summaryrefslogtreecommitdiff
path: root/Lib/test/test_math.py
blob: 48f84ba73241dac011c80e10183c2f18c1a6a73d (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
# Python test set -- math module
# XXXX Should not do tests around zero only

from test.support import run_unittest, verbose, requires_IEEE_754
from test import support
import unittest
import math
import os
import platform
import sys
import struct
import sysconfig

eps = 1E-05
NAN = float('nan')
INF = float('inf')
NINF = float('-inf')

# detect evidence of double-rounding: fsum is not always correctly
# rounded on machines that suffer from double rounding.
x, y = 1e16, 2.9999 # use temporary values to defeat peephole optimizer
HAVE_DOUBLE_ROUNDING = (x + y == 1e16 + 4)

# locate file with test values
if __name__ == '__main__':
    file = sys.argv[0]
else:
    file = __file__
test_dir = os.path.dirname(file) or os.curdir
math_testcases = os.path.join(test_dir, 'math_testcases.txt')
test_file = os.path.join(test_dir, 'cmath_testcases.txt')

def to_ulps(x):
    """Convert a non-NaN float x to an integer, in such a way that
    adjacent floats are converted to adjacent integers.  Then
    abs(ulps(x) - ulps(y)) gives the difference in ulps between two
    floats.

    The results from this function will only make sense on platforms
    where C doubles are represented in IEEE 754 binary64 format.

    """
    n = struct.unpack('<q', struct.pack('<d', x))[0]
    if n < 0:
        n = ~(n+2**63)
    return n

def ulps_check(expected, got, ulps=20):
    """Given non-NaN floats `expected` and `got`,
    check that they're equal to within the given number of ulps.

    Returns None on success and an error message on failure."""

    ulps_error = to_ulps(got) - to_ulps(expected)
    if abs(ulps_error) <= ulps:
        return None
    return "error = {} ulps; permitted error = {} ulps".format(ulps_error,
                                                               ulps)

# Here's a pure Python version of the math.factorial algorithm, for
# documentation and comparison purposes.
#
# Formula:
#
#   factorial(n) = factorial_odd_part(n) << (n - count_set_bits(n))
#
# where
#
#   factorial_odd_part(n) = product_{i >= 0} product_{0 < j <= n >> i; j odd} j
#
# The outer product above is an infinite product, but once i >= n.bit_length,
# (n >> i) < 1 and the corresponding term of the product is empty.  So only the
# finitely many terms for 0 <= i < n.bit_length() contribute anything.
#
# We iterate downwards from i == n.bit_length() - 1 to i == 0.  The inner
# product in the formula above starts at 1 for i == n.bit_length(); for each i
# < n.bit_length() we get the inner product for i from that for i + 1 by
# multiplying by all j in {n >> i+1 < j <= n >> i; j odd}.  In Python terms,
# this set is range((n >> i+1) + 1 | 1, (n >> i) + 1 | 1, 2).

def count_set_bits(n):
    """Number of '1' bits in binary expansion of a nonnnegative integer."""
    return 1 + count_set_bits(n & n - 1) if n else 0

def partial_product(start, stop):
    """Product of integers in range(start, stop, 2), computed recursively.
    start and stop should both be odd, with start <= stop.

    """
    numfactors = (stop - start) >> 1
    if not numfactors:
        return 1
    elif numfactors == 1:
        return start
    else:
        mid = (start + numfactors) | 1
        return partial_product(start, mid) * partial_product(mid, stop)

def py_factorial(n):
    """Factorial of nonnegative integer n, via "Binary Split Factorial Formula"
    described at http://www.luschny.de/math/factorial/binarysplitfact.html

    """
    inner = outer = 1
    for i in reversed(range(n.bit_length())):
        inner *= partial_product((n >> i + 1) + 1 | 1, (n >> i) + 1 | 1)
        outer *= inner
    return outer << (n - count_set_bits(n))

def acc_check(expected, got, rel_err=2e-15, abs_err = 5e-323):
    """Determine whether non-NaN floats a and b are equal to within a
    (small) rounding error.  The default values for rel_err and
    abs_err are chosen to be suitable for platforms where a float is
    represented by an IEEE 754 double.  They allow an error of between
    9 and 19 ulps."""

    # need to special case infinities, since inf - inf gives nan
    if math.isinf(expected) and got == expected:
        return None

    error = got - expected

    permitted_error = max(abs_err, rel_err * abs(expected))
    if abs(error) < permitted_error:
        return None
    return "error = {}; permitted error = {}".format(error,
                                                     permitted_error)

def parse_mtestfile(fname):
    """Parse a file with test values

    -- starts a comment
    blank lines, or lines containing only a comment, are ignored
    other lines are expected to have the form
      id fn arg -> expected [flag]*

    """
    with open(fname) as fp:
        for line in fp:
            # strip comments, and skip blank lines
            if '--' in line:
                line = line[:line.index('--')]
            if not line.strip():
                continue

            lhs, rhs = line.split('->')
            id, fn, arg = lhs.split()
            rhs_pieces = rhs.split()
            exp = rhs_pieces[0]
            flags = rhs_pieces[1:]

            yield (id, fn, float(arg), float(exp), flags)

def parse_testfile(fname):
    """Parse a file with test values

    Empty lines or lines starting with -- are ignored
    yields id, fn, arg_real, arg_imag, exp_real, exp_imag
    """
    with open(fname) as fp:
        for line in fp:
            # skip comment lines and blank lines
            if line.startswith('--') or not line.strip():
                continue

            lhs, rhs = line.split('->')
            id, fn, arg_real, arg_imag = lhs.split()
            rhs_pieces = rhs.split()
            exp_real, exp_imag = rhs_pieces[0], rhs_pieces[1]
            flags = rhs_pieces[2:]

            yield (id, fn,
                   float(arg_real), float(arg_imag),
                   float(exp_real), float(exp_imag),
                   flags
                  )

class MathTests(unittest.TestCase):

    def ftest(self, name, value, expected):
        if abs(value-expected) > eps:
            # Use %r instead of %f so the error message
            # displays full precision. Otherwise discrepancies
            # in the last few bits will lead to very confusing
            # error messages
            self.fail('%s returned %r, expected %r' %
                      (name, value, expected))

    def testConstants(self):
        self.ftest('pi', math.pi, 3.1415926)
        self.ftest('e', math.e, 2.7182818)

    def testAcos(self):
        self.assertRaises(TypeError, math.acos)
        self.ftest('acos(-1)', math.acos(-1), math.pi)
        self.ftest('acos(0)', math.acos(0), math.pi/2)
        self.ftest('acos(1)', math.acos(1), 0)
        self.assertRaises(ValueError, math.acos, INF)
        self.assertRaises(ValueError, math.acos, NINF)
        self.assertTrue(math.isnan(math.acos(NAN)))

