-
-
Notifications
You must be signed in to change notification settings - Fork 30.7k
/
dtoa.c
2841 lines (2566 loc) · 78.6 KB
/
dtoa.c
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/****************************************************************
*
* The author of this software is David M. Gay.
*
* Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
*
* Permission to use, copy, modify, and distribute this software for any
* purpose without fee is hereby granted, provided that this entire notice
* is included in all copies of any software which is or includes a copy
* or modification of this software and in all copies of the supporting
* documentation for such software.
*
* THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
* WARRANTY. IN PARTICULAR, NEITHER THE AUTHOR NOR LUCENT MAKES ANY
* REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
* OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
*
***************************************************************/
/****************************************************************
* This is dtoa.c by David M. Gay, downloaded from
* http://www.netlib.org/fp/dtoa.c on April 15, 2009 and modified for
* inclusion into the Python core by Mark E. T. Dickinson and Eric V. Smith.
*
* Please remember to check http://www.netlib.org/fp regularly (and especially
* before any Python release) for bugfixes and updates.
*
* The major modifications from Gay's original code are as follows:
*
* 0. The original code has been specialized to Python's needs by removing
* many of the #ifdef'd sections. In particular, code to support VAX and
* IBM floating-point formats, hex NaNs, hex floats, locale-aware
* treatment of the decimal point, and setting of the inexact flag have
* been removed.
*
* 1. We use PyMem_Malloc and PyMem_Free in place of malloc and free.
*
* 2. The public functions strtod, dtoa and freedtoa all now have
* a _Py_dg_ prefix.
*
* 3. Instead of assuming that PyMem_Malloc always succeeds, we thread
* PyMem_Malloc failures through the code. The functions
*
* Balloc, multadd, s2b, i2b, mult, pow5mult, lshift, diff, d2b
*
* of return type *Bigint all return NULL to indicate a malloc failure.
* Similarly, rv_alloc and nrv_alloc (return type char *) return NULL on
* failure. bigcomp now has return type int (it used to be void) and
* returns -1 on failure and 0 otherwise. _Py_dg_dtoa returns NULL
* on failure. _Py_dg_strtod indicates failure due to malloc failure
* by returning -1.0, setting errno=ENOMEM and *se to s00.
*
* 4. The static variable dtoa_result has been removed. Callers of
* _Py_dg_dtoa are expected to call _Py_dg_freedtoa to free
* the memory allocated by _Py_dg_dtoa.
*
* 5. The code has been reformatted to better fit with Python's
* C style guide (PEP 7).
*
* 6. A bug in the memory allocation has been fixed: to avoid FREEing memory
* that hasn't been MALLOC'ed, private_mem should only be used when k <=
* Kmax.
*
* 7. _Py_dg_strtod has been modified so that it doesn't accept strings with
* leading whitespace.
*
* 8. A corner case where _Py_dg_dtoa didn't strip trailing zeros has been
* fixed. (bugs.python.org/issue40780)
*
***************************************************************/
/* Please send bug reports for the original dtoa.c code to David M. Gay (dmg
* at acm dot org, with " at " changed at "@" and " dot " changed to ".").
* Please report bugs for this modified version using the Python issue tracker
* as detailed at (https://devguide.python.org/triage/issue-tracker/). */
/* On a machine with IEEE extended-precision registers, it is
* necessary to specify double-precision (53-bit) rounding precision
* before invoking strtod or dtoa. If the machine uses (the equivalent
* of) Intel 80x87 arithmetic, the call
* _control87(PC_53, MCW_PC);
* does this with many compilers. Whether this or another call is
* appropriate depends on the compiler; for this to work, it may be
* necessary to #include "float.h" or another system-dependent header
* file.
*/
/* strtod for IEEE-, VAX-, and IBM-arithmetic machines.
*
* This strtod returns a nearest machine number to the input decimal
* string (or sets errno to ERANGE). With IEEE arithmetic, ties are
* broken by the IEEE round-even rule. Otherwise ties are broken by
* biased rounding (add half and chop).
*
* Inspired loosely by William D. Clinger's paper "How to Read Floating
* Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101].
