-
-
Notifications
You must be signed in to change notification settings - Fork 34.4k
Expand file tree
/
Copy pathcmathmodule.c
More file actions
1316 lines (1114 loc) · 39.1 KB
/
cmathmodule.c
File metadata and controls
1316 lines (1114 loc) · 39.1 KB
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
/* Complex math module */
/* much code borrowed from mathmodule.c */
#ifndef Py_BUILD_CORE_BUILTIN
# define Py_BUILD_CORE_MODULE 1
#endif
#include "Python.h"
#include "pycore_complexobject.h" // _Py_c_neg()
#include "pycore_pymath.h" // _PY_SHORT_FLOAT_REPR
/* we need DBL_MAX, DBL_MIN, DBL_EPSILON, DBL_MANT_DIG and FLT_RADIX from
float.h. We assume that FLT_RADIX is either 2 or 16. */
#include <float.h>
/* For _Py_log1p with workarounds for buggy handling of zeros. */
#include "_math.h"
#include "clinic/cmathmodule.c.h"
/*[clinic input]
module cmath
[clinic start generated code]*/
/*[clinic end generated code: output=da39a3ee5e6b4b0d input=308d6839f4a46333]*/
/*[python input]
class Py_complex_protected_converter(Py_complex_converter):
def modify(self):
return 'errno = 0;'
class Py_complex_protected_return_converter(CReturnConverter):
type = "Py_complex"
def render(self, function, data):
self.declare(data)
data.return_conversion.append("""
if (errno == EDOM) {
PyErr_SetString(PyExc_ValueError, "math domain error");
goto exit;
}
else if (errno == ERANGE) {
PyErr_SetString(PyExc_OverflowError, "math range error");
goto exit;
}
else {
return_value = PyComplex_FromCComplex(_return_value);
}
""".strip())
[python start generated code]*/
/*[python end generated code: output=da39a3ee5e6b4b0d input=8b27adb674c08321]*/
#if (FLT_RADIX != 2 && FLT_RADIX != 16)
#error "Modules/cmathmodule.c expects FLT_RADIX to be 2 or 16"
#endif
#ifndef M_LN2
#define M_LN2 (0.6931471805599453094) /* natural log of 2 */
#endif
#ifndef M_LN10
#define M_LN10 (2.302585092994045684) /* natural log of 10 */
#endif
/*
CM_LARGE_DOUBLE is used to avoid spurious overflow in the sqrt, log,
inverse trig and inverse hyperbolic trig functions. Its log is used in the
evaluation of exp, cos, cosh, sin, sinh, tan, and tanh to avoid unnecessary
overflow.
*/
#define CM_LARGE_DOUBLE (DBL_MAX/4.)
#define CM_SQRT_LARGE_DOUBLE (sqrt(CM_LARGE_DOUBLE))
#define CM_LOG_LARGE_DOUBLE (log(CM_LARGE_DOUBLE))
#define CM_SQRT_DBL_MIN (sqrt(DBL_MIN))
/*
CM_SCALE_UP is an odd integer chosen such that multiplication by
2**CM_SCALE_UP is sufficient to turn a subnormal into a normal.
CM_SCALE_DOWN is (-(CM_SCALE_UP+1)/2). These scalings are used to compute
square roots accurately when the real and imaginary parts of the argument
are subnormal.
*/
#if FLT_RADIX==2
#define CM_SCALE_UP (2*(DBL_MANT_DIG/2) + 1)
#elif FLT_RADIX==16
#define CM_SCALE_UP (4*DBL_MANT_DIG+1)
#endif
#define CM_SCALE_DOWN (-(CM_SCALE_UP+1)/2)
/* forward declarations */
static Py_complex cmath_asinh_impl(PyObject *, Py_complex);
static Py_complex cmath_atanh_impl(PyObject *, Py_complex);
static Py_complex cmath_cosh_impl(PyObject *, Py_complex);
static Py_complex cmath_sinh_impl(PyObject *, Py_complex);
static Py_complex cmath_sqrt_impl(PyObject *, Py_complex);
static Py_complex cmath_tanh_impl(PyObject *, Py_complex);
static PyObject * math_error(void);
/* Code to deal with special values (infinities, NaNs, etc.). */
/* special_type takes a double and returns an integer code indicating
the type of the double as follows:
*/
enum special_types {
ST_NINF, /* 0, negative infinity */
ST_NEG, /* 1, negative finite number (nonzero) */
ST_NZERO, /* 2, -0. */
ST_PZERO, /* 3, +0. */
ST_POS, /* 4, positive finite number (nonzero) */
ST_PINF, /* 5, positive infinity */
ST_NAN /* 6, Not a Number */
};
static enum special_types
special_type(double d)
{
if (isfinite(d)) {
if (d != 0) {
if (copysign(1., d) == 1.)