    def testAcosh(self):
        self.assertRaises(TypeError, math.acosh)
        self.ftest('acosh(1)', math.acosh(1), 0)
        self.ftest('acosh(2)', math.acosh(2), 1.3169578969248168)
        self.assertRaises(ValueError, math.acosh, 0)
        self.assertRaises(ValueError, math.acosh, -1)
        self.assertEqual(math.acosh(INF), INF)
        self.assertRaises(ValueError, math.acosh, NINF)
        self.assertTrue(math.isnan(math.acosh(NAN)))

    def testAsin(self):
        self.assertRaises(TypeError, math.asin)
        self.ftest('asin(-1)', math.asin(-1), -math.pi/2)
        self.ftest('asin(0)', math.asin(0), 0)
        self.ftest('asin(1)', math.asin(1), math.pi/2)
        self.assertRaises(ValueError, math.asin, INF)
        self.assertRaises(ValueError, math.asin, NINF)
        self.assertTrue(math.isnan(math.asin(NAN)))

    def testAsinh(self):
        self.assertRaises(TypeError, math.asinh)
        self.ftest('asinh(0)', math.asinh(0), 0)
        self.ftest('asinh(1)', math.asinh(1), 0.88137358701954305)
        self.ftest('asinh(-1)', math.asinh(-1), -0.88137358701954305)
        self.assertEqual(math.asinh(INF), INF)
        self.assertEqual(math.asinh(NINF), NINF)
        self.assertTrue(math.isnan(math.asinh(NAN)))

    def testAtan(self):
        self.assertRaises(TypeError, math.atan)
        self.ftest('atan(-1)', math.atan(-1), -math.pi/4)
        self.ftest('atan(0)', math.atan(0), 0)
        self.ftest('atan(1)', math.atan(1), math.pi/4)
        self.ftest('atan(inf)', math.atan(INF), math.pi/2)
        self.ftest('atan(-inf)', math.atan(NINF), -math.pi/2)
        self.assertTrue(math.isnan(math.atan(NAN)))

    def testAtanh(self):
        self.assertRaises(TypeError, math.atan)
        self.ftest('atanh(0)', math.atanh(0), 0)
        self.ftest('atanh(0.5)', math.atanh(0.5), 0.54930614433405489)
        self.ftest('atanh(-0.5)', math.atanh(-0.5), -0.54930614433405489)
        self.assertRaises(ValueError, math.atanh, 1)
        self.assertRaises(ValueError, math.atanh, -1)
        self.assertRaises(ValueError, math.atanh, INF)
        self.assertRaises(ValueError, math.atanh, NINF)
        self.assertTrue(math.isnan(math.atanh(NAN)))

    def testAtan2(self):
        self.assertRaises(TypeError, math.atan2)
        self.ftest('atan2(-1, 0)', math.atan2(-1, 0), -math.pi/2)
        self.ftest('atan2(-1, 1)', math.atan2(-1, 1), -math.pi/4)
        self.ftest('atan2(0, 1)', math.atan2(0, 1), 0)
        self.ftest('atan2(1, 1)', math.atan2(1, 1), math.pi/4)
        self.ftest('atan2(1, 0)', math.atan2(1, 0), math.pi/2)

        # math.atan2(0, x)
        self.ftest('atan2(0., -inf)', math.atan2(0., NINF), math.pi)
        self.ftest('atan2(0., -2.3)', math.atan2(0., -2.3), math.pi)
        self.ftest('atan2(0., -0.)', math.atan2(0., -0.), math.pi)
        self.assertEqual(math.atan2(0., 0.), 0.)
        self.assertEqual(math.atan2(0., 2.3), 0.)
        self.assertEqual(math.atan2(0., INF), 0.)
        self.assertTrue(math.isnan(math.atan2(0., NAN)))
        # math.atan2(-0, x)
        self.ftest('atan2(-0., -inf)', math.atan2(-0., NINF), -math.pi)
        self.ftest('atan2(-0., -2.3)', math.atan2(-0., -2.3), -math.pi)
        self.ftest('atan2(-0., -0.)', math.atan2(-0., -0.), -math.pi)
        self.assertEqual(math.atan2(-0., 0.), -0.)
        self.assertEqual(math.atan2(-0., 2.3), -0.)
        self.assertEqual(math.atan2(-0., INF), -0.)
        self.assertTrue(math.isnan(math.atan2(-0., NAN)))
        # math.atan2(INF, x)
        self.ftest('atan2(inf, -inf)', math.atan2(INF, NINF), math.pi*3/4)
        self.ftest('atan2(inf, -2.3)', math.atan2(INF, -2.3), math.pi/2)
        self.ftest('atan2(inf, -0.)', math.atan2(INF, -0.0), math.pi/2)
        self.ftest('atan2(inf, 0.)', math.atan2(INF, 0.0), math.pi/2)
        self.ftest('atan2(inf, 2.3)', math.atan2(INF, 2.3), math.pi/2)
        self.ftest('atan2(inf, inf)', math.atan2(INF, INF), math.pi/4)
        self.assertTrue(math.isnan(math.atan2(INF, NAN)))
        # math.atan2(NINF, x)
        self.ftest('atan2(-inf, -inf)', math.atan2(NINF, NINF), -math.pi*3/4)
        self.ftest('atan2(-inf, -2.3)', math.atan2(NINF, -2.3), -math.pi/2)
        self.ftest('atan2(-inf, -0.)', math.atan2(NINF, -0.0), -math.pi/2)
        self.ftest('atan2(-inf, 0.)', math.atan2(NINF, 0.0), -math.pi/2)
        self.ftest('atan2(-inf, 2.3)', math.atan2(NINF, 2.3), -math.pi/2)
        self.ftest('atan2(-inf, inf)', math.atan2(NINF, INF), -math.pi/4)
        self.assertTrue(math.isnan(math.atan2(NINF, NAN)))
        # math.atan2(+finite, x)
        self.ftest('atan2(2.3, -inf)', math.atan2(2.3, NINF), math.pi)
        self.ftest('atan2(2.3, -0.)', math.atan2(2.3, -0.), math.pi/2)
        self.ftest('atan2(2.3, 0.)', math.atan2(2.3, 0.), math.pi/2)
        self.assertEqual(math.atan2(2.3, INF), 0.)
        self.assertTrue(math.isnan(math.atan2(2.3, NAN)))
        # math.atan2(-finite, x)
        self.ftest('atan2(-2.3, -inf)', math.atan2(-2.3, NINF), -math.pi)
        self.ftest('atan2(-2.3, -0.)', math.atan2(-2.3, -0.), -math.pi/2)
        self.ftest('atan2(-2.3, 0.)', math.atan2(-2.3, 0.), -math.pi/2)
        self.assertEqual(math.atan2(-2.3, INF), -0.)
        self.assertTrue(math.isnan(math.atan2(-2.3, NAN)))
        # math.atan2(NAN, x)
        self.assertTrue(math.isnan(math.atan2(NAN, NINF)))
        self.assertTrue(math.isnan(math.atan2(NAN, -2.3)))
        self.assertTrue(math.isnan(math.atan2(NAN, -0.)))
        self.assertTrue(math.isnan(math.atan2(NAN, 0.)))
        self.assertTrue(math.isnan(math.atan2(NAN, 2.3)))
        self.assertTrue(math.isnan(math.atan2(NAN, INF)))
        self.assertTrue(math.isnan(math.atan2(NAN, NAN)))

