summaryrefslogtreecommitdiffstats
path: root/Modules/mathmodule.c
blob: 50dd43ef9f3d08ea3f089360238f9b238c5357ad (plain)
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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
/* Math module -- standard C math library functions, pi and e */

/* Here are some comments from Tim Peters, extracted from the
   discussion attached to http://bugs.python.org/issue1640.  They
   describe the general aims of the math module with respect to
   special values, IEEE-754 floating-point exceptions, and Python
   exceptions.

These are the "spirit of 754" rules:

1. If the mathematical result is a real number, but of magnitude too
large to approximate by a machine float, overflow is signaled and the
result is an infinity (with the appropriate sign).

2. If the mathematical result is a real number, but of magnitude too
small to approximate by a machine float, underflow is signaled and the
result is a zero (with the appropriate sign).

3. At a singularity (a value x such that the limit of f(y) as y
approaches x exists and is an infinity), "divide by zero" is signaled
and the result is an infinity (with the appropriate sign).  This is
complicated a little by that the left-side and right-side limits may
not be the same; e.g., 1/x approaches +inf or -inf as x approaches 0
from the positive or negative directions.  In that specific case, the
sign of the zero determines the result of 1/0.

4. At a point where a function has no defined result in the extended
reals (i.e., the reals plus an infinity or two), invalid operation is
signaled and a NaN is returned.

And these are what Python has historically /tried/ to do (but not
always successfully, as platform libm behavior varies a lot):

For #1, raise OverflowError.

For #2, return a zero (with the appropriate sign if that happens by
accident ;-)).

For #3 and #4, raise ValueError.  It may have made sense to raise
Python's ZeroDivisionError in #3, but historically that's only been
raised for division by zero and mod by zero.

*/

/*
   In general, on an IEEE-754 platform the aim is to follow the C99
   standard, including Annex 'F', whenever possible.  Where the
   standard recommends raising the 'divide-by-zero' or 'invalid'
   floating-point exceptions, Python should raise a ValueError.  Where
   the standard recommends raising 'overflow', Python should raise an
   OverflowError.  In all other circumstances a value should be
   returned.
 */

#include "Python.h"
#include "longintrepr.h" /* just for SHIFT */

#ifdef _OSF_SOURCE
/* OSF1 5.1 doesn't make this available with XOPEN_SOURCE_EXTENDED defined */
extern double copysign(double, double);
#endif

/* Call is_error when errno != 0, and where x is the result libm
 * returned.  is_error will usually set up an exception and return
 * true (1), but may return false (0) without setting up an exception.
 */
static int
is_error(double x)
{
    int result = 1;     /* presumption of guilt */
    assert(errno);      /* non-zero errno is a precondition for calling */
    if (errno == EDOM)
        PyErr_SetString(PyExc_ValueError, "math domain error");

    else if (errno == ERANGE) {
        /* ANSI C generally requires libm functions to set ERANGE
         * on overflow, but also generally *allows* them to set
         * ERANGE on underflow too.  There's no consistency about
         * the latter across platforms.
         * Alas, C99 never requires that errno be set.
         * Here we suppress the underflow errors (libm functions
         * should return a zero on underflow, and +- HUGE_VAL on
         * overflow, so testing the result for zero suffices to
         * distinguish the cases).
         *
         * On some platforms (Ubuntu/ia64) it seems that errno can be
         * set to ERANGE for subnormal results that do *not* underflow
         * to zero.  So to be safe, we'll ignore ERANGE whenever the
         * function result is less than one in absolute value.
         */
        if (fabs(x) < 1.0)
            result = 0;
        else
            PyErr_SetString(PyExc_OverflowError,
                            "math range error");
    }
    else
        /* Unexpected math error */
        PyErr_SetFromErrno(PyExc_ValueError);
    return result;
}

/*
   wrapper for atan2 that deals directly with special cases before
   delegating to the platform libm for the remaining cases.  This
   is necessary to get consistent behaviour across platforms.
   Windows, FreeBSD and alpha Tru64 are amongst platforms that don't
   always follow C99.
*/

static double
m_atan2(double y, double x)
{
    if (Py_IS_NAN(x) || Py_IS_NAN(y))
        return Py_NAN;
    if (Py_IS_INFINITY(y)) {
        if (Py_IS_INFINITY(x)) {
            if (copysign(1., x) == 1.)
                /* atan2(+-inf, +inf) == +-pi/4 */
                return copysign(0.25*Py_MATH_PI, y);
            else
                /* atan2(+-inf, -inf) == +-pi*3/4 */
                return copysign(0.75*Py_MATH_PI, y);
        }
        /* atan2(+-inf, x) == +-pi/2 for finite x */
        return copysign(0.5*Py_MATH_PI, y);
    }
    if (Py_IS_INFINITY(x) || y == 0.) {
        if (copysign(1., x) == 1.)
            /* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */
            return copysign(0., y);
        else
            /* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */
            return copysign(Py_MATH_PI, y);
    }
    return atan2(y, x);
}