return ST_POS;
else
return ST_NEG;
}
else {
if (copysign(1., d) == 1.)
return ST_PZERO;
else
return ST_NZERO;
}
}
if (isnan(d))
return ST_NAN;
if (copysign(1., d) == 1.)
return ST_PINF;
else
return ST_NINF;
}
#define SPECIAL_VALUE(z, table) \
if (!isfinite((z).real) || !isfinite((z).imag)) { \
errno = 0; \
return table[special_type((z).real)] \
[special_type((z).imag)]; \
}
#define P Py_MATH_PI
#define P14 0.25*Py_MATH_PI
#define P12 0.5*Py_MATH_PI
#define P34 0.75*Py_MATH_PI
#define INF INFINITY
#define N Py_NAN
#define U -9.5426319407711027e33 /* unlikely value, used as placeholder */
/* First, the C functions that do the real work. Each of the c_*
functions computes and returns the C99 Annex G recommended result
and also sets errno as follows: errno = 0 if no floating-point
exception is associated with the result; errno = EDOM if C99 Annex
G recommends raising divide-by-zero or invalid for this result; and
errno = ERANGE where the overflow floating-point signal should be
raised.
*/
static Py_complex acos_special_values[7][7] = {
{ {P34,INF}, {P,INF}, {P,INF}, {P,-INF}, {P,-INF}, {P34,-INF}, {N,INF} },
{ {P12,INF}, {U,U}, {U,U}, {U,U}, {U,U}, {P12,-INF}, {N,N} },
{ {P12,INF}, {U,U}, {P12,0.}, {P12,-0.}, {U,U}, {P12,-INF}, {P12,N} },
{ {P12,INF}, {U,U}, {P12,0.}, {P12,-0.}, {U,U}, {P12,-INF}, {P12,N} },
{ {P12,INF}, {U,U}, {U,U}, {U,U}, {U,U}, {P12,-INF}, {N,N} },
{ {P14,INF}, {0.,INF}, {0.,INF}, {0.,-INF}, {0.,-INF}, {P14,-INF}, {N,INF} },
{ {N,INF}, {N,N}, {N,N}, {N,N}, {N,N}, {N,-INF}, {N,N} }
};
/*[clinic input]
cmath.acos -> Py_complex_protected
z: Py_complex_protected
/
Return the arc cosine of z.
[clinic start generated code]*/
static Py_complex
cmath_acos_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=40bd42853fd460ae input=bd6cbd78ae851927]*/
{
Py_complex s1, s2, r;
SPECIAL_VALUE(z, acos_special_values);
if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
/* avoid unnecessary overflow for large arguments */
r.real = atan2(fabs(z.imag), z.real);
r.imag = -copysign(log(hypot(z.real/2., z.imag/2.)) +
M_LN2*2., z.imag);
} else {
s1.real = 1.-z.real;
s1.imag = -z.imag;
s1 = cmath_sqrt_impl(module, s1);
s2.real = 1.+z.real;
s2.imag = z.imag;
s2 = cmath_sqrt_impl(module, s2);
r.real = 2.*atan2(s1.real, s2.real);
r.imag = asinh(s2.real*s1.imag - s2.imag*s1.real);
}
errno = 0;
return r;
}
static Py_complex acosh_special_values[7][7] = {
{ {INF,-P34}, {INF,-P}, {INF,-P}, {INF,P}, {INF,P}, {INF,P34}, {INF,N} },
{ {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {0.,-P12}, {0.,P12}, {U,U}, {INF,P12}, {N,P12} },
{ {INF,-P12}, {U,U}, {0.,-P12}, {0.,P12}, {U,U}, {INF,P12}, {N,P12} },
{ {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} },
{ {INF,N}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,N}, {N,N} }
};
/*[clinic input]
cmath.acosh = cmath.acos
Return the inverse hyperbolic cosine of z.