*
* Modifications:
*
* 1. We only require IEEE, IBM, or VAX double-precision
* arithmetic (not IEEE double-extended).
* 2. We get by with floating-point arithmetic in a case that
* Clinger missed -- when we're computing d * 10^n
* for a small integer d and the integer n is not too
* much larger than 22 (the maximum integer k for which
* we can represent 10^k exactly), we may be able to
* compute (d*10^k) * 10^(e-k) with just one roundoff.
* 3. Rather than a bit-at-a-time adjustment of the binary
* result in the hard case, we use floating-point
* arithmetic to determine the adjustment to within
* one bit; only in really hard cases do we need to
* compute a second residual.
* 4. Because of 3., we don't need a large table of powers of 10
* for ten-to-e (just some small tables, e.g. of 10^k
* for 0 <= k <= 22).
*/
/* Linking of Python's #defines to Gay's #defines starts here. */
#include "Python.h"
#include "pycore_dtoa.h" // _PY_SHORT_FLOAT_REPR
#include "pycore_pystate.h" // _PyInterpreterState_GET()
#include <stdlib.h> // exit()
/* if _PY_SHORT_FLOAT_REPR == 0, then don't even try to compile
the following code */
#if _PY_SHORT_FLOAT_REPR == 1
#include "float.h"
#define MALLOC PyMem_Malloc
#define FREE PyMem_Free
/* This code should also work for ARM mixed-endian format on little-endian
machines, where doubles have byte order 45670123 (in increasing address
order, 0 being the least significant byte). */
#ifdef DOUBLE_IS_LITTLE_ENDIAN_IEEE754
# define IEEE_8087
#endif
#if defined(DOUBLE_IS_BIG_ENDIAN_IEEE754) || \
defined(DOUBLE_IS_ARM_MIXED_ENDIAN_IEEE754)
# define IEEE_MC68k
#endif
#if defined(IEEE_8087) + defined(IEEE_MC68k) != 1
#error "Exactly one of IEEE_8087 or IEEE_MC68k should be defined."
#endif
/* The code below assumes that the endianness of integers matches the
endianness of the two 32-bit words of a double. Check this. */
#if defined(WORDS_BIGENDIAN) && (defined(DOUBLE_IS_LITTLE_ENDIAN_IEEE754) || \
defined(DOUBLE_IS_ARM_MIXED_ENDIAN_IEEE754))
#error "doubles and ints have incompatible endianness"
#endif
#if !defined(WORDS_BIGENDIAN) && defined(DOUBLE_IS_BIG_ENDIAN_IEEE754)
#error "doubles and ints have incompatible endianness"
#endif
// ULong is defined in pycore_dtoa.h.
typedef int32_t Long;
typedef uint64_t ULLong;
#undef DEBUG
#ifdef Py_DEBUG
#define DEBUG
#endif
/* End Python #define linking */
#ifdef DEBUG
#define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
#endif
typedef union { double d; ULong L[2]; } U;
#ifdef IEEE_8087
#define word0(x) (x)->L[1]
#define word1(x) (x)->L[0]
#else
#define word0(x) (x)->L[0]
#define word1(x) (x)->L[1]
#endif
#define dval(x) (x)->d
#ifndef STRTOD_DIGLIM
#define STRTOD_DIGLIM 40
#endif
/* maximum permitted exponent value for strtod; exponents larger than
MAX_ABS_EXP in absolute value get truncated to +-MAX_ABS_EXP. MAX_ABS_EXP
should fit into an int. */
#ifndef MAX_ABS_EXP
#define MAX_ABS_EXP 1100000000U
#endif
/* Bound on length of pieces of input strings in _Py_dg_strtod; specifically,
this is used to bound the total number of digits ignoring leading zeros and
the number of digits that follow the decimal point. Ideally, MAX_DIGITS
should satisfy MAX_DIGITS + 400 < MAX_ABS_EXP; that ensures that the
exponent clipping in _Py_dg_strtod can't affect the value of the output. */
#ifndef MAX_DIGITS
#define MAX_DIGITS 1000000000U
#endif
/* Guard against trying to use the above values on unusual platforms with ints
* of width less than 32 bits. */
#if MAX_ABS_EXP > INT_MAX
#error "MAX_ABS_EXP should fit in an int"
#endif
#if MAX_DIGITS > INT_MAX
#error "MAX_DIGITS should fit in an int"
#endif
/* The following definition of Storeinc is appropriate for MIPS processors.