    def testCeil(self):
        self.assertRaises(TypeError, math.ceil)
        self.assertEqual(int, type(math.ceil(0.5)))
        self.ftest('ceil(0.5)', math.ceil(0.5), 1)
        self.ftest('ceil(1.0)', math.ceil(1.0), 1)
        self.ftest('ceil(1.5)', math.ceil(1.5), 2)
        self.ftest('ceil(-0.5)', math.ceil(-0.5), 0)
        self.ftest('ceil(-1.0)', math.ceil(-1.0), -1)
        self.ftest('ceil(-1.5)', math.ceil(-1.5), -1)
        #self.assertEqual(math.ceil(INF), INF)
        #self.assertEqual(math.ceil(NINF), NINF)
        #self.assertTrue(math.isnan(math.ceil(NAN)))

        class TestCeil:
            def __ceil__(self):
                return 42
        class TestNoCeil:
            pass
        self.ftest('ceil(TestCeil())', math.ceil(TestCeil()), 42)
        self.assertRaises(TypeError, math.ceil, TestNoCeil())

        t = TestNoCeil()
        t.__ceil__ = lambda *args: args
        self.assertRaises(TypeError, math.ceil, t)
        self.assertRaises(TypeError, math.ceil, t, 0)

    @requires_IEEE_754
    def testCopysign(self):
        self.assertEqual(math.copysign(1, 42), 1.0)
        self.assertEqual(math.copysign(0., 42), 0.0)
        self.assertEqual(math.copysign(1., -42), -1.0)
        self.assertEqual(math.copysign(3, 0.), 3.0)
        self.assertEqual(math.copysign(4., -0.), -4.0)

        self.assertRaises(TypeError, math.copysign)
        # copysign should let us distinguish signs of zeros
        self.assertEqual(math.copysign(1., 0.), 1.)
        self.assertEqual(math.copysign(1., -0.), -1.)
        self.assertEqual(math.copysign(INF, 0.), INF)
        self.assertEqual(math.copysign(INF, -0.), NINF)
        self.assertEqual(math.copysign(NINF, 0.), INF)
        self.assertEqual(math.copysign(NINF, -0.), NINF)
        # and of infinities
        self.assertEqual(math.copysign(1., INF), 1.)
        self.assertEqual(math.copysign(1., NINF), -1.)
        self.assertEqual(math.copysign(INF, INF), INF)
        self.assertEqual(math.copysign(INF, NINF), NINF)
        self.assertEqual(math.copysign(NINF, INF), INF)
        self.assertEqual(math.copysign(NINF, NINF), NINF)
        self.assertTrue(math.isnan(math.copysign(NAN, 1.)))
        self.assertTrue(math.isnan(math.copysign(NAN, INF)))
        self.assertTrue(math.isnan(math.copysign(NAN, NINF)))
        self.assertTrue(math.isnan(math.copysign(NAN, NAN)))
        # copysign(INF, NAN) may be INF or it may be NINF, since
        # we don't know whether the sign bit of NAN is set on any
        # given platform.
        self.assertTrue(math.isinf(math.copysign(INF, NAN)))
        # similarly, copysign(2., NAN) could be 2. or -2.
        self.assertEqual(abs(math.copysign(2., NAN)), 2.)

    def testCos(self):
        self.assertRaises(TypeError, math.cos)
        self.ftest('cos(-pi/2)', math.cos(-math.pi/2), 0)
        self.ftest('cos(0)', math.cos(0), 1)
        self.ftest('cos(pi/2)', math.cos(math.pi/2), 0)
        self.ftest('cos(pi)', math.cos(math.pi), -1)
        try:
            self.assertTrue(math.isnan(math.cos(INF)))
            self.assertTrue(math.isnan(math.cos(NINF)))
        except ValueError:
            self.assertRaises(ValueError, math.cos, INF)
            self.assertRaises(ValueError, math.cos, NINF)
        self.assertTrue(math.isnan(math.cos(NAN)))

    def testCosh(self):
        self.assertRaises(TypeError, math.cosh)
        self.ftest('cosh(0)', math.cosh(0), 1)
        self.ftest('cosh(2)-2*cosh(1)**2', math.cosh(2)-2*math.cosh(1)**2, -1) # Thanks to Lambert
        self.assertEqual(math.cosh(INF), INF)
        self.assertEqual(math.cosh(NINF), INF)
        self.assertTrue(math.isnan(math.cosh(NAN)))

    def testDegrees(self):
        self.assertRaises(TypeError, math.degrees)
        self.ftest('degrees(pi)', math.degrees(math.pi), 180.0)
        self.ftest('degrees(pi/2)', math.degrees(math.pi/2), 90.0)
        self.ftest('degrees(-pi/4)', math.degrees(-math.pi/4), -45.0)

    def testExp(self):
        self.assertRaises(TypeError, math.exp)
        self.ftest('exp(-1)', math.exp(-1), 1/math.e)
        self.ftest('exp(0)', math.exp(0), 1)
        self.ftest('exp(1)', math.exp(1), math.e)
        self.assertEqual(math.exp(INF), INF)
        self.assertEqual(math.exp(NINF), 0.)
        self.assertTrue(math.isnan(math.exp(NAN)))

    def testFabs(self):
        self.assertRaises(TypeError, math.fabs)
        self.ftest('fabs(-1)', math.fabs(-1), 1)
        self.ftest('fabs(0)', math.fabs(0), 0)
        self.ftest('fabs(1)', math.fabs(1), 1)