[clinic start generated code]*/
static Py_complex
cmath_acosh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=3e2454d4fcf404ca input=3f61bee7d703e53c]*/
{
Py_complex s1, s2, r;
SPECIAL_VALUE(z, acosh_special_values);
if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
/* avoid unnecessary overflow for large arguments */
r.real = log(hypot(z.real/2., z.imag/2.)) + M_LN2*2.;
r.imag = atan2(z.imag, z.real);
} else {
s1.real = z.real - 1.;
s1.imag = z.imag;
s1 = cmath_sqrt_impl(module, s1);
s2.real = z.real + 1.;
s2.imag = z.imag;
s2 = cmath_sqrt_impl(module, s2);
r.real = asinh(s1.real*s2.real + s1.imag*s2.imag);
r.imag = 2.*atan2(s1.imag, s2.real);
}
errno = 0;
return r;
}
/*[clinic input]
cmath.asin = cmath.acos
Return the arc sine of z.
[clinic start generated code]*/
static Py_complex
cmath_asin_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=3b264cd1b16bf4e1 input=be0bf0cfdd5239c5]*/
{
/* asin(z) = -i asinh(iz) */
Py_complex s, r;
s.real = -z.imag;
s.imag = z.real;
s = cmath_asinh_impl(module, s);
r.real = s.imag;
r.imag = -s.real;
return r;
}
static Py_complex asinh_special_values[7][7] = {
{ {-INF,-P14}, {-INF,-0.}, {-INF,-0.}, {-INF,0.}, {-INF,0.}, {-INF,P14}, {-INF,N} },
{ {-INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {-INF,P12}, {N,N} },
{ {-INF,-P12}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} },
{ {INF,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {INF,N}, {N,N} }
};
/*[clinic input]
cmath.asinh = cmath.acos
Return the inverse hyperbolic sine of z.
[clinic start generated code]*/
static Py_complex
cmath_asinh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=733d8107841a7599 input=5c09448fcfc89a79]*/
{
Py_complex s1, s2, r;
SPECIAL_VALUE(z, asinh_special_values);
if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
if (z.imag >= 0.) {
r.real = copysign(log(hypot(z.real/2., z.imag/2.)) +
M_LN2*2., z.real);
} else {
r.real = -copysign(log(hypot(z.real/2., z.imag/2.)) +
M_LN2*2., -z.real);
}
r.imag = atan2(z.imag, fabs(z.real));
} else {
s1.real = 1.+z.imag;
s1.imag = -z.real;
s1 = cmath_sqrt_impl(module, s1);
s2.real = 1.-z.imag;
s2.imag = z.real;
s2 = cmath_sqrt_impl(module, s2);
r.real = asinh(s1.real*s2.imag-s2.real*s1.imag);
r.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag);
}
errno = 0;
return r;
}
/*[clinic input]
cmath.atan = cmath.acos
Return the arc tangent of z.
[clinic start generated code]*/
static Py_complex
cmath_atan_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=b6bfc497058acba4 input=3b21ff7d5eac632a]*/
{
/* atan(z) = -i atanh(iz) */
Py_complex s, r;
s.real = -z.imag;
s.imag = z.real;
s = cmath_atanh_impl(module, s);
r.real = s.imag;
r.imag = -s.real;
return r;
}
static Py_complex atanh_special_values[7][7] = {
{ {-0.,-P12}, {-0.,-P12}, {-0.,-P12}, {-0.,P12}, {-0.,P12}, {-0.,P12}, {-0.,N} },
{ {-0.,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {-0.,P12}, {N,N} },
{ {-0.,-P12}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-0.,P12}, {-0.,N} },
{ {0.,-P12}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,P12}, {0.,N} },
{ {0.,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {0.,P12}, {N,N} },
{ {0.,-P12}, {0.,-P12}, {0.,-P12}, {0.,P12}, {0.,P12}, {0.,P12}, {0.,N} },
{ {0.,-P12}, {N,N}, {N,N}, {N,N}, {N,N}, {0.,P12}, {N,N} }
};
/*[clinic input]
cmath.atanh = cmath.acos
Return the inverse hyperbolic tangent of z.