* An alternative that might be better on some machines is
* #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
*/
#if defined(IEEE_8087)
#define Storeinc(a,b,c) (((unsigned short *)a)[1] = (unsigned short)b, \
((unsigned short *)a)[0] = (unsigned short)c, a++)
#else
#define Storeinc(a,b,c) (((unsigned short *)a)[0] = (unsigned short)b, \
((unsigned short *)a)[1] = (unsigned short)c, a++)
#endif
/* #define P DBL_MANT_DIG */
/* Ten_pmax = floor(P*log(2)/log(5)) */
/* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
/* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
/* Int_max = floor(P*log(FLT_RADIX)/log(10) - 1) */
#define Exp_shift 20
#define Exp_shift1 20
#define Exp_msk1 0x100000
#define Exp_msk11 0x100000
#define Exp_mask 0x7ff00000
#define P 53
#define Nbits 53
#define Bias 1023
#define Emax 1023
#define Emin (-1022)
#define Etiny (-1074) /* smallest denormal is 2**Etiny */
#define Exp_1 0x3ff00000
#define Exp_11 0x3ff00000
#define Ebits 11
#define Frac_mask 0xfffff
#define Frac_mask1 0xfffff
#define Ten_pmax 22
#define Bletch 0x10
#define Bndry_mask 0xfffff
#define Bndry_mask1 0xfffff
#define Sign_bit 0x80000000
#define Log2P 1
#define Tiny0 0
#define Tiny1 1
#define Quick_max 14
#define Int_max 14
#ifndef Flt_Rounds
#ifdef FLT_ROUNDS
#define Flt_Rounds FLT_ROUNDS
#else
#define Flt_Rounds 1
#endif
#endif /*Flt_Rounds*/
#define Rounding Flt_Rounds
#define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
#define Big1 0xffffffff
/* Bits of the representation of positive infinity. */
#define POSINF_WORD0 0x7ff00000
#define POSINF_WORD1 0
/* struct BCinfo is used to pass information from _Py_dg_strtod to bigcomp */
typedef struct BCinfo BCinfo;
struct
BCinfo {
int e0, nd, nd0, scale;
};
#define FFFFFFFF 0xffffffffUL
/* struct Bigint is used to represent arbitrary-precision integers. These
integers are stored in sign-magnitude format, with the magnitude stored as
an array of base 2**32 digits. Bigints are always normalized: if x is a
Bigint then x->wds >= 1, and either x->wds == 1 or x[wds-1] is nonzero.
The Bigint fields are as follows:
- next is a header used by Balloc and Bfree to keep track of lists
of freed Bigints; it's also used for the linked list of
powers of 5 of the form 5**2**i used by pow5mult.
- k indicates which pool this Bigint was allocated from
- maxwds is the maximum number of words space was allocated for
(usually maxwds == 2**k)
- sign is 1 for negative Bigints, 0 for positive. The sign is unused
(ignored on inputs, set to 0 on outputs) in almost all operations
involving Bigints: a notable exception is the diff function, which
ignores signs on inputs but sets the sign of the output correctly.
- wds is the actual number of significant words
- x contains the vector of words (digits) for this Bigint, from least
significant (x[0]) to most significant (x[wds-1]).
*/
// struct Bigint is defined in pycore_dtoa.h.
typedef struct Bigint Bigint;
#if !defined(Py_GIL_DISABLED) && !defined(Py_USING_MEMORY_DEBUGGER)
/* Memory management: memory is allocated from, and returned to, Kmax+1 pools
of memory, where pool k (0 <= k <= Kmax) is for Bigints b with b->maxwds ==
1 << k. These pools are maintained as linked lists, with freelist[k]
pointing to the head of the list for pool k.
On allocation, if there's no free slot in the appropriate pool, MALLOC is
called to get more memory. This memory is not returned to the system until
Python quits. There's also a private memory pool that's allocated from
in preference to using MALLOC.
