(b
,S
= multadd(PASS_STATE S
,5,0)) < 0) {
3071 /* no digits, fcvt style */
3073 /* MOZILLA CHANGE: Always return a non-empty string. */
3085 mhi
= lshift(PASS_STATE mhi
, m2
);
3087 /* Compute mlo -- check for special case
3088 * that d is a normalized power of 2.
3093 mhi
= Balloc(PASS_STATE mhi
->k
);
3095 mhi
= lshift(PASS_STATE mhi
, Log2P
);
3099 dig
= quorem(b
,S
) + '0';
3100 /* Do we yet have the shortest decimal string
3101 * that will round to d?
3104 delta
= diff(PASS_STATE S
, mhi
);
3105 j1
= delta
->sign
? 1 : cmp(b
, delta
);
3106 Bfree(PASS_STATE delta
);
3107 #ifndef ROUND_BIASED
3108 if (j1
== 0 && mode
!= 1 && !(word1(d
) & 1)
3109 #ifdef Honor_FLT_ROUNDS
3118 else if (!b
->x
[0] && b
->wds
<= 1)
3125 if (j
< 0 || (j
== 0 && mode
!= 1
3126 #ifndef ROUND_BIASED
3130 if (!b
->x
[0] && b
->wds
<= 1) {
3136 #ifdef Honor_FLT_ROUNDS
3139 case 0: goto accept_dig
;
3140 case 2: goto keep_dig
;
3142 #endif /*Honor_FLT_ROUNDS*/
3144 b
= lshift(PASS_STATE b
, 1);
3146 if ((j1
> 0 || (j1
== 0 && dig
& 1))
3155 #ifdef Honor_FLT_ROUNDS
3159 if (dig
== '9') { /* possible if i == 1 */
3167 #ifdef Honor_FLT_ROUNDS
3173 b
= multadd(PASS_STATE b
, 10, 0);
3175 mlo
= mhi
= multadd(PASS_STATE mhi
, 10, 0);
3177 mlo
= multadd(PASS_STATE mlo
, 10, 0);
3178 mhi
= multadd(PASS_STATE mhi
, 10, 0);
3184 *s
++ = dig
= quorem(b
,S
) + '0';
3185 if (!b
->x
[0] && b
->wds
<= 1) {
3193 b
= multadd(PASS_STATE b
, 10, 0);
3196 /* Round off last digit */
3198 #ifdef Honor_FLT_ROUNDS
3200 case 0: goto trimzeros
;
3201 case 2: goto roundoff
;
3204 b
= lshift(PASS_STATE b
, 1);
3206 if (j
>= 0) { /* ECMA compatible rounding needed by Spidermonkey */
3217 #ifdef Honor_FLT_ROUNDS
3224 Bfree(PASS_STATE S
);
3226 if (mlo
&& mlo
!= mhi
)
3227 Bfree(PASS_STATE mlo
);
3228 Bfree(PASS_STATE mhi
);
3234 word0(d
) = Exp_1
+ (70 << Exp_shift
);
3239 else if (!oldinexact
)
3242 Bfree(PASS_STATE b
);