[clinic start generated code]*/
static Py_complex
cmath_atanh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=e83355f93a989c9e input=2b3fdb82fb34487b]*/
{
Py_complex r;
double ay, h;
SPECIAL_VALUE(z, atanh_special_values);
/* Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z). */
if (z.real < 0.) {
return _Py_c_neg(cmath_atanh_impl(module, _Py_c_neg(z)));
}
ay = fabs(z.imag);
if (z.real > CM_SQRT_LARGE_DOUBLE || ay > CM_SQRT_LARGE_DOUBLE) {
/*
if abs(z) is large then we use the approximation
atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign
of z.imag)
*/
h = hypot(z.real/2., z.imag/2.); /* safe from overflow */
r.real = z.real/4./h/h;
r.imag = copysign(Py_MATH_PI/2., z.imag);
errno = 0;
} else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) {
/* C99 standard says: atanh(1+/-0.) should be inf +/- 0i */
if (ay == 0.) {
r.real = INF;
r.imag = z.imag;
errno = EDOM;
} else {
r.real = -log(sqrt(ay)/sqrt(hypot(ay, 2.)));
r.imag = copysign(atan2(2., -ay)/2, z.imag);
errno = 0;
}
} else {
r.real = m_log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.;
r.imag = -atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.;
errno = 0;
}
return r;
}
/*[clinic input]
cmath.cos = cmath.acos
Return the cosine of z.
[clinic start generated code]*/
static Py_complex
cmath_cos_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=fd64918d5b3186db input=6022e39b77127ac7]*/
{
/* cos(z) = cosh(iz) */
Py_complex r;
r.real = -z.imag;
r.imag = z.real;
r = cmath_cosh_impl(module, r);
return r;
}
/* cosh(infinity + i*y) needs to be dealt with specially */
static Py_complex cosh_special_values[7][7] = {
{ {INF,N}, {U,U}, {INF,0.}, {INF,-0.}, {U,U}, {INF,N}, {INF,N} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {N,0.}, {U,U}, {1.,0.}, {1.,-0.}, {U,U}, {N,0.}, {N,0.} },
{ {N,0.}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,0.}, {N,0.} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} },
{ {N,N}, {N,N}, {N,0.}, {N,0.}, {N,N}, {N,N}, {N,N} }
};
/*[clinic input]
cmath.cosh = cmath.acos
Return the hyperbolic cosine of z.
[clinic start generated code]*/
static Py_complex
cmath_cosh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=2e969047da601bdb input=d6b66339e9cc332b]*/
{
Py_complex r;
double x_minus_one;
/* special treatment for cosh(+/-inf + iy) if y is not a NaN */
if (!isfinite(z.real) || !isfinite(z.imag)) {
if (isinf(z.real) && isfinite(z.imag) &&
(z.imag != 0.)) {
if (z.real > 0) {
r.real = copysign(INF, cos(z.imag));
r.imag = copysign(INF, sin(z.imag));
}
else {
r.real = copysign(INF, cos(z.imag));
r.imag = -copysign(INF, sin(z.imag));
}
}
else {
r = cosh_special_values[special_type(z.real)]
[special_type(z.imag)];
}
/* need to set errno = EDOM if y is +/- infinity and x is not
a NaN */
if (isinf(z.imag) && !isnan(z.real))
errno = EDOM;
else
errno = 0;
return r;
}
if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
/* deal correctly with cases where cosh(z.real) overflows but
cosh(z) does not. */
x_minus_one = z.real - copysign(1., z.real);
r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E;
r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E;
} else {
r.real = cos(z.imag) * cosh(z.real);
r.imag = sin(z.imag) * sinh(z.real);
}
/* detect overflow, and set errno accordingly */
if (isinf(r.real) || isinf(r.imag))
errno = ERANGE;
else
errno = 0;
return r;
}
/* exp(infinity + i*y) and exp(-infinity + i*y) need special treatment for
finite y */
static Py_complex exp_special_values[7][7] = {
{ {0.,0.}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,0.}, {0.,0.} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {N,N}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,N}, {N,N} },
{ {N,N}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,N}, {N,N} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} },
{ {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} }
};
/*[clinic input]
cmath.exp = cmath.acos
Return the exponential value e**z.