For Bigints with more than (1 << Kmax) digits (which implies at least 1233
decimal digits), memory is directly allocated using MALLOC, and freed using
FREE.
XXX: it would be easy to bypass this memory-management system and
translate each call to Balloc into a call to PyMem_Malloc, and each
Bfree to PyMem_Free. Investigate whether this has any significant
performance on impact. */
#define freelist interp->dtoa.freelist
#define private_mem interp->dtoa.preallocated
#define pmem_next interp->dtoa.preallocated_next
/* Allocate space for a Bigint with up to 1<<k digits */
static Bigint *
Balloc(int k)
{
int x;
Bigint *rv;
unsigned int len;
PyInterpreterState *interp = _PyInterpreterState_GET();
if (k <= Bigint_Kmax && (rv = freelist[k]))
freelist[k] = rv->next;
else {
x = 1 << k;
len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
/sizeof(double);
if (k <= Bigint_Kmax &&
pmem_next - private_mem + len <= (Py_ssize_t)Bigint_PREALLOC_SIZE
) {
rv = (Bigint*)pmem_next;
pmem_next += len;
}
else {
rv = (Bigint*)MALLOC(len*sizeof(double));
if (rv == NULL)
return NULL;
}
rv->k = k;
rv->maxwds = x;
}
rv->sign = rv->wds = 0;
return rv;
}
/* Free a Bigint allocated with Balloc */
static void
Bfree(Bigint *v)
{
if (v) {
if (v->k > Bigint_Kmax)
FREE((void*)v);
else {
PyInterpreterState *interp = _PyInterpreterState_GET();
v->next = freelist[v->k];
freelist[v->k] = v;
}
}
}
#undef pmem_next
#undef private_mem
#undef freelist
#else
/* Alternative versions of Balloc and Bfree that use PyMem_Malloc and
PyMem_Free directly in place of the custom memory allocation scheme above.
These are provided for the benefit of memory debugging tools like
Valgrind. */
/* Allocate space for a Bigint with up to 1<<k digits */
static Bigint *
Balloc(int k)
{
int x;
Bigint *rv;
unsigned int len;
x = 1 << k;
len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
/sizeof(double);
rv = (Bigint*)MALLOC(len*sizeof(double));
if (rv == NULL)
return NULL;
rv->k = k;
rv->maxwds = x;
rv->sign = rv->wds = 0;
return rv;
}
/* Free a Bigint allocated with Balloc */
static void
Bfree(Bigint *v)
{
if (v) {
FREE((void*)v);
}
}
#endif /* !defined(Py_GIL_DISABLED) && !defined(Py_USING_MEMORY_DEBUGGER) */
#define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
y->wds*sizeof(Long) + 2*sizeof(int))
/* Multiply a Bigint b by m and add a. Either modifies b in place and returns
a pointer to the modified b, or Bfrees b and returns a pointer to a copy.