/*
    Various platforms (Solaris, OpenBSD) do nonstandard things for log(0),
    log(-ve), log(NaN).  Here are wrappers for log and log10 that deal with
    special values directly, passing positive non-special values through to
    the system log/log10.
 */

static double
m_log(double x)
{
    if (Py_IS_FINITE(x)) {
        if (x > 0.0)
            return log(x);
        errno = EDOM;
        if (x == 0.0)
            return -Py_HUGE_VAL; /* log(0) = -inf */
        else
            return Py_NAN; /* log(-ve) = nan */
    }
    else if (Py_IS_NAN(x))
        return x; /* log(nan) = nan */
    else if (x > 0.0)
        return x; /* log(inf) = inf */
    else {
        errno = EDOM;
        return Py_NAN; /* log(-inf) = nan */
    }
}

static double
m_log10(double x)
{
    if (Py_IS_FINITE(x)) {
        if (x > 0.0)
            return log10(x);
        errno = EDOM;
        if (x == 0.0)
            return -Py_HUGE_VAL; /* log10(0) = -inf */
        else
            return Py_NAN; /* log10(-ve) = nan */
    }
    else if (Py_IS_NAN(x))
        return x; /* log10(nan) = nan */
    else if (x > 0.0)
        return x; /* log10(inf) = inf */
    else {
        errno = EDOM;
        return Py_NAN; /* log10(-inf) = nan */
    }
}


/*
   math_1 is used to wrap a libm function f that takes a double
   arguments and returns a double.

   The error reporting follows these rules, which are designed to do
   the right thing on C89/C99 platforms and IEEE 754/non IEEE 754
   platforms.

   - a NaN result from non-NaN inputs causes ValueError to be raised
   - an infinite result from finite inputs causes OverflowError to be
     raised if can_overflow is 1, or raises ValueError if can_overflow
     is 0.
   - if the result is finite and errno == EDOM then ValueError is
     raised
   - if the result is finite and nonzero and errno == ERANGE then
     OverflowError is raised

   The last rule is used to catch overflow on platforms which follow
   C89 but for which HUGE_VAL is not an infinity.

   For the majority of one-argument functions these rules are enough
   to ensure that Python's functions behave as specified in 'Annex F'
   of the C99 standard, with the 'invalid' and 'divide-by-zero'
   floating-point exceptions mapping to Python's ValueError and the
   'overflow' floating-point exception mapping to OverflowError.
   math_1 only works for functions that don't have singularities *and*
   the possibility of overflow; fortunately, that covers everything we
   care about right now.
*/

static PyObject *
math_1_to_whatever(PyObject *arg, double (*func) (double),
                   PyObject *(*from_double_func) (double),
                   int can_overflow)
{
    double x, r;
    x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    errno = 0;
    PyFPE_START_PROTECT("in math_1", return 0);
    r = (*func)(x);
    PyFPE_END_PROTECT(r);
    if (Py_IS_NAN(r) && !Py_IS_NAN(x)) {
        PyErr_SetString(PyExc_ValueError,
                        "math domain error"); /* invalid arg */
        return NULL;
    }
    if (Py_IS_INFINITY(r) && Py_IS_FINITE(x)) {
                    if (can_overflow)
                            PyErr_SetString(PyExc_OverflowError,
                                    "math range error"); /* overflow */
            else
                PyErr_SetString(PyExc_ValueError,
                    "math domain error"); /* singularity */
            return NULL;
    }
    if (Py_IS_FINITE(r) && errno && is_error(r))
        /* this branch unnecessary on most platforms */
        return NULL;

    return (*from_double_func)(r);
}

/*
   math_2 is used to wrap a libm function f that takes two double
   arguments and returns a double.

   The error reporting follows these rules, which are designed to do
   the right thing on C89/C99 platforms and IEEE 754/non IEEE 754
   platforms.

   - a NaN result from non-NaN inputs causes ValueError to be raised
   - an infinite result from finite inputs causes OverflowError to be
     raised.
   - if the result is finite and errno == EDOM then ValueError is
     raised
   - if the result is finite and nonzero and errno == ERANGE then
     OverflowError is raised

   The last rule is used to catch overflow on platforms which follow
   C89 but for which HUGE_VAL is not an infinity.

   For most two-argument functions (copysign, fmod, hypot, atan2)
   these rules are enough to ensure that Python's functions behave as
   specified in 'Annex F' of the C99 standard, with the 'invalid' and
   'divide-by-zero' floating-point exceptions mapping to Python's
   ValueError and the 'overflow' floating-point exception mapping to
   OverflowError.
*/

static PyObject *
math_1(PyObject *arg, double (*func) (double), int can_overflow)
{
    return math_1_to_whatever(arg, func, PyFloat_FromDouble, can_overflow);
}

static PyObject *
math_1_to_int(PyObject *arg, double (*func) (double), int can_overflow)
{
    return math_1_to_whatever(arg, func, PyLong_FromDouble, can_overflow);
}

static PyObject *
math_2(PyObject *args, double (*func) (double, double), char *funcname)
{
    PyObject *ox, *oy;
    double x, y, r;
    if (! PyArg_UnpackTuple(args, funcname, 2, 2, &ox, &oy))
        return NULL;
    x = PyFloat_AsDouble(ox);
    y = PyFloat_AsDouble(oy);
    if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
        return NULL;
    errno = 0;
    PyFPE_START_PROTECT("in math_2", return 0);
    r = (*func)(x, y);
    PyFPE_END_PROTECT(r);
    if (Py_IS_NAN(r)) {
        if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
            errno = EDOM;
        else
            errno = 0;
    }
    else if (Py_IS_INFINITY(r)) {
        if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
            errno = ERANGE;
        else
            errno = 0;
    }
    if (errno && is_error(r))
        return NULL;
    else
        return PyFloat_FromDouble(r);
}