    def testFactorial(self):
        self.assertEqual(math.factorial(0), 1)
        self.assertEqual(math.factorial(0.0), 1)
        total = 1
        for i in range(1, 1000):
            total *= i
            self.assertEqual(math.factorial(i), total)
            self.assertEqual(math.factorial(float(i)), total)
            self.assertEqual(math.factorial(i), py_factorial(i))
        self.assertRaises(ValueError, math.factorial, -1)
        self.assertRaises(ValueError, math.factorial, -1.0)
        self.assertRaises(ValueError, math.factorial, math.pi)
        self.assertRaises(OverflowError, math.factorial, sys.maxsize+1)
        self.assertRaises(OverflowError, math.factorial, 10e100)

    def testFloor(self):
        self.assertRaises(TypeError, math.floor)
        self.assertEqual(int, type(math.floor(0.5)))
        self.ftest('floor(0.5)', math.floor(0.5), 0)
        self.ftest('floor(1.0)', math.floor(1.0), 1)
        self.ftest('floor(1.5)', math.floor(1.5), 1)
        self.ftest('floor(-0.5)', math.floor(-0.5), -1)
        self.ftest('floor(-1.0)', math.floor(-1.0), -1)
        self.ftest('floor(-1.5)', math.floor(-1.5), -2)
        # pow() relies on floor() to check for integers
        # This fails on some platforms - so check it here
        self.ftest('floor(1.23e167)', math.floor(1.23e167), 1.23e167)
        self.ftest('floor(-1.23e167)', math.floor(-1.23e167), -1.23e167)
        #self.assertEqual(math.ceil(INF), INF)
        #self.assertEqual(math.ceil(NINF), NINF)
        #self.assertTrue(math.isnan(math.floor(NAN)))

        class TestFloor:
            def __floor__(self):
                return 42
        class TestNoFloor:
            pass
        self.ftest('floor(TestFloor())', math.floor(TestFloor()), 42)
        self.assertRaises(TypeError, math.floor, TestNoFloor())

        t = TestNoFloor()
        t.__floor__ = lambda *args: args
        self.assertRaises(TypeError, math.floor, t)
        self.assertRaises(TypeError, math.floor, t, 0)

    def testFmod(self):
        self.assertRaises(TypeError, math.fmod)
        self.ftest('fmod(10, 1)', math.fmod(10, 1), 0.0)
        self.ftest('fmod(10, 0.5)', math.fmod(10, 0.5), 0.0)
        self.ftest('fmod(10, 1.5)', math.fmod(10, 1.5), 1.0)
        self.ftest('fmod(-10, 1)', math.fmod(-10, 1), -0.0)
        self.ftest('fmod(-10, 0.5)', math.fmod(-10, 0.5), -0.0)
        self.ftest('fmod(-10, 1.5)', math.fmod(-10, 1.5), -1.0)
        self.assertTrue(math.isnan(math.fmod(NAN, 1.)))
        self.assertTrue(math.isnan(math.fmod(1., NAN)))
        self.assertTrue(math.isnan(math.fmod(NAN, NAN)))
        self.assertRaises(ValueError, math.fmod, 1., 0.)
        self.assertRaises(ValueError, math.fmod, INF, 1.)
        self.assertRaises(ValueError, math.fmod, NINF, 1.)
        self.assertRaises(ValueError, math.fmod, INF, 0.)
        self.assertEqual(math.fmod(3.0, INF), 3.0)
        self.assertEqual(math.fmod(-3.0, INF), -3.0)
        self.assertEqual(math.fmod(3.0, NINF), 3.0)
        self.assertEqual(math.fmod(-3.0, NINF), -3.0)
        self.assertEqual(math.fmod(0.0, 3.0), 0.0)
        self.assertEqual(math.fmod(0.0, NINF), 0.0)

    def testFrexp(self):
        self.assertRaises(TypeError, math.frexp)

        def testfrexp(name, result, expected):
            (mant, exp), (emant, eexp) = result, expected
            if abs(mant-emant) > eps or exp != eexp:
                self.fail('%s returned %r, expected %r'%\
                          (name, result, expected))

        testfrexp('frexp(-1)', math.frexp(-1), (-0.5, 1))
        testfrexp('frexp(0)', math.frexp(0), (0, 0))
        testfrexp('frexp(1)', math.frexp(1), (0.5, 1))
        testfrexp('frexp(2)', math.frexp(2), (0.5, 2))

        self.assertEqual(math.frexp(INF)[0], INF)
        self.assertEqual(math.frexp(NINF)[0], NINF)
        self.assertTrue(math.isnan(math.frexp(NAN)[0]))

    @requires_IEEE_754
    @unittest.skipIf(HAVE_DOUBLE_ROUNDING,
                         "fsum is not exact on machines with double rounding")
    def testFsum(self):
        # math.fsum relies on exact rounding for correct operation.
        # There's a known problem with IA32 floating-point that causes
        # inexact rounding in some situations, and will cause the
        # math.fsum tests below to fail; see issue #2937.  On non IEEE
        # 754 platforms, and on IEEE 754 platforms that exhibit the
        # problem described in issue #2937, we simply skip the whole
        # test.

        # Python version of math.fsum, for comparison.  Uses a
        # different algorithm based on frexp, ldexp and integer
        # arithmetic.
        from sys import float_info
        mant_dig = float_info.mant_dig
        etiny = float_info.min_exp - mant_dig

        def msum(iterable):
            """Full precision summation.  Compute sum(iterable) without any
            intermediate accumulation of error.  Based on the 'lsum' function
            at http://code.activestate.com/recipes/393090/

            """
            tmant, texp = 0, 0
            for x in iterable:
                mant, exp = math.frexp(x)
                mant, exp = int(math.ldexp(mant, mant_dig)), exp - mant_dig
                if texp > exp:
                    tmant <<= texp-exp
                    texp = exp
                else:
                    mant <<= exp-texp
                tmant += mant
            # Round tmant * 2**texp to a float.  The original recipe
            # used float(str(tmant)) * 2.0**texp for this, but that's
            # a little unsafe because str -> float conversion can't be
            # relied upon to do correct rounding on all platforms.
            tail = max(len(bin(abs(tmant)))-2 - mant_dig, etiny - texp)
            if tail > 0:
                h = 1 << (tail-1)
                tmant = tmant // (2*h) + bool(tmant & h and tmant & 3*h-1)
                texp += tail
            return math.ldexp(tmant, texp)