On failure, return NULL. In this case, b will have been already freed. */
static Bigint *
multadd(Bigint *b, int m, int a) /* multiply by m and add a */
{
int i, wds;
ULong *x;
ULLong carry, y;
Bigint *b1;
wds = b->wds;
x = b->x;
i = 0;
carry = a;
do {
y = *x * (ULLong)m + carry;
carry = y >> 32;
*x++ = (ULong)(y & FFFFFFFF);
}
while(++i < wds);
if (carry) {
if (wds >= b->maxwds) {
b1 = Balloc(b->k+1);
if (b1 == NULL){
Bfree(b);
return NULL;
}
Bcopy(b1, b);
Bfree(b);
b = b1;
}
b->x[wds++] = (ULong)carry;
b->wds = wds;
}
return b;
}
/* convert a string s containing nd decimal digits (possibly containing a
decimal separator at position nd0, which is ignored) to a Bigint. This
function carries on where the parsing code in _Py_dg_strtod leaves off: on
entry, y9 contains the result of converting the first 9 digits. Returns
NULL on failure. */
static Bigint *
s2b(const char *s, int nd0, int nd, ULong y9)
{
Bigint *b;
int i, k;
Long x, y;
x = (nd + 8) / 9;
for(k = 0, y = 1; x > y; y <<= 1, k++) ;
b = Balloc(k);
if (b == NULL)
return NULL;
b->x[0] = y9;
b->wds = 1;
if (nd <= 9)
return b;
s += 9;
for (i = 9; i < nd0; i++) {
b = multadd(b, 10, *s++ - '0');
if (b == NULL)
return NULL;
}
s++;
for(; i < nd; i++) {
b = multadd(b, 10, *s++ - '0');
if (b == NULL)
return NULL;
}
return b;
}
/* count leading 0 bits in the 32-bit integer x. */
static int
hi0bits(ULong x)
{
int k = 0;
if (!(x & 0xffff0000)) {
k = 16;
x <<= 16;
}
if (!(x & 0xff000000)) {
k += 8;
x <<= 8;
}
if (!(x & 0xf0000000)) {
k += 4;
x <<= 4;
}
if (!(x & 0xc0000000)) {
k += 2;
x <<= 2;
}
if (!(x & 0x80000000)) {
k++;
if (!(x & 0x40000000))
return 32;
}
return k;
}
/* count trailing 0 bits in the 32-bit integer y, and shift y right by that
number of bits. */
static int
lo0bits(ULong *y)
{
int k;
ULong x = *y;
if (x & 7) {
if (x & 1)
return 0;
if (x & 2) {
*y = x >> 1;
return 1;
}
*y = x >> 2;
return 2;
}
k = 0;
if (!(x & 0xffff)) {
k = 16;
x >>= 16;
}
if (!(x & 0xff)) {
k += 8;
x >>= 8;
}
if (!(x & 0xf)) {
k += 4;
x >>= 4;
}
if (!(x & 0x3)) {
k += 2;
x >>= 2;
}
if (!(x & 1)) {
k++;
x >>= 1;
if (!x)
return 32;
}
*y = x;
return k;
}
/* convert a small nonnegative integer to a Bigint */
static Bigint *
i2b(int i)
{
Bigint *b;
b = Balloc(1);
if (b == NULL)
return NULL;
b->x[0] = i;
b->wds = 1;
return b;
}
/* multiply two Bigints. Returns a new Bigint, or NULL on failure. Ignores
the signs of a and b. */
static Bigint *
mult(Bigint *a, Bigint *b)
{
Bigint *c;
int k, wa, wb, wc;
ULong *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
ULong y;
ULLong carry, z;
if ((!a->x[0] && a->wds == 1) || (!b->x[0] && b->wds == 1)) {
c = Balloc(0);
if (c == NULL)
return NULL;
c->wds = 1;
c->x[0] = 0;
return c;
}
if (a->wds < b->wds) {
c = a;
a = b;
b = c;
}
k = a->k;
wa = a->wds;
wb = b->wds;
wc = wa + wb;
if (wc > a->maxwds)
k++;
c = Balloc(k);
if (c == NULL)
return NULL;
for(x = c->x, xa = x + wc; x < xa; x++)
*x = 0;
xa = a->x;
xae = xa + wa;
xb = b->x;
xbe = xb + wb;
xc0 = c->x;
for(; xb < xbe; xc0++) {
if ((y = *xb++)) {
x = xa;
xc = xc0;
carry = 0;
do {
z = *x++ * (ULLong)y + *xc + carry;
carry = z >> 32;
*xc++ = (ULong)(z & FFFFFFFF);
}
while(x < xae);
*xc = (ULong)carry;
}
}
for(xc0 = c->x, xc = xc0 + wc; wc > 0 && !*--xc; --wc) ;
c->wds = wc;
return c;
}
#ifndef Py_USING_MEMORY_DEBUGGER
/* multiply the Bigint b by 5**k. Returns a pointer to the result, or NULL on
failure; if the returned pointer is distinct from b then the original
Bigint b will have been Bfree'd. Ignores the sign of b. */
static Bigint *
pow5mult(Bigint *b, int k)
{
Bigint *b1, *p5, **p5s;
int i;
static const int p05[3] = { 5, 25, 125 };
// For double-to-string conversion, the maximum value of k is limited by
// DBL_MAX_10_EXP (308), the maximum decimal base-10 exponent for binary64.