#define FUNC1(funcname, func, can_overflow, docstring)                  \
    static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
        return math_1(args, func, can_overflow);                            \
    }\
    PyDoc_STRVAR(math_##funcname##_doc, docstring);

#define FUNC2(funcname, func, docstring) \
    static PyObject * math_##funcname(PyObject *self, PyObject *args) { \
        return math_2(args, func, #funcname); \
    }\
    PyDoc_STRVAR(math_##funcname##_doc, docstring);

FUNC1(acos, acos, 0,
      "acos(x)\n\nReturn the arc cosine (measured in radians) of x.")
FUNC1(acosh, acosh, 0,
      "acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.")
FUNC1(asin, asin, 0,
      "asin(x)\n\nReturn the arc sine (measured in radians) of x.")
FUNC1(asinh, asinh, 0,
      "asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.")
FUNC1(atan, atan, 0,
      "atan(x)\n\nReturn the arc tangent (measured in radians) of x.")
FUNC2(atan2, m_atan2,
      "atan2(y, x)\n\nReturn the arc tangent (measured in radians) of y/x.\n"
      "Unlike atan(y/x), the signs of both x and y are considered.")
FUNC1(atanh, atanh, 0,
      "atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.")

static PyObject * math_ceil(PyObject *self, PyObject *number) {
    static PyObject *ceil_str = NULL;
    PyObject *method;

    if (ceil_str == NULL) {
        ceil_str = PyUnicode_InternFromString("__ceil__");
        if (ceil_str == NULL)
            return NULL;
    }

    method = _PyType_Lookup(Py_TYPE(number), ceil_str);
    if (method == NULL)
        return math_1_to_int(number, ceil, 0);
    else
        return PyObject_CallFunction(method, "O", number);
}

PyDoc_STRVAR(math_ceil_doc,
             "ceil(x)\n\nReturn the ceiling of x as an int.\n"
             "This is the smallest integral value >= x.");

FUNC2(copysign, copysign,
      "copysign(x, y)\n\nReturn x with the sign of y.")
FUNC1(cos, cos, 0,
      "cos(x)\n\nReturn the cosine of x (measured in radians).")
FUNC1(cosh, cosh, 1,
      "cosh(x)\n\nReturn the hyperbolic cosine of x.")
FUNC1(exp, exp, 1,
      "exp(x)\n\nReturn e raised to the power of x.")
FUNC1(fabs, fabs, 0,
      "fabs(x)\n\nReturn the absolute value of the float x.")

static PyObject * math_floor(PyObject *self, PyObject *number) {
    static PyObject *floor_str = NULL;
    PyObject *method;

    if (floor_str == NULL) {
        floor_str = PyUnicode_InternFromString("__floor__");
        if (floor_str == NULL)
            return NULL;
    }

    method = _PyType_Lookup(Py_TYPE(number), floor_str);
    if (method == NULL)
        return math_1_to_int(number, floor, 0);
    else
        return PyObject_CallFunction(method, "O", number);
}

PyDoc_STRVAR(math_floor_doc,
             "floor(x)\n\nReturn the floor of x as an int.\n"
             "This is the largest integral value <= x.");

FUNC1(log1p, log1p, 1,
      "log1p(x)\n\nReturn the natural logarithm of 1+x (base e).\n"
      "The result is computed in a way which is accurate for x near zero.")
FUNC1(sin, sin, 0,
      "sin(x)\n\nReturn the sine of x (measured in radians).")
FUNC1(sinh, sinh, 1,
      "sinh(x)\n\nReturn the hyperbolic sine of x.")
FUNC1(sqrt, sqrt, 0,
      "sqrt(x)\n\nReturn the square root of x.")
FUNC1(tan, tan, 0,
      "tan(x)\n\nReturn the tangent of x (measured in radians).")
FUNC1(tanh, tanh, 0,
      "tanh(x)\n\nReturn the hyperbolic tangent of x.")

/* Precision summation function as msum() by Raymond Hettinger in
   <http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/393090>,
   enhanced with the exact partials sum and roundoff from Mark
   Dickinson's post at <http://bugs.python.org/file10357/msum4.py>.
   See those links for more details, proofs and other references.

   Note 1: IEEE 754R floating point semantics are assumed,
   but the current implementation does not re-establish special
   value semantics across iterations (i.e. handling -Inf + Inf).

   Note 2:  No provision is made for intermediate overflow handling;
   therefore, sum([1e+308, 1e-308, 1e+308]) returns 1e+308 while
   sum([1e+308, 1e+308, 1e-308]) raises an OverflowError due to the
   overflow of the first partial sum.