        test_values = [
            ([], 0.0),
            ([0.0], 0.0),
            ([1e100, 1.0, -1e100, 1e-100, 1e50, -1.0, -1e50], 1e-100),
            ([2.0**53, -0.5, -2.0**-54], 2.0**53-1.0),
            ([2.0**53, 1.0, 2.0**-100], 2.0**53+2.0),
            ([2.0**53+10.0, 1.0, 2.0**-100], 2.0**53+12.0),
            ([2.0**53-4.0, 0.5, 2.0**-54], 2.0**53-3.0),
            ([1./n for n in range(1, 1001)],
             float.fromhex('0x1.df11f45f4e61ap+2')),
            ([(-1.)**n/n for n in range(1, 1001)],
             float.fromhex('-0x1.62a2af1bd3624p-1')),
            ([1.7**(i+1)-1.7**i for i in range(1000)] + [-1.7**1000], -1.0),
            ([1e16, 1., 1e-16], 10000000000000002.0),
            ([1e16-2., 1.-2.**-53, -(1e16-2.), -(1.-2.**-53)], 0.0),
            # exercise code for resizing partials array
            ([2.**n - 2.**(n+50) + 2.**(n+52) for n in range(-1074, 972, 2)] +
             [-2.**1022],
             float.fromhex('0x1.5555555555555p+970')),
            ]

        for i, (vals, expected) in enumerate(test_values):
            try:
                actual = math.fsum(vals)
            except OverflowError:
                self.fail("test %d failed: got OverflowError, expected %r "
                          "for math.fsum(%.100r)" % (i, expected, vals))
            except ValueError:
                self.fail("test %d failed: got ValueError, expected %r "
                          "for math.fsum(%.100r)" % (i, expected, vals))
            self.assertEqual(actual, expected)

        from random import random, gauss, shuffle
        for j in range(1000):
            vals = [7, 1e100, -7, -1e100, -9e-20, 8e-20] * 10
            s = 0
            for i in range(200):
                v = gauss(0, random()) ** 7 - s
                s += v
                vals.append(v)
            shuffle(vals)

            s = msum(vals)
            self.assertEqual(msum(vals), math.fsum(vals))

    def testHypot(self):
        self.assertRaises(TypeError, math.hypot)
        self.ftest('hypot(0,0)', math.hypot(0,0), 0)
        self.ftest('hypot(3,4)', math.hypot(3,4), 5)
        self.assertEqual(math.hypot(NAN, INF), INF)
        self.assertEqual(math.hypot(INF, NAN), INF)
        self.assertEqual(math.hypot(NAN, NINF), INF)
        self.assertEqual(math.hypot(NINF, NAN), INF)
        self.assertTrue(math.isnan(math.hypot(1.0, NAN)))
        self.assertTrue(math.isnan(math.hypot(NAN, -2.0)))

    def testLdexp(self):
        self.assertRaises(TypeError, math.ldexp)
        self.ftest('ldexp(0,1)', math.ldexp(0,1), 0)
        self.ftest('ldexp(1,1)', math.ldexp(1,1), 2)
        self.ftest('ldexp(1,-1)', math.ldexp(1,-1), 0.5)
        self.ftest('ldexp(-1,1)', math.ldexp(-1,1), -2)
        self.assertRaises(OverflowError, math.ldexp, 1., 1000000)
        self.assertRaises(OverflowError, math.ldexp, -1., 1000000)
        self.assertEqual(math.ldexp(1., -1000000), 0.)
        self.assertEqual(math.ldexp(-1., -1000000), -0.)
        self.assertEqual(math.ldexp(INF, 30), INF)
        self.assertEqual(math.ldexp(NINF, -213), NINF)
        self.assertTrue(math.isnan(math.ldexp(NAN, 0)))

        # large second argument
        for n in [10**5, 10**10, 10**20, 10**40]:
            self.assertEqual(math.ldexp(INF, -n), INF)
            self.assertEqual(math.ldexp(NINF, -n), NINF)
            self.assertEqual(math.ldexp(1., -n), 0.)
            self.assertEqual(math.ldexp(-1., -n), -0.)
            self.assertEqual(math.ldexp(0., -n), 0.)
            self.assertEqual(math.ldexp(-0., -n), -0.)
            self.assertTrue(math.isnan(math.ldexp(NAN, -n)))

            self.assertRaises(OverflowError, math.ldexp, 1., n)
            self.assertRaises(OverflowError, math.ldexp, -1., n)
            self.assertEqual(math.ldexp(0., n), 0.)
            self.assertEqual(math.ldexp(-0., n), -0.)
            self.assertEqual(math.ldexp(INF, n), INF)
            self.assertEqual(math.ldexp(NINF, n), NINF)
            self.assertTrue(math.isnan(math.ldexp(NAN, n)))

    def testLog(self):
        self.assertRaises(TypeError, math.log)
        self.ftest('log(1/e)', math.log(1/math.e), -1)
        self.ftest('log(1)', math.log(1), 0)
        self.ftest('log(e)', math.log(math.e), 1)
        self.ftest('log(32,2)', math.log(32,2), 5)
        self.ftest('log(10**40, 10)', math.log(10**40, 10), 40)
        self.ftest('log(10**40, 10**20)', math.log(10**40, 10**20), 2)
        self.ftest('log(10**1000)', math.log(10**1000),
                   2302.5850929940457)
        self.assertRaises(ValueError, math.log, -1.5)
        self.assertRaises(ValueError, math.log, -10**1000)
        self.assertRaises(ValueError, math.log, NINF)
        self.assertEqual(math.log(INF), INF)
        self.assertTrue(math.isnan(math.log(NAN)))

    def testLog1p(self):
        self.assertRaises(TypeError, math.log1p)
        n= 2**90
        self.assertAlmostEqual(math.log1p(n), math.log1p(float(n)))

    @requires_IEEE_754
    def testLog2(self):
        self.assertRaises(TypeError, math.log2)

        # Check some integer values
        self.assertEqual(math.log2(1), 0.0)
        self.assertEqual(math.log2(2), 1.0)
        self.assertEqual(math.log2(4), 2.0)

        # Large integer values
        self.assertEqual(math.log2(2**1023), 1023.0)
        self.assertEqual(math.log2(2**1024), 1024.0)
        self.assertEqual(math.log2(2**2000), 2000.0)

        self.assertRaises(ValueError, math.log2, -1.5)
        self.assertRaises(ValueError, math.log2, NINF)
        self.assertTrue(math.isnan(math.log2(NAN)))