// For string-to-double conversion, the extreme case is constrained by our
// hardcoded exponent limit before we underflow of -512, adjusted by
// STRTOD_DIGLIM-DBL_DIG-1, giving a maximum of k=535.
assert(0 <= k && k < 1024);
if ((i = k & 3)) {
b = multadd(b, p05[i-1], 0);
if (b == NULL)
return NULL;
}
if (!(k >>= 2))
return b;
PyInterpreterState *interp = _PyInterpreterState_GET();
p5s = interp->dtoa.p5s;
for(;;) {
assert(p5s != interp->dtoa.p5s + Bigint_Pow5size);
p5 = *p5s;
p5s++;
if (k & 1) {
b1 = mult(b, p5);
Bfree(b);
b = b1;
if (b == NULL)
return NULL;
}
if (!(k >>= 1))
break;
}
return b;
}
#else
/* Version of pow5mult that doesn't cache powers of 5. Provided for
the benefit of memory debugging tools like Valgrind. */
static Bigint *
pow5mult(Bigint *b, int k)
{
Bigint *b1, *p5, *p51;
int i;
static const int p05[3] = { 5, 25, 125 };
if ((i = k & 3)) {
b = multadd(b, p05[i-1], 0);
if (b == NULL)
return NULL;
}
if (!(k >>= 2))
return b;
p5 = i2b(625);
if (p5 == NULL) {
Bfree(b);
return NULL;
}
for(;;) {
if (k & 1) {
b1 = mult(b, p5);
Bfree(b);
b = b1;
if (b == NULL) {
Bfree(p5);
return NULL;
}
}
if (!(k >>= 1))
break;
p51 = mult(p5, p5);
Bfree(p5);
p5 = p51;
if (p5 == NULL) {
Bfree(b);
return NULL;
}
}
Bfree(p5);
return b;
}
#endif /* Py_USING_MEMORY_DEBUGGER */
/* shift a Bigint b left by k bits. Return a pointer to the shifted result,
or NULL on failure. If the returned pointer is distinct from b then the
original b will have been Bfree'd. Ignores the sign of b. */
static Bigint *
lshift(Bigint *b, int k)
{
int i, k1, n, n1;
Bigint *b1;
ULong *x, *x1, *xe, z;
if (!k || (!b->x[0] && b->wds == 1))
return b;
n = k >> 5;
k1 = b->k;
n1 = n + b->wds + 1;
for(i = b->maxwds; n1 > i; i <<= 1)
k1++;
b1 = Balloc(k1);
if (b1 == NULL) {
Bfree(b);
return NULL;
}
x1 = b1->x;
for(i = 0; i < n; i++)
*x1++ = 0;
x = b->x;
xe = x + b->wds;
if (k &= 0x1f) {
k1 = 32 - k;
z = 0;
do {
*x1++ = *x << k | z;
z = *x++ >> k1;
}
while(x < xe);
if ((*x1 = z))
++n1;
}
else do
*x1++ = *x++;
while(x < xe);
b1->wds = n1 - 1;
Bfree(b);
return b1;
}
/* Do a three-way compare of a and b, returning -1 if a < b, 0 if a == b and
1 if a > b. Ignores signs of a and b. */
static int
cmp(Bigint *a, Bigint *b)
{
ULong *xa, *xa0, *xb, *xb0;
int i, j;
i = a->wds;
j = b->wds;
#ifdef DEBUG
if (i > 1 && !a->x[i-1])
Bug("cmp called with a->x[a->wds-1] == 0");
if (j > 1 && !b->x[j-1])
Bug("cmp called with b->x[b->wds-1] == 0");
#endif
if (i -= j)
return i;
xa0 = a->x;
xa = xa0 + j;
xb0 = b->x;
xb = xb0 + j;
for(;;) {
if (*--xa != *--xb)
return *xa < *xb ? -1 : 1;
if (xa <= xa0)
break;
}
return 0;
}
/* Take the difference of Bigints a and b, returning a new Bigint. Returns
NULL on failure. The signs of a and b are ignored, but the sign of the
result is set appropriately. */
static Bigint *
diff(Bigint *a, Bigint *b)
{