   Note 3: The intermediate values lo, yr, and hi are declared volatile so
   aggressive compilers won't algebraically reduce lo to always be exactly 0.0.
   Also, the volatile declaration forces the values to be stored in memory as
   regular doubles instead of extended long precision (80-bit) values.  This
   prevents double rounding because any addition or subtraction of two doubles
   can be resolved exactly into double-sized hi and lo values.  As long as the
   hi value gets forced into a double before yr and lo are computed, the extra
   bits in downstream extended precision operations (x87 for example) will be
   exactly zero and therefore can be losslessly stored back into a double,
   thereby preventing double rounding.

   Note 4: A similar implementation is in Modules/cmathmodule.c.
   Be sure to update both when making changes.

   Note 5: The signature of math.fsum() differs from __builtin__.sum()
   because the start argument doesn't make sense in the context of
   accurate summation.  Since the partials table is collapsed before
   returning a result, sum(seq2, start=sum(seq1)) may not equal the
   accurate result returned by sum(itertools.chain(seq1, seq2)).
*/

#define NUM_PARTIALS  32  /* initial partials array size, on stack */

/* Extend the partials array p[] by doubling its size. */
static int                          /* non-zero on error */
_fsum_realloc(double **p_ptr, Py_ssize_t  n,
             double  *ps,    Py_ssize_t *m_ptr)
{
    void *v = NULL;
    Py_ssize_t m = *m_ptr;

    m += m;  /* double */
    if (n < m && m < (PY_SSIZE_T_MAX / sizeof(double))) {
        double *p = *p_ptr;
        if (p == ps) {
            v = PyMem_Malloc(sizeof(double) * m);
            if (v != NULL)
                memcpy(v, ps, sizeof(double) * n);
        }
        else
            v = PyMem_Realloc(p, sizeof(double) * m);
    }
    if (v == NULL) {        /* size overflow or no memory */
        PyErr_SetString(PyExc_MemoryError, "math.fsum partials");
        return 1;
    }
    *p_ptr = (double*) v;
    *m_ptr = m;
    return 0;
}

/* Full precision summation of a sequence of floats.

   def msum(iterable):
       partials = []  # sorted, non-overlapping partial sums
       for x in iterable:
       i = 0
       for y in partials:
           if abs(x) < abs(y):
           x, y = y, x
           hi = x + y
           lo = y - (hi - x)
           if lo:
           partials[i] = lo
           i += 1
           x = hi
       partials[i:] = [x]
       return sum_exact(partials)

   Rounded x+y stored in hi with the roundoff stored in lo.  Together hi+lo
   are exactly equal to x+y.  The inner loop applies hi/lo summation to each
   partial so that the list of partial sums remains exact.

   Sum_exact() adds the partial sums exactly and correctly rounds the final
   result (using the round-half-to-even rule).  The items in partials remain
   non-zero, non-special, non-overlapping and strictly increasing in
   magnitude, but possibly not all having the same sign.

   Depends on IEEE 754 arithmetic guarantees and half-even rounding.
*/

static PyObject*
math_fsum(PyObject *self, PyObject *seq)
{
    PyObject *item, *iter, *sum = NULL;
    Py_ssize_t i, j, n = 0, m = NUM_PARTIALS;
    double x, y, t, ps[NUM_PARTIALS], *p = ps;
    double xsave, special_sum = 0.0, inf_sum = 0.0;
    volatile double hi, yr, lo;

    iter = PyObject_GetIter(seq);
    if (iter == NULL)
        return NULL;

    PyFPE_START_PROTECT("fsum", Py_DECREF(iter); return NULL)

    for(;;) {           /* for x in iterable */
        assert(0 <= n && n <= m);
        assert((m == NUM_PARTIALS && p == ps) ||
               (m >  NUM_PARTIALS && p != NULL));

        item = PyIter_Next(iter);
        if (item == NULL) {
            if (PyErr_Occurred())
                goto _fsum_error;
            break;
        }
        x = PyFloat_AsDouble(item);
        Py_DECREF(item);
        if (PyErr_Occurred())
            goto _fsum_error;

        xsave = x;
        for (i = j = 0; j < n; j++) {       /* for y in partials */
            y = p[j];
            if (fabs(x) < fabs(y)) {
                t = x; x = y; y = t;
            }
            hi = x + y;
            yr = hi - x;
            lo = y - yr;
            if (lo != 0.0)
                p[i++] = lo;
            x = hi;
        }

        n = i;                              /* ps[i:] = [x] */
        if (x != 0.0) {
            if (! Py_IS_FINITE(x)) {
                /* a nonfinite x could arise either as
                   a result of intermediate overflow, or
                   as a result of a nan or inf in the
                   summands */
                if (Py_IS_FINITE(xsave)) {
                    PyErr_SetString(PyExc_OverflowError,
                          "intermediate overflow in fsum");
                    goto _fsum_error;
                }
                if (Py_IS_INFINITY(xsave))
                    inf_sum += xsave;
                special_sum += xsave;
                /* reset partials */
                n = 0;
            }
            else if (n >= m && _fsum_realloc(&p, n, ps, &m))
                goto _fsum_error;
            else
                p[n++] = x;
        }
    }