[clinic start generated code]*/
static Py_complex
cmath_exp_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=edcec61fb9dfda6c input=8b9e6cf8a92174c3]*/
{
Py_complex r;
double l;
if (!isfinite(z.real) || !isfinite(z.imag)) {
if (isinf(z.real) && isfinite(z.imag)
&& (z.imag != 0.)) {
if (z.real > 0) {
r.real = copysign(INF, cos(z.imag));
r.imag = copysign(INF, sin(z.imag));
}
else {
r.real = copysign(0., cos(z.imag));
r.imag = copysign(0., sin(z.imag));
}
}
else {
r = exp_special_values[special_type(z.real)]
[special_type(z.imag)];
}
/* need to set errno = EDOM if y is +/- infinity and x is not
a NaN and not -infinity */
if (isinf(z.imag) &&
(isfinite(z.real) ||
(isinf(z.real) && z.real > 0)))
errno = EDOM;
else
errno = 0;
return r;
}
if (z.real > CM_LOG_LARGE_DOUBLE) {
l = exp(z.real-1.);
r.real = l*cos(z.imag)*Py_MATH_E;
r.imag = l*sin(z.imag)*Py_MATH_E;
} else {
l = exp(z.real);
r.real = l*cos(z.imag);
r.imag = l*sin(z.imag);
}
/* detect overflow, and set errno accordingly */
if (isinf(r.real) || isinf(r.imag))
errno = ERANGE;
else
errno = 0;
return r;
}
static Py_complex log_special_values[7][7] = {
{ {INF,-P34}, {INF,-P}, {INF,-P}, {INF,P}, {INF,P}, {INF,P34}, {INF,N} },
{ {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {-INF,-P}, {-INF,P}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {-INF,-0.}, {-INF,0.}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} },
{ {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} },
{ {INF,N}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,N}, {N,N} }
};
static Py_complex
c_log(Py_complex z)
{
/*
The usual formula for the real part is log(hypot(z.real, z.imag)).
There are four situations where this formula is potentially
problematic:
(1) the absolute value of z is subnormal. Then hypot is subnormal,
so has fewer than the usual number of bits of accuracy, hence may
have large relative error. This then gives a large absolute error
in the log. This can be solved by rescaling z by a suitable power
of 2.
(2) the absolute value of z is greater than DBL_MAX (e.g. when both
z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX)
Again, rescaling solves this.
(3) the absolute value of z is close to 1. In this case it's
difficult to achieve good accuracy, at least in part because a
change of 1ulp in the real or imaginary part of z can result in a
change of billions of ulps in the correctly rounded answer.
(4) z = 0. The simplest thing to do here is to call the
floating-point log with an argument of 0, and let its behaviour
(returning -infinity, signaling a floating-point exception, setting
errno, or whatever) determine that of c_log. So the usual formula
is fine here.
*/
Py_complex r;
double ax, ay, am, an, h;
SPECIAL_VALUE(z, log_special_values);
ax = fabs(z.real);
ay = fabs(z.imag);
if (ax > CM_LARGE_DOUBLE || ay > CM_LARGE_DOUBLE) {
r.real = log(hypot(ax/2., ay/2.)) + M_LN2;
} else if (ax < DBL_MIN && ay < DBL_MIN) {
if (ax > 0. || ay > 0.) {
/* catch cases where hypot(ax, ay) is subnormal */
r.real = log(hypot(ldexp(ax, DBL_MANT_DIG),
ldexp(ay, DBL_MANT_DIG))) - DBL_MANT_DIG*M_LN2;
}
else {
/* log(+/-0. +/- 0i) */
r.real = -INF;
r.imag = atan2(z.imag, z.real);
errno = EDOM;
return r;
}
} else {
h = hypot(ax, ay);
if (0.71 <= h && h <= 1.73) {
am = ax > ay ? ax : ay; /* max(ax, ay) */
an = ax > ay ? ay : ax; /* min(ax, ay) */
r.real = m_log1p((am-1)*(am+1)+an*an)/2.;
} else {
r.real = log(h);
}
}
r.imag = atan2(z.imag, z.real);
errno = 0;
return r;
}
/*[clinic input]
cmath.log10 = cmath.acos
Return the base-10 logarithm of z.
[clinic start generated code]*/
static Py_complex
cmath_log10_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=2922779a7c38cbe1 input=cff5644f73c1519c]*/
{
Py_complex r;
int errno_save;
r = c_log(z);
errno_save = errno; /* just in case the divisions affect errno */
r.real = r.real / M_LN10;
r.imag = r.imag / M_LN10;
errno = errno_save;
return r;
}
/*[clinic input]
cmath.sin = cmath.acos
Return the sine of z.