Bigint *c;
int i, wa, wb;
ULong *xa, *xae, *xb, *xbe, *xc;
ULLong borrow, y;
i = cmp(a,b);
if (!i) {
c = Balloc(0);
if (c == NULL)
return NULL;
c->wds = 1;
c->x[0] = 0;
return c;
}
if (i < 0) {
c = a;
a = b;
b = c;
i = 1;
}
else
i = 0;
c = Balloc(a->k);
if (c == NULL)
return NULL;
c->sign = i;
wa = a->wds;
xa = a->x;
xae = xa + wa;
wb = b->wds;
xb = b->x;
xbe = xb + wb;
xc = c->x;
borrow = 0;
do {
y = (ULLong)*xa++ - *xb++ - borrow;
borrow = y >> 32 & (ULong)1;
*xc++ = (ULong)(y & FFFFFFFF);
}
while(xb < xbe);
while(xa < xae) {
y = *xa++ - borrow;
borrow = y >> 32 & (ULong)1;
*xc++ = (ULong)(y & FFFFFFFF);
}
while(!*--xc)
wa--;
c->wds = wa;
return c;
}
/* Given a positive normal double x, return the difference between x and the
next double up. Doesn't give correct results for subnormals. */
static double
ulp(U *x)
{
Long L;
U u;
L = (word0(x) & Exp_mask) - (P-1)*Exp_msk1;
word0(&u) = L;
word1(&u) = 0;
return dval(&u);
}
/* Convert a Bigint to a double plus an exponent */
static double
b2d(Bigint *a, int *e)
{
ULong *xa, *xa0, w, y, z;
int k;
U d;
xa0 = a->x;
xa = xa0 + a->wds;
y = *--xa;
#ifdef DEBUG
if (!y) Bug("zero y in b2d");
#endif
k = hi0bits(y);
*e = 32 - k;
if (k < Ebits) {
word0(&d) = Exp_1 | y >> (Ebits - k);
w = xa > xa0 ? *--xa : 0;
word1(&d) = y << ((32-Ebits) + k) | w >> (Ebits - k);
goto ret_d;
}
z = xa > xa0 ? *--xa : 0;
if (k -= Ebits) {
word0(&d) = Exp_1 | y << k | z >> (32 - k);
y = xa > xa0 ? *--xa : 0;
word1(&d) = z << k | y >> (32 - k);
}
else {
word0(&d) = Exp_1 | y;
word1(&d) = z;
}
ret_d:
return dval(&d);
}
/* Convert a scaled double to a Bigint plus an exponent. Similar to d2b,
except that it accepts the scale parameter used in _Py_dg_strtod (which
should be either 0 or 2*P), and the normalization for the return value is
different (see below). On input, d should be finite and nonnegative, and d
/ 2**scale should be exactly representable as an IEEE 754 double.
Returns a Bigint b and an integer e such that
dval(d) / 2**scale = b * 2**e.
Unlike d2b, b is not necessarily odd: b and e are normalized so
that either 2**(P-1) <= b < 2**P and e >= Etiny, or b < 2**P
and e == Etiny. This applies equally to an input of 0.0: in that
case the return values are b = 0 and e = Etiny.
The above normalization ensures that for all possible inputs d,
2**e gives ulp(d/2**scale).
Returns NULL on failure.
*/
static Bigint *
sd2b(U *d, int scale, int *e)
{
Bigint *b;
b = Balloc(1);
if (b == NULL)
return NULL;
/* First construct b and e assuming that scale == 0. */
b->wds = 2;
b->x[0] = word1(d);
b->x[1] = word0(d) & Frac_mask;
*e = Etiny - 1 + (int)((word0(d) & Exp_mask) >> Exp_shift);
if (*e < Etiny)
*e = Etiny;
else
b->x[1] |= Exp_msk1;
/* Now adjust for scale, provided that b != 0. */
if (scale && (b->x[0] || b->x[1])) {
*e -= scale;
if (*e < Etiny) {
scale = Etiny - *e;