    if (special_sum != 0.0) {
        if (Py_IS_NAN(inf_sum))
            PyErr_SetString(PyExc_ValueError,
                            "-inf + inf in fsum");
        else
            sum = PyFloat_FromDouble(special_sum);
        goto _fsum_error;
    }

    hi = 0.0;
    if (n > 0) {
        hi = p[--n];
        /* sum_exact(ps, hi) from the top, stop when the sum becomes
           inexact. */
        while (n > 0) {
            x = hi;
            y = p[--n];
            assert(fabs(y) < fabs(x));
            hi = x + y;
            yr = hi - x;
            lo = y - yr;
            if (lo != 0.0)
                break;
        }
        /* Make half-even rounding work across multiple partials.
           Needed so that sum([1e-16, 1, 1e16]) will round-up the last
           digit to two instead of down to zero (the 1e-16 makes the 1
           slightly closer to two).  With a potential 1 ULP rounding
           error fixed-up, math.fsum() can guarantee commutativity. */
        if (n > 0 && ((lo < 0.0 && p[n-1] < 0.0) ||
                      (lo > 0.0 && p[n-1] > 0.0))) {
            y = lo * 2.0;
            x = hi + y;
            yr = x - hi;
            if (y == yr)
                hi = x;
        }
    }
    sum = PyFloat_FromDouble(hi);

_fsum_error:
    PyFPE_END_PROTECT(hi)
    Py_DECREF(iter);
    if (p != ps)
        PyMem_Free(p);
    return sum;
}

#undef NUM_PARTIALS

PyDoc_STRVAR(math_fsum_doc,
"fsum(iterable)\n\n\
Return an accurate floating point sum of values in the iterable.\n\
Assumes IEEE-754 floating point arithmetic.");

static PyObject *
math_factorial(PyObject *self, PyObject *arg)
{
    long i, x;
    PyObject *result, *iobj, *newresult;

    if (PyFloat_Check(arg)) {
        double dx = PyFloat_AS_DOUBLE((PyFloatObject *)arg);
        if (dx != floor(dx)) {
            PyErr_SetString(PyExc_ValueError,
                "factorial() only accepts integral values");
            return NULL;
        }
    }

    x = PyLong_AsLong(arg);
    if (x == -1 && PyErr_Occurred())
        return NULL;
    if (x < 0) {
        PyErr_SetString(PyExc_ValueError,
            "factorial() not defined for negative values");
        return NULL;
    }

    result = (PyObject *)PyLong_FromLong(1);
    if (result == NULL)
        return NULL;
    for (i=1 ; i<=x ; i++) {
        iobj = (PyObject *)PyLong_FromLong(i);
        if (iobj == NULL)
            goto error;
        newresult = PyNumber_Multiply(result, iobj);
        Py_DECREF(iobj);
        if (newresult == NULL)
            goto error;
        Py_DECREF(result);
        result = newresult;
    }
    return result;

error:
    Py_DECREF(result);
    return NULL;
}

PyDoc_STRVAR(math_factorial_doc,
"factorial(x) -> Integral\n"
"\n"
"Find x!. Raise a ValueError if x is negative or non-integral.");

static PyObject *
math_trunc(PyObject *self, PyObject *number)
{
    static PyObject *trunc_str = NULL;
    PyObject *trunc;

    if (Py_TYPE(number)->tp_dict == NULL) {
        if (PyType_Ready(Py_TYPE(number)) < 0)
            return NULL;
    }

    if (trunc_str == NULL) {
        trunc_str = PyUnicode_InternFromString("__trunc__");
        if (trunc_str == NULL)
            return NULL;
    }

    trunc = _PyType_Lookup(Py_TYPE(number), trunc_str);
    if (trunc == NULL) {
        PyErr_Format(PyExc_TypeError,
                     "type %.100s doesn't define __trunc__ method",
                     Py_TYPE(number)->tp_name);
        return NULL;
    }
    return PyObject_CallFunctionObjArgs(trunc, number, NULL);
}

PyDoc_STRVAR(math_trunc_doc,
"trunc(x:Real) -> Integral\n"
"\n"
"Truncates x to the nearest Integral toward 0. Uses the __trunc__ magic method.");

static PyObject *
math_frexp(PyObject *self, PyObject *arg)
{
    int i;
    double x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    /* deal with special cases directly, to sidestep platform
       differences */
    if (Py_IS_NAN(x) || Py_IS_INFINITY(x) || !x) {
        i = 0;
    }
    else {
        PyFPE_START_PROTECT("in math_frexp", return 0);
        x = frexp(x, &i);
        PyFPE_END_PROTECT(x);
    }
    return Py_BuildValue("(di)", x, i);
}