    @requires_IEEE_754
    # log2() is not accurate enough on Mac OS X Tiger (10.4)
    @support.requires_mac_ver(10, 5)
    def testLog2Exact(self):
        # Check that we get exact equality for log2 of powers of 2.
        actual = [math.log2(math.ldexp(1.0, n)) for n in range(-1074, 1024)]
        expected = [float(n) for n in range(-1074, 1024)]
        self.assertEqual(actual, expected)

    def testLog10(self):
        self.assertRaises(TypeError, math.log10)
        self.ftest('log10(0.1)', math.log10(0.1), -1)
        self.ftest('log10(1)', math.log10(1), 0)
        self.ftest('log10(10)', math.log10(10), 1)
        self.ftest('log10(10**1000)', math.log10(10**1000), 1000.0)
        self.assertRaises(ValueError, math.log10, -1.5)
        self.assertRaises(ValueError, math.log10, -10**1000)
        self.assertRaises(ValueError, math.log10, NINF)
        self.assertEqual(math.log(INF), INF)
        self.assertTrue(math.isnan(math.log10(NAN)))

    def testModf(self):
        self.assertRaises(TypeError, math.modf)

        def testmodf(name, result, expected):
            (v1, v2), (e1, e2) = result, expected
            if abs(v1-e1) > eps or abs(v2-e2):
                self.fail('%s returned %r, expected %r'%\
                          (name, result, expected))

        testmodf('modf(1.5)', math.modf(1.5), (0.5, 1.0))
        testmodf('modf(-1.5)', math.modf(-1.5), (-0.5, -1.0))

        self.assertEqual(math.modf(INF), (0.0, INF))
        self.assertEqual(math.modf(NINF), (-0.0, NINF))

        modf_nan = math.modf(NAN)
        self.assertTrue(math.isnan(modf_nan[0]))
        self.assertTrue(math.isnan(modf_nan[1]))

    def testPow(self):
        self.assertRaises(TypeError, math.pow)
        self.ftest('pow(0,1)', math.pow(0,1), 0)
        self.ftest('pow(1,0)', math.pow(1,0), 1)
        self.ftest('pow(2,1)', math.pow(2,1), 2)
        self.ftest('pow(2,-1)', math.pow(2,-1), 0.5)
        self.assertEqual(math.pow(INF, 1), INF)
        self.assertEqual(math.pow(NINF, 1), NINF)
        self.assertEqual((math.pow(1, INF)), 1.)
        self.assertEqual((math.pow(1, NINF)), 1.)
        self.assertTrue(math.isnan(math.pow(NAN, 1)))
        self.assertTrue(math.isnan(math.pow(2, NAN)))
        self.assertTrue(math.isnan(math.pow(0, NAN)))
        self.assertEqual(math.pow(1, NAN), 1)

        # pow(0., x)
        self.assertEqual(math.pow(0., INF), 0.)
        self.assertEqual(math.pow(0., 3.), 0.)
        self.assertEqual(math.pow(0., 2.3), 0.)
        self.assertEqual(math.pow(0., 2.), 0.)
        self.assertEqual(math.pow(0., 0.), 1.)
        self.assertEqual(math.pow(0., -0.), 1.)
        self.assertRaises(ValueError, math.pow, 0., -2.)
        self.assertRaises(ValueError, math.pow, 0., -2.3)
        self.assertRaises(ValueError, math.pow, 0., -3.)
        self.assertRaises(ValueError, math.pow, 0., NINF)
        self.assertTrue(math.isnan(math.pow(0., NAN)))

        # pow(INF, x)
        self.assertEqual(math.pow(INF, INF), INF)
        self.assertEqual(math.pow(INF, 3.), INF)
        self.assertEqual(math.pow(INF, 2.3), INF)
        self.assertEqual(math.pow(INF, 2.), INF)
        self.assertEqual(math.pow(INF, 0.), 1.)
        self.assertEqual(math.pow(INF, -0.), 1.)
        self.assertEqual(math.pow(INF, -2.), 0.)
        self.assertEqual(math.pow(INF, -2.3), 0.)
        self.assertEqual(math.pow(INF, -3.), 0.)
        self.assertEqual(math.pow(INF, NINF), 0.)
        self.assertTrue(math.isnan(math.pow(INF, NAN)))

        # pow(-0., x)
        self.assertEqual(math.pow(-0., INF), 0.)
        self.assertEqual(math.pow(-0., 3.), -0.)
        self.assertEqual(math.pow(-0., 2.3), 0.)
        self.assertEqual(math.pow(-0., 2.), 0.)
        self.assertEqual(math.pow(-0., 0.), 1.)
        self.assertEqual(math.pow(-0., -0.), 1.)
        self.assertRaises(ValueError, math.pow, -0., -2.)
        self.assertRaises(ValueError, math.pow, -0., -2.3)
        self.assertRaises(ValueError, math.pow, -0., -3.)
        self.assertRaises(ValueError, math.pow, -0., NINF)
        self.assertTrue(math.isnan(math.pow(-0., NAN)))

        # pow(NINF, x)
        self.assertEqual(math.pow(NINF, INF), INF)
        self.assertEqual(math.pow(NINF, 3.), NINF)
        self.assertEqual(math.pow(NINF, 2.3), INF)
        self.assertEqual(math.pow(NINF, 2.), INF)
        self.assertEqual(math.pow(NINF, 0.), 1.)
        self.assertEqual(math.pow(NINF, -0.), 1.)
        self.assertEqual(math.pow(NINF, -2.), 0.)
        self.assertEqual(math.pow(NINF, -2.3), 0.)
        self.assertEqual(math.pow(NINF, -3.), -0.)
        self.assertEqual(math.pow(NINF, NINF), 0.)
        self.assertTrue(math.isnan(math.pow(NINF, NAN)))

        # pow(-1, x)
        self.assertEqual(math.pow(-1., INF), 1.)
        self.assertEqual(math.pow(-1., 3.), -1.)
        self.assertRaises(ValueError, math.pow, -1., 2.3)
        self.assertEqual(math.pow(-1., 2.), 1.)
        self.assertEqual(math.pow(-1., 0.), 1.)
        self.assertEqual(math.pow(-1., -0.), 1.)
        self.assertEqual(math.pow(-1., -2.), 1.)
        self.assertRaises(ValueError, math.pow, -1., -2.3)
        self.assertEqual(math.pow(-1., -3.), -1.)
        self.assertEqual(math.pow(-1., NINF), 1.)
        self.assertTrue(math.isnan(math.pow(-1., NAN)))

        # pow(1, x)
        self.assertEqual(math.pow(1., INF), 1.)
        self.assertEqual(math.pow(1., 3.), 1.)
        self.assertEqual(math.pow(1., 2.3), 1.)
        self.assertEqual(math.pow(1., 2.), 1.)
        self.assertEqual(math.pow(1., 0.), 1.)
        self.assertEqual(math.pow(1., -0.), 1.)
        self.assertEqual(math.pow(1., -2.), 1.)
        self.assertEqual(math.pow(1., -2.3), 1.)
        self.assertEqual(math.pow(1., -3.), 1.)
        self.assertEqual(math.pow(1., NINF), 1.)
        self.assertEqual(math.pow(1., NAN), 1.)

