Source

cpython-withatomic / Modules / mathmodule.c

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
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
/* 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(PyObject *arg, 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)) {
		if (!Py_IS_NAN(x))
			errno = EDOM;
		else
			errno = 0;
	}
	else if (Py_IS_INFINITY(r)) {
		if (Py_IS_FINITE(x))
			errno = can_overflow ? ERANGE : EDOM;
		else
			errno = 0;
	}
	if (errno && is_error(r))
		return NULL;
	else
		return PyFloat_FromDouble(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_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.")
FUNC1(ceil, ceil, 0,
      "ceil(x)\n\nReturn the ceiling of x as a float.\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.")
FUNC1(floor, floor, 0,
      "floor(x)\n\nReturn the floor of x as a float.\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,
"sum(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 = PyInt_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 *)PyInt_FromLong(1);
	if (result == NULL)
		return NULL;
	for (i=1 ; i<=x ; i++) {
		iobj = (PyObject *)PyInt_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)
{
	return PyObject_CallMethod(number, "__trunc__", 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 if (PyInt_Check(oexp)) {
		exp = PyInt_AS_LONG(oexp);
	}
	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) -> 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_Divide(num, den);
	Py_DECREF(num);
	Py_DECREF(den);
	return ans;
}

PyDoc_STRVAR(math_log_doc,
"log(x[, base]) -> 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) -> 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) -> converts 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) -> converts 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\
Checks 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\
Checks 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.");

PyMODINIT_FUNC
initmath(void)
{
	PyObject *m;

	m = Py_InitModule3("math", math_methods, module_doc);
	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;
}
Tip: Filter by directory path e.g. /media app.js to search for public/media/app.js.
Tip: Use camelCasing e.g. ProjME to search for ProjectModifiedEvent.java.
Tip: Filter by extension type e.g. /repo .js to search for all .js files in the /repo directory.
Tip: Separate your search with spaces e.g. /ssh pom.xml to search for src/ssh/pom.xml.
Tip: Use ↑ and ↓ arrow keys to navigate and return to view the file.
Tip: You can also navigate files with Ctrl+j (next) and Ctrl+k (previous) and view the file with Ctrl+o.
Tip: You can also navigate files with Alt+j (next) and Alt+k (previous) and view the file with Alt+o.