[clinic start generated code]*/
static Py_complex
cmath_sin_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=980370d2ff0bb5aa input=2d3519842a8b4b85]*/
{
/* sin(z) = -i sin(iz) */
Py_complex s, r;
s.real = -z.imag;
s.imag = z.real;
s = cmath_sinh_impl(module, s);
r.real = s.imag;
r.imag = -s.real;
return r;
}
/* sinh(infinity + i*y) needs to be dealt with specially */
static Py_complex sinh_special_values[7][7] = {
{ {INF,N}, {U,U}, {-INF,-0.}, {-INF,0.}, {U,U}, {INF,N}, {INF,N} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {0.,N}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {0.,N}, {0.,N} },
{ {0.,N}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,N}, {0.,N} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} },
{ {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} }
};
/*[clinic input]
cmath.sinh = cmath.acos
Return the hyperbolic sine of z.
[clinic start generated code]*/
static Py_complex
cmath_sinh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=38b0a6cce26f3536 input=d2d3fc8c1ddfd2dd]*/
{
Py_complex r;
double x_minus_one;
/* special treatment for sinh(+/-inf + iy) if y is finite and
nonzero */
if (!isfinite(z.real) || !isfinite(z.imag)) {
if (isinf(z.real) && isfinite(z.imag)
&& (z.imag != 0.)) {
if (z.real > 0) {
r.real = copysign(INF, cos(z.imag));
r.imag = copysign(INF, sin(z.imag));
}
else {
r.real = -copysign(INF, cos(z.imag));
r.imag = copysign(INF, sin(z.imag));
}
}
else {
r = sinh_special_values[special_type(z.real)]
[special_type(z.imag)];
}
/* need to set errno = EDOM if y is +/- infinity and x is not
a NaN */
if (isinf(z.imag) && !isnan(z.real))
errno = EDOM;
else
errno = 0;
return r;
}
if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
x_minus_one = z.real - copysign(1., z.real);
r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E;
r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E;
} else {
r.real = cos(z.imag) * sinh(z.real);
r.imag = sin(z.imag) * cosh(z.real);
}
/* detect overflow, and set errno accordingly */
if (isinf(r.real) || isinf(r.imag))
errno = ERANGE;
else
errno = 0;
return r;
}
static Py_complex sqrt_special_values[7][7] = {
{ {INF,-INF}, {0.,-INF}, {0.,-INF}, {0.,INF}, {0.,INF}, {INF,INF}, {N,INF} },
{ {INF,-INF}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,INF}, {N,N} },
{ {INF,-INF}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,INF}, {N,N} },
{ {INF,-INF}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,INF}, {N,N} },
{ {INF,-INF}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,INF}, {N,N} },
{ {INF,-INF}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,INF}, {INF,N} },
{ {INF,-INF}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,INF}, {N,N} }
};
/*[clinic input]
cmath.sqrt = cmath.acos
Return the square root of z.
[clinic start generated code]*/
static Py_complex
cmath_sqrt_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=b6507b3029c339fc input=7088b166fc9a58c7]*/
{
/*
Method: use symmetries to reduce to the case when x = z.real and y
= z.imag are nonnegative. Then the real part of the result is
given by
s = sqrt((x + hypot(x, y))/2)
and the imaginary part is
d = (y/2)/s
If either x or y is very large then there's a risk of overflow in
computation of the expression x + hypot(x, y). We can avoid this
by rewriting the formula for s as:
s = 2*sqrt(x/8 + hypot(x/8, y/8))
This costs us two extra multiplications/divisions, but avoids the
overhead of checking for x and y large.
If both x and y are subnormal then hypot(x, y) may also be
subnormal, so will lack full precision. We solve this by rescaling
x and y by a sufficiently large power of 2 to ensure that x and y
are normal.
*/
Py_complex r;
double s,d;
double ax, ay;
SPECIAL_VALUE(z, sqrt_special_values);
if (z.real == 0. && z.imag == 0.) {
r.real = 0.;
r.imag = z.imag;
return r;
}
ax = fabs(z.real);
ay = fabs(z.imag);
if (ax < DBL_MIN && ay < DBL_MIN) {
/* here we catch cases where hypot(ax, ay) is subnormal */
ax = ldexp(ax, CM_SCALE_UP);
s = ldexp(sqrt(ax + hypot(ax, ldexp(ay, CM_SCALE_UP))),
CM_SCALE_DOWN);
} else {
ax /= 8.;
s = 2.*sqrt(ax + hypot(ax, ay/8.));
}
d = ay/(2.*s);
if (z.real >= 0.) {
r.real = s;
r.imag = copysign(d, z.imag);
} else {
r.real = d;
r.imag = copysign(s, z.imag);
}
errno = 0;
return r;
}
/*[clinic input]
cmath.tan = cmath.acos
Return the tangent of z.