        # pow(x, 0) should be 1 for any x
        self.assertEqual(math.pow(2.3, 0.), 1.)
        self.assertEqual(math.pow(-2.3, 0.), 1.)
        self.assertEqual(math.pow(NAN, 0.), 1.)
        self.assertEqual(math.pow(2.3, -0.), 1.)
        self.assertEqual(math.pow(-2.3, -0.), 1.)
        self.assertEqual(math.pow(NAN, -0.), 1.)

        # pow(x, y) is invalid if x is negative and y is not integral
        self.assertRaises(ValueError, math.pow, -1., 2.3)
        self.assertRaises(ValueError, math.pow, -15., -3.1)

        # pow(x, NINF)
        self.assertEqual(math.pow(1.9, NINF), 0.)
        self.assertEqual(math.pow(1.1, NINF), 0.)
        self.assertEqual(math.pow(0.9, NINF), INF)
        self.assertEqual(math.pow(0.1, NINF), INF)
        self.assertEqual(math.pow(-0.1, NINF), INF)
        self.assertEqual(math.pow(-0.9, NINF), INF)
        self.assertEqual(math.pow(-1.1, NINF), 0.)
        self.assertEqual(math.pow(-1.9, NINF), 0.)

        # pow(x, INF)
        self.assertEqual(math.pow(1.9, INF), INF)
        self.assertEqual(math.pow(1.1, INF), INF)
        self.assertEqual(math.pow(0.9, INF), 0.)
        self.assertEqual(math.pow(0.1, INF), 0.)
        self.assertEqual(math.pow(-0.1, INF), 0.)
        self.assertEqual(math.pow(-0.9, INF), 0.)
        self.assertEqual(math.pow(-1.1, INF), INF)
        self.assertEqual(math.pow(-1.9, INF), INF)

        # pow(x, y) should work for x negative, y an integer
        self.ftest('(-2.)**3.', math.pow(-2.0, 3.0), -8.0)
        self.ftest('(-2.)**2.', math.pow(-2.0, 2.0), 4.0)
        self.ftest('(-2.)**1.', math.pow(-2.0, 1.0), -2.0)
        self.ftest('(-2.)**0.', math.pow(-2.0, 0.0), 1.0)
        self.ftest('(-2.)**-0.', math.pow(-2.0, -0.0), 1.0)
        self.ftest('(-2.)**-1.', math.pow(-2.0, -1.0), -0.5)
        self.ftest('(-2.)**-2.', math.pow(-2.0, -2.0), 0.25)
        self.ftest('(-2.)**-3.', math.pow(-2.0, -3.0), -0.125)
        self.assertRaises(ValueError, math.pow, -2.0, -0.5)
        self.assertRaises(ValueError, math.pow, -2.0, 0.5)

        # the following tests have been commented out since they don't
        # really belong here:  the implementation of ** for floats is
        # independent of the implementation of math.pow
        #self.assertEqual(1**NAN, 1)
        #self.assertEqual(1**INF, 1)
        #self.assertEqual(1**NINF, 1)
        #self.assertEqual(1**0, 1)
        #self.assertEqual(1.**NAN, 1)
        #self.assertEqual(1.**INF, 1)
        #self.assertEqual(1.**NINF, 1)
        #self.assertEqual(1.**0, 1)

    def testRadians(self):
        self.assertRaises(TypeError, math.radians)
        self.ftest('radians(180)', math.radians(180), math.pi)
        self.ftest('radians(90)', math.radians(90), math.pi/2)
        self.ftest('radians(-45)', math.radians(-45), -math.pi/4)

    def testSin(self):
        self.assertRaises(TypeError, math.sin)
        self.ftest('sin(0)', math.sin(0), 0)
        self.ftest('sin(pi/2)', math.sin(math.pi/2), 1)
        self.ftest('sin(-pi/2)', math.sin(-math.pi/2), -1)
        try:
            self.assertTrue(math.isnan(math.sin(INF)))
            self.assertTrue(math.isnan(math.sin(NINF)))
        except ValueError:
            self.assertRaises(ValueError, math.sin, INF)
            self.assertRaises(ValueError, math.sin, NINF)
        self.assertTrue(math.isnan(math.sin(NAN)))

    def testSinh(self):
        self.assertRaises(TypeError, math.sinh)
        self.ftest('sinh(0)', math.sinh(0), 0)
        self.ftest('sinh(1)**2-cosh(1)**2', math.sinh(1)**2-math.cosh(1)**2, -1)
        self.ftest('sinh(1)+sinh(-1)', math.sinh(1)+math.sinh(-1), 0)
        self.assertEqual(math.sinh(INF), INF)
        self.assertEqual(math.sinh(NINF), NINF)
        self.assertTrue(math.isnan(math.sinh(NAN)))

    def testSqrt(self):
        self.assertRaises(TypeError, math.sqrt)
        self.ftest('sqrt(0)', math.sqrt(0), 0)
        self.ftest('sqrt(1)', math.sqrt(1), 1)
        self.ftest('sqrt(4)', math.sqrt(4), 2)
        self.assertEqual(math.sqrt(INF), INF)
        self.assertRaises(ValueError, math.sqrt, NINF)
        self.assertTrue(math.isnan(math.sqrt(NAN)))

    def testTan(self):
        self.assertRaises(TypeError, math.tan)
        self.ftest('tan(0)', math.tan(0), 0)
        self.ftest('tan(pi/4)', math.tan(math.pi/4), 1)
        self.ftest('tan(-pi/4)', math.tan(-math.pi/4), -1)
        try:
            self.assertTrue(math.isnan(math.tan(INF)))
            self.assertTrue(math.isnan(math.tan(NINF)))
        except:
            self.assertRaises(ValueError, math.tan, INF)
            self.assertRaises(ValueError, math.tan, NINF)
        self.assertTrue(math.isnan(math.tan(NAN)))

    def testTanh(self):
        self.assertRaises(TypeError, math.tanh)
        self.ftest('tanh(0)', math.tanh(0), 0)
        self.ftest('tanh(1)+tanh(-1)', math.tanh(1)+math.tanh(-1), 0)
        self.ftest('tanh(inf)', math.tanh(INF), 1)
        self.ftest('tanh(-inf)', math.tanh(NINF), -1)
        self.assertTrue(math.isnan(math.tanh(NAN)))

    @requires_IEEE_754
    @unittest.skipIf(sysconfig.get_config_var('TANH_PRESERVES_ZERO_SIGN') == 0,
                     "system tanh() function doesn't copy the sign")
    def testTanhSign(self):
        # check that tanh(-0.) == -0. on IEEE 754 systems
        self.assertEqual(math.tanh(-0.), -0.)
        self.assertEqual(math.copysign(1., math.tanh(-0.)),
                         math.copysign(1., -0.))