PyDoc_STRVAR(math_frexp_doc,
"frexp(x)\n"
"\n"
"Return the mantissa and exponent of x, as pair (m, e).\n"
"m is a float and e is an int, such that x = m * 2.**e.\n"
"If x is 0, m and e are both 0.  Else 0.5 <= abs(m) < 1.0.");

static PyObject *
math_ldexp(PyObject *self, PyObject *args)
{
    double x, r;
    PyObject *oexp;
    long exp;
    if (! PyArg_ParseTuple(args, "dO:ldexp", &x, &oexp))
        return NULL;

    if (PyLong_Check(oexp)) {
        /* on overflow, replace exponent with either LONG_MAX
           or LONG_MIN, depending on the sign. */
        exp = PyLong_AsLong(oexp);
        if (exp == -1 && PyErr_Occurred()) {
            if (PyErr_ExceptionMatches(PyExc_OverflowError)) {
                if (Py_SIZE(oexp) < 0) {
                    exp = LONG_MIN;
                }
                else {
                    exp = LONG_MAX;
                }
                PyErr_Clear();
            }
            else {
                /* propagate any unexpected exception */
                return NULL;
            }
        }
    }
    else {
        PyErr_SetString(PyExc_TypeError,
                        "Expected an int or long as second argument "
                        "to ldexp.");
        return NULL;
    }

    if (x == 0. || !Py_IS_FINITE(x)) {
        /* NaNs, zeros and infinities are returned unchanged */
        r = x;
        errno = 0;
    } else if (exp > INT_MAX) {
        /* overflow */
        r = copysign(Py_HUGE_VAL, x);
        errno = ERANGE;
    } else if (exp < INT_MIN) {
        /* underflow to +-0 */
        r = copysign(0., x);
        errno = 0;
    } else {
        errno = 0;
        PyFPE_START_PROTECT("in math_ldexp", return 0);
        r = ldexp(x, (int)exp);
        PyFPE_END_PROTECT(r);
        if (Py_IS_INFINITY(r))
            errno = ERANGE;
    }

    if (errno && is_error(r))
        return NULL;
    return PyFloat_FromDouble(r);
}

PyDoc_STRVAR(math_ldexp_doc,
"ldexp(x, i)\n\n\
Return x * (2**i).");

static PyObject *
math_modf(PyObject *self, PyObject *arg)
{
    double y, x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    /* some platforms don't do the right thing for NaNs and
       infinities, so we take care of special cases directly. */
    if (!Py_IS_FINITE(x)) {
        if (Py_IS_INFINITY(x))
            return Py_BuildValue("(dd)", copysign(0., x), x);
        else if (Py_IS_NAN(x))
            return Py_BuildValue("(dd)", x, x);
    }

    errno = 0;
    PyFPE_START_PROTECT("in math_modf", return 0);
    x = modf(x, &y);
    PyFPE_END_PROTECT(x);
    return Py_BuildValue("(dd)", x, y);
}

PyDoc_STRVAR(math_modf_doc,
"modf(x)\n"
"\n"
"Return the fractional and integer parts of x.  Both results carry the sign\n"
"of x and are floats.");

/* A decent logarithm is easy to compute even for huge longs, but libm can't
   do that by itself -- loghelper can.  func is log or log10, and name is
   "log" or "log10".  Note that overflow isn't possible:  a long can contain
   no more than INT_MAX * SHIFT bits, so has value certainly less than
   2**(2**64 * 2**16) == 2**2**80, and log2 of that is 2**80, which is
   small enough to fit in an IEEE single.  log and log10 are even smaller.
*/

static PyObject*
loghelper(PyObject* arg, double (*func)(double), char *funcname)
{
    /* If it is long, do it ourselves. */
    if (PyLong_Check(arg)) {
        double x;
        int e;
        x = _PyLong_AsScaledDouble(arg, &e);
        if (x <= 0.0) {
            PyErr_SetString(PyExc_ValueError,
                            "math domain error");
            return NULL;
        }
        /* Value is ~= x * 2**(e*PyLong_SHIFT), so the log ~=
           log(x) + log(2) * e * PyLong_SHIFT.
           CAUTION:  e*PyLong_SHIFT may overflow using int arithmetic,
           so force use of double. */
        x = func(x) + (e * (double)PyLong_SHIFT) * func(2.0);
        return PyFloat_FromDouble(x);
    }

    /* Else let libm handle it by itself. */
    return math_1(arg, func, 0);
}

static PyObject *
math_log(PyObject *self, PyObject *args)
{
    PyObject *arg;
    PyObject *base = NULL;
    PyObject *num, *den;
    PyObject *ans;

    if (!PyArg_UnpackTuple(args, "log", 1, 2, &arg, &base))
        return NULL;

    num = loghelper(arg, m_log, "log");
    if (num == NULL || base == NULL)
        return num;

    den = loghelper(base, m_log, "log");
    if (den == NULL) {
        Py_DECREF(num);
        return NULL;
    }

    ans = PyNumber_TrueDivide(num, den);
    Py_DECREF(num);
    Py_DECREF(den);
    return ans;
}

PyDoc_STRVAR(math_log_doc,
"log(x[, base])\n\n\
Return the logarithm of x to the given base.\n\
If the base not specified, returns the natural logarithm (base e) of x.");

static PyObject *
math_log10(PyObject *self, PyObject *arg)
{
    return loghelper(arg, m_log10, "log10");
}