[clinic start generated code]*/
static Py_complex
cmath_tan_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=7c5f13158a72eb13 input=fc167e528767888e]*/
{
/* tan(z) = -i tanh(iz) */
Py_complex s, r;
s.real = -z.imag;
s.imag = z.real;
s = cmath_tanh_impl(module, s);
r.real = s.imag;
r.imag = -s.real;
return r;
}
/* tanh(infinity + i*y) needs to be dealt with specially */
static Py_complex tanh_special_values[7][7] = {
{ {-1.,0.}, {U,U}, {-1.,-0.}, {-1.,0.}, {U,U}, {-1.,0.}, {-1.,0.} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {-0.0,N}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-0.0,N}, {-0.,N} },
{ {0.0,N}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.0,N}, {0.,N} },
{ {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} },
{ {1.,0.}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {1.,0.}, {1.,0.} },
{ {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} }
};
/*[clinic input]
cmath.tanh = cmath.acos
Return the hyperbolic tangent of z.
[clinic start generated code]*/
static Py_complex
cmath_tanh_impl(PyObject *module, Py_complex z)
/*[clinic end generated code: output=36d547ef7aca116c input=22f67f9dc6d29685]*/
{
/* Formula:
tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) /
(1+tan(y)^2 tanh(x)^2)
To avoid excessive roundoff error, 1-tanh(x)^2 is better computed
as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2
by 4 exp(-2*x) instead, to avoid possible overflow in the
computation of cosh(x).
*/
Py_complex r;
double tx, ty, cx, txty, denom;
/* special treatment for tanh(+/-inf + iy) if y is finite and
nonzero */
if (!isfinite(z.real) || !isfinite(z.imag)) {
if (isinf(z.real) && isfinite(z.imag)
&& (z.imag != 0.)) {
if (z.real > 0) {
r.real = 1.0;
r.imag = copysign(0.,
2.*sin(z.imag)*cos(z.imag));
}
else {
r.real = -1.0;
r.imag = copysign(0.,
2.*sin(z.imag)*cos(z.imag));
}
}
else {
r = tanh_special_values[special_type(z.real)]
[special_type(z.imag)];
}
/* need to set errno = EDOM if z.imag is +/-infinity and
z.real is finite */
if (isinf(z.imag) && isfinite(z.real))
errno = EDOM;
else
errno = 0;
return r;
}
/* danger of overflow in 2.*z.imag !*/
if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
r.real = copysign(1., z.real);
r.imag = 4.*sin(z.imag)*cos(z.imag)*exp(-2.*fabs(z.real));
} else {
tx = tanh(z.real);
ty = tan(z.imag);
cx = 1./cosh(z.real);
txty = tx*ty;
denom = 1. + txty*txty;
r.real = tx*(1.+ty*ty)/denom;
r.imag = ((ty/denom)*cx)*cx;
}
errno = 0;
return r;
}
/*[clinic input]
cmath.log
z as x: Py_complex
base as y_obj: object = NULL
/
log(z[, base]) -> the logarithm of z to the given base.
If the base is not specified, returns the natural logarithm (base e) of z.
[clinic start generated code]*/
static PyObject *
cmath_log_impl(PyObject *module, Py_complex x, PyObject *y_obj)
/*[clinic end generated code: output=4effdb7d258e0d94 input=e1f81d4fcfd26497]*/
{
Py_complex y;
errno = 0;
x = c_log(x);
if (y_obj != NULL) {
y = PyComplex_AsCComplex(y_obj);
if (PyErr_Occurred()) {
return NULL;
}
y = c_log(y);
x = _Py_c_quot(x, y);
}
if (errno != 0)
return math_error();
return PyComplex_FromCComplex(x);
}
/* And now the glue to make them available from Python: */
static PyObject *
math_error(void)
{
if (errno == EDOM)
PyErr_SetString(PyExc_ValueError, "math domain error");
else if (errno == ERANGE)
PyErr_SetString(PyExc_OverflowError, "math range error");
else /* Unexpected math error */
PyErr_SetFromErrno(PyExc_ValueError);
return NULL;
}
/*[clinic input]
cmath.phase
z: Py_complex
/
Return argument, also known as the phase angle, of a complex.
[clinic start generated code]*/
static PyObject *