    def test_trunc(self):
        self.assertEqual(math.trunc(1), 1)
        self.assertEqual(math.trunc(-1), -1)
        self.assertEqual(type(math.trunc(1)), int)
        self.assertEqual(type(math.trunc(1.5)), int)
        self.assertEqual(math.trunc(1.5), 1)
        self.assertEqual(math.trunc(-1.5), -1)
        self.assertEqual(math.trunc(1.999999), 1)
        self.assertEqual(math.trunc(-1.999999), -1)
        self.assertEqual(math.trunc(-0.999999), -0)
        self.assertEqual(math.trunc(-100.999), -100)

        class TestTrunc(object):
            def __trunc__(self):
                return 23

        class TestNoTrunc(object):
            pass

        self.assertEqual(math.trunc(TestTrunc()), 23)

        self.assertRaises(TypeError, math.trunc)
        self.assertRaises(TypeError, math.trunc, 1, 2)
        self.assertRaises(TypeError, math.trunc, TestNoTrunc())

    def testIsfinite(self):
        self.assertTrue(math.isfinite(0.0))
        self.assertTrue(math.isfinite(-0.0))
        self.assertTrue(math.isfinite(1.0))
        self.assertTrue(math.isfinite(-1.0))
        self.assertFalse(math.isfinite(float("nan")))
        self.assertFalse(math.isfinite(float("inf")))
        self.assertFalse(math.isfinite(float("-inf")))

    def testIsnan(self):
        self.assertTrue(math.isnan(float("nan")))
        self.assertTrue(math.isnan(float("inf")* 0.))
        self.assertFalse(math.isnan(float("inf")))
        self.assertFalse(math.isnan(0.))
        self.assertFalse(math.isnan(1.))

    def testIsinf(self):
        self.assertTrue(math.isinf(float("inf")))
        self.assertTrue(math.isinf(float("-inf")))
        self.assertTrue(math.isinf(1E400))
        self.assertTrue(math.isinf(-1E400))
        self.assertFalse(math.isinf(float("nan")))
        self.assertFalse(math.isinf(0.))
        self.assertFalse(math.isinf(1.))

    # RED_FLAG 16-Oct-2000 Tim
    # While 2.0 is more consistent about exceptions than previous releases, it
    # still fails this part of the test on some platforms.  For now, we only
    # *run* test_exceptions() in verbose mode, so that this isn't normally
    # tested.
    @unittest.skipUnless(verbose, 'requires verbose mode')
    def test_exceptions(self):
        try:
            x = math.exp(-1000000000)
        except:
            # mathmodule.c is failing to weed out underflows from libm, or
            # we've got an fp format with huge dynamic range
            self.fail("underflowing exp() should not have raised "
                        "an exception")
        if x != 0:
            self.fail("underflowing exp() should have returned 0")

        # If this fails, probably using a strict IEEE-754 conforming libm, and x
        # is +Inf afterwards.  But Python wants overflows detected by default.
        try:
            x = math.exp(1000000000)
        except OverflowError:
            pass
        else:
            self.fail("overflowing exp() didn't trigger OverflowError")

        # If this fails, it could be a puzzle.  One odd possibility is that
        # mathmodule.c's macros are getting confused while comparing
        # Inf (HUGE_VAL) to a NaN, and artificially setting errno to ERANGE
        # as a result (and so raising OverflowError instead).
        try:
            x = math.sqrt(-1.0)
        except ValueError:
            pass
        else:
            self.fail("sqrt(-1) didn't raise ValueError")

    @requires_IEEE_754
    def test_testfile(self):
        for id, fn, ar, ai, er, ei, flags in parse_testfile(test_file):
            # Skip if either the input or result is complex, or if
            # flags is nonempty
            if ai != 0. or ei != 0. or flags:
                continue
            if fn in ['rect', 'polar']:
                # no real versions of rect, polar
                continue
            func = getattr(math, fn)
            try:
                result = func(ar)
            except ValueError as exc:
                message = (("Unexpected ValueError: %s\n        " +
                           "in test %s:%s(%r)\n") % (exc.args[0], id, fn, ar))
                self.fail(message)
            except OverflowError:
                message = ("Unexpected OverflowError in " +
                           "test %s:%s(%r)\n" % (id, fn, ar))
                self.fail(message)
            self.ftest("%s:%s(%r)" % (id, fn, ar), result, er)

    @requires_IEEE_754
    def test_mtestfile(self):
        fail_fmt = "{}:{}({!r}): expected {!r}, got {!r}"

        failures = []
        for id, fn, arg, expected, flags in parse_mtestfile(math_testcases):
            func = getattr(math, fn)

            if 'invalid' in flags or 'divide-by-zero' in flags:
                expected = 'ValueError'
            elif 'overflow' in flags:
                expected = 'OverflowError'

            try:
                got = func(arg)
            except ValueError:
                got = 'ValueError'
            except OverflowError:
                got = 'OverflowError'

            accuracy_failure = None
            if isinstance(got, float) and isinstance(expected, float):
                if math.isnan(expected) and math.isnan(got):
                    continue
                if not math.isnan(expected) and not math.isnan(got):
                    if fn == 'lgamma':
                        # we use a weaker accuracy test for lgamma;
                        # lgamma only achieves an absolute error of
                        # a few multiples of the machine accuracy, in
                        # general.
                        accuracy_failure = acc_check(expected, got,
                                                  rel_err = 5e-15,
                                                  abs_err = 5e-15)
                    elif fn == 'erfc':
                        # erfc has less-than-ideal accuracy for large
                        # arguments (x ~ 25 or so), mainly due to the
                        # error involved in computing exp(-x*x).
                        #
                        # XXX Would be better to weaken this test only
                        # for large x, instead of for all x.
                        accuracy_failure = ulps_check(expected, got, 2000)

                    else:
                        accuracy_failure = ulps_check(expected, got, 20)
                    if accuracy_failure is None:
                        continue

            if isinstance(got, str) and isinstance(expected, str):
                if got == expected:
                    continue

            fail_msg = fail_fmt.format(id, fn, arg, expected, got)
            if accuracy_failure is not None:
                fail_msg += ' ({})'.format(accuracy_failure)
            failures.append(fail_msg)

        if failures:
            self.fail('Failures in test_mtestfile:\n  ' +
                      '\n  '.join(failures))


def test_main():
    from doctest import DocFileSuite
    suite = unittest.TestSuite()
    suite.addTest(unittest.makeSuite(MathTests))
    suite.addTest(DocFileSuite("ieee754.txt"))
    run_unittest(suite)

if __name__ == '__main__':
    test_main()