PyDoc_STRVAR(math_log10_doc,
"log10(x)\n\nReturn the base 10 logarithm of x.");

static PyObject *
math_fmod(PyObject *self, PyObject *args)
{
    PyObject *ox, *oy;
    double r, x, y;
    if (! PyArg_UnpackTuple(args, "fmod", 2, 2, &ox, &oy))
        return NULL;
    x = PyFloat_AsDouble(ox);
    y = PyFloat_AsDouble(oy);
    if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
        return NULL;
    /* fmod(x, +/-Inf) returns x for finite x. */
    if (Py_IS_INFINITY(y) && Py_IS_FINITE(x))
        return PyFloat_FromDouble(x);
    errno = 0;
    PyFPE_START_PROTECT("in math_fmod", return 0);
    r = fmod(x, y);
    PyFPE_END_PROTECT(r);
    if (Py_IS_NAN(r)) {
        if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
            errno = EDOM;
        else
            errno = 0;
    }
    if (errno && is_error(r))
        return NULL;
    else
        return PyFloat_FromDouble(r);
}

PyDoc_STRVAR(math_fmod_doc,
"fmod(x, y)\n\nReturn fmod(x, y), according to platform C."
"  x % y may differ.");

static PyObject *
math_hypot(PyObject *self, PyObject *args)
{
    PyObject *ox, *oy;
    double r, x, y;
    if (! PyArg_UnpackTuple(args, "hypot", 2, 2, &ox, &oy))
        return NULL;
    x = PyFloat_AsDouble(ox);
    y = PyFloat_AsDouble(oy);
    if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
        return NULL;
    /* hypot(x, +/-Inf) returns Inf, even if x is a NaN. */
    if (Py_IS_INFINITY(x))
        return PyFloat_FromDouble(fabs(x));
    if (Py_IS_INFINITY(y))
        return PyFloat_FromDouble(fabs(y));
    errno = 0;
    PyFPE_START_PROTECT("in math_hypot", return 0);
    r = hypot(x, y);
    PyFPE_END_PROTECT(r);
    if (Py_IS_NAN(r)) {
        if (!Py_IS_NAN(x) && !Py_IS_NAN(y))
            errno = EDOM;
        else
            errno = 0;
    }
    else if (Py_IS_INFINITY(r)) {
        if (Py_IS_FINITE(x) && Py_IS_FINITE(y))
            errno = ERANGE;
        else
            errno = 0;
    }
    if (errno && is_error(r))
        return NULL;
    else
        return PyFloat_FromDouble(r);
}

PyDoc_STRVAR(math_hypot_doc,
"hypot(x, y)\n\nReturn the Euclidean distance, sqrt(x*x + y*y).");

/* pow can't use math_2, but needs its own wrapper: the problem is
   that an infinite result can arise either as a result of overflow
   (in which case OverflowError should be raised) or as a result of
   e.g. 0.**-5. (for which ValueError needs to be raised.)
*/

static PyObject *
math_pow(PyObject *self, PyObject *args)
{
    PyObject *ox, *oy;
    double r, x, y;
    int odd_y;

    if (! PyArg_UnpackTuple(args, "pow", 2, 2, &ox, &oy))
        return NULL;
    x = PyFloat_AsDouble(ox);
    y = PyFloat_AsDouble(oy);
    if ((x == -1.0 || y == -1.0) && PyErr_Occurred())
        return NULL;

    /* deal directly with IEEE specials, to cope with problems on various
       platforms whose semantics don't exactly match C99 */
    r = 0.; /* silence compiler warning */
    if (!Py_IS_FINITE(x) || !Py_IS_FINITE(y)) {
        errno = 0;
        if (Py_IS_NAN(x))
            r = y == 0. ? 1. : x; /* NaN**0 = 1 */
        else if (Py_IS_NAN(y))
            r = x == 1. ? 1. : y; /* 1**NaN = 1 */
        else if (Py_IS_INFINITY(x)) {
            odd_y = Py_IS_FINITE(y) && fmod(fabs(y), 2.0) == 1.0;
            if (y > 0.)
                r = odd_y ? x : fabs(x);
            else if (y == 0.)
                r = 1.;
            else /* y < 0. */
                r = odd_y ? copysign(0., x) : 0.;
        }
        else if (Py_IS_INFINITY(y)) {
            if (fabs(x) == 1.0)
                r = 1.;
            else if (y > 0. && fabs(x) > 1.0)
                r = y;
            else if (y < 0. && fabs(x) < 1.0) {
                r = -y; /* result is +inf */
                if (x == 0.) /* 0**-inf: divide-by-zero */
                    errno = EDOM;
            }
            else
                r = 0.;
        }
    }
    else {
        /* let libm handle finite**finite */
        errno = 0;
        PyFPE_START_PROTECT("in math_pow", return 0);
        r = pow(x, y);
        PyFPE_END_PROTECT(r);
        /* a NaN result should arise only from (-ve)**(finite
           non-integer); in this case we want to raise ValueError. */
        if (!Py_IS_FINITE(r)) {
            if (Py_IS_NAN(r)) {
                errno = EDOM;
            }
            /*
               an infinite result here arises either from:
               (A) (+/-0.)**negative (-> divide-by-zero)
               (B) overflow of x**y with x and y finite
            */
            else if (Py_IS_INFINITY(r)) {
                if (x == 0.)
                    errno = EDOM;
                else
                    errno = ERANGE;
            }
        }
    }

    if (errno && is_error(r))
        return NULL;
    else
        return PyFloat_FromDouble(r);
}

PyDoc_STRVAR(math_pow_doc,
"pow(x, y)\n\nReturn x**y (x to the power of y).");

static const double degToRad = Py_MATH_PI / 180.0;
static const double radToDeg = 180.0 / Py_MATH_PI;

static PyObject *
math_degrees(PyObject *self, PyObject *arg)
{
    double x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    return PyFloat_FromDouble(x * radToDeg);
}

PyDoc_STRVAR(math_degrees_doc,
"degrees(x)\n\n\
Convert angle x from radians to degrees.");

static PyObject *
math_radians(PyObject *self, PyObject *arg)
{
    double x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    return PyFloat_FromDouble(x * degToRad);
}

PyDoc_STRVAR(math_radians_doc,
"radians(x)\n\n\
Convert angle x from degrees to radians.");

static PyObject *
math_isnan(PyObject *self, PyObject *arg)
{
    double x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    return PyBool_FromLong((long)Py_IS_NAN(x));
}

PyDoc_STRVAR(math_isnan_doc,
"isnan(x) -> bool\n\n\
Check if float x is not a number (NaN).");

static PyObject *
math_isinf(PyObject *self, PyObject *arg)
{
    double x = PyFloat_AsDouble(arg);
    if (x == -1.0 && PyErr_Occurred())
        return NULL;
    return PyBool_FromLong((long)Py_IS_INFINITY(x));
}

PyDoc_STRVAR(math_isinf_doc,
"isinf(x) -> bool\n\n\
Check if float x is infinite (positive or negative).");

static PyMethodDef math_methods[] = {
    {"acos",            math_acos,      METH_O,         math_acos_doc},
    {"acosh",           math_acosh,     METH_O,         math_acosh_doc},
    {"asin",            math_asin,      METH_O,         math_asin_doc},
    {"asinh",           math_asinh,     METH_O,         math_asinh_doc},
    {"atan",            math_atan,      METH_O,         math_atan_doc},
    {"atan2",           math_atan2,     METH_VARARGS,   math_atan2_doc},
    {"atanh",           math_atanh,     METH_O,         math_atanh_doc},
    {"ceil",            math_ceil,      METH_O,         math_ceil_doc},
    {"copysign",        math_copysign,  METH_VARARGS,   math_copysign_doc},
    {"cos",             math_cos,       METH_O,         math_cos_doc},
    {"cosh",            math_cosh,      METH_O,         math_cosh_doc},
    {"degrees",         math_degrees,   METH_O,         math_degrees_doc},
    {"exp",             math_exp,       METH_O,         math_exp_doc},
    {"fabs",            math_fabs,      METH_O,         math_fabs_doc},
    {"factorial",       math_factorial, METH_O,         math_factorial_doc},
    {"floor",           math_floor,     METH_O,         math_floor_doc},
    {"fmod",            math_fmod,      METH_VARARGS,   math_fmod_doc},
    {"frexp",           math_frexp,     METH_O,         math_frexp_doc},
    {"fsum",            math_fsum,      METH_O,         math_fsum_doc},
    {"hypot",           math_hypot,     METH_VARARGS,   math_hypot_doc},
    {"isinf",           math_isinf,     METH_O,         math_isinf_doc},
    {"isnan",           math_isnan,     METH_O,         math_isnan_doc},
    {"ldexp",           math_ldexp,     METH_VARARGS,   math_ldexp_doc},
    {"log",             math_log,       METH_VARARGS,   math_log_doc},
    {"log1p",           math_log1p,     METH_O,         math_log1p_doc},
    {"log10",           math_log10,     METH_O,         math_log10_doc},
    {"modf",            math_modf,      METH_O,         math_modf_doc},
    {"pow",             math_pow,       METH_VARARGS,   math_pow_doc},
    {"radians",         math_radians,   METH_O,         math_radians_doc},
    {"sin",             math_sin,       METH_O,         math_sin_doc},
    {"sinh",            math_sinh,      METH_O,         math_sinh_doc},
    {"sqrt",            math_sqrt,      METH_O,         math_sqrt_doc},
    {"tan",             math_tan,       METH_O,         math_tan_doc},
    {"tanh",            math_tanh,      METH_O,         math_tanh_doc},
    {"trunc",           math_trunc,     METH_O,         math_trunc_doc},
    {NULL,              NULL}           /* sentinel */
};


PyDoc_STRVAR(module_doc,
"This module is always available.  It provides access to the\n"
"mathematical functions defined by the C standard.");


static struct PyModuleDef mathmodule = {
    PyModuleDef_HEAD_INIT,
    "math",
    module_doc,
    -1,
    math_methods,
    NULL,
    NULL,
    NULL,
    NULL
};

PyMODINIT_FUNC
PyInit_math(void)
{
    PyObject *m;

    m = PyModule_Create(&mathmodule);
    if (m == NULL)
        goto finally;

    PyModule_AddObject(m, "pi", PyFloat_FromDouble(Py_MATH_PI));
    PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));

    finally:
    return m;
}