cpython/Lib/test/math_testcases.txt

-- Testcases for functions in math.
--
-- Each line takes the form:
--
-- <testid> <function> <input_value> -> <output_value> <flags>
--
-- where:
--
--   <testid> is a short name identifying the test,
--
--   <function> is the function to be tested (exp, cos, asinh, ...),
--
--   <input_value> is a string representing a floating-point value
--
--   <output_value> is the expected (ideal) output value, again
--     represented as a string.
--
--   <flags> is a list of the floating-point flags required by C99
--
-- The possible flags are:
--
--   divide-by-zero : raised when a finite input gives a
--     mathematically infinite result.
--
--   overflow : raised when a finite input gives a finite result that
--     is too large to fit in the usual range of an IEEE 754 double.
--
--   invalid : raised for invalid inputs (e.g., sqrt(-1))
--
--   ignore-sign : indicates that the sign of the result is
--     unspecified; e.g., if the result is given as inf,
--     then both -inf and inf should be accepted as correct.
--
-- Flags may appear in any order.
--
-- Lines beginning with '--' (like this one) start a comment, and are
-- ignored.  Blank lines, or lines containing only whitespace, are also
-- ignored.

-- Many of the values below were computed with the help of
-- version 2.4 of the MPFR library for multiple-precision
-- floating-point computations with correct rounding.  All output
-- values in this file are (modulo yet-to-be-discovered bugs)
-- correctly rounded, provided that each input and output decimal
-- floating-point value below is interpreted as a representation of
-- the corresponding nearest IEEE 754 double-precision value.  See the
-- MPFR homepage at http://www.mpfr.org for more information about the
-- MPFR project.


-------------------------
-- erf: error function --
-------------------------

erf0000 erf 0.0 -> 0.0
erf0001 erf -0.0 -> -0.0
erf0002 erf inf -> 1.0
erf0003 erf -inf -> -1.0
erf0004 erf nan -> nan

-- tiny values
erf0010 erf 1e-308 -> 1.1283791670955125e-308
erf0011 erf 5e-324 -> 4.9406564584124654e-324
erf0012 erf 1e-10 -> 1.1283791670955126e-10

-- small integers
erf0020 erf 1 -> 0.84270079294971489
erf0021 erf 2 -> 0.99532226501895271
erf0022 erf 3 -> 0.99997790950300136
erf0023 erf 4 -> 0.99999998458274209
erf0024 erf 5 -> 0.99999999999846256
erf0025 erf 6 -> 1.0

erf0030 erf -1 -> -0.84270079294971489
erf0031 erf -2 -> -0.99532226501895271
erf0032 erf -3 -> -0.99997790950300136
erf0033 erf -4 -> -0.99999998458274209
erf0034 erf -5 -> -0.99999999999846256
erf0035 erf -6 -> -1.0

-- huge values should all go to +/-1, depending on sign
erf0040 erf -40 -> -1.0
erf0041 erf 1e16 -> 1.0
erf0042 erf -1e150 -> -1.0
erf0043 erf 1.7e308 -> 1.0


----------------------------------------
-- erfc: complementary error function --
----------------------------------------

erfc0000 erfc 0.0 -> 1.0
erfc0001 erfc -0.0 -> 1.0
erfc0002 erfc inf -> 0.0
erfc0003 erfc -inf -> 2.0
erfc0004 erfc nan -> nan

-- tiny values
erfc0010 erfc 1e-308 -> 1.0
erfc0011 erfc 5e-324 -> 1.0
erfc0012 erfc 1e-10 -> 0.99999999988716204

-- small integers
erfc0020 erfc 1 -> 0.15729920705028513
erfc0021 erfc 2 -> 0.0046777349810472662
erfc0022 erfc 3 -> 2.2090496998585441e-05
erfc0023 erfc 4 -> 1.541725790028002e-08
erfc0024 erfc 5 -> 1.5374597944280349e-12
erfc0025 erfc 6 -> 2.1519736712498913e-17

erfc0030 erfc -1 -> 1.8427007929497148
erfc0031 erfc -2 -> 1.9953222650189528
erfc0032 erfc -3 -> 1.9999779095030015
erfc0033 erfc -4 -> 1.9999999845827421
erfc0034 erfc -5 -> 1.9999999999984626
erfc0035 erfc -6 -> 2.0

-- as x -> infinity, erfc(x) behaves like exp(-x*x)/x/sqrt(pi)
erfc0040 erfc 20 -> 5.3958656116079012e-176
erfc0041 erfc 25 -> 8.3001725711965228e-274
erfc0042 erfc 27 -> 5.2370464393526292e-319
erfc0043 erfc 28 -> 0.0

-- huge values
erfc0050 erfc -40 -> 2.0
erfc0051 erfc 1e16 -> 0.0
erfc0052 erfc -1e150 -> 2.0
erfc0053 erfc 1.7e308 -> 0.0


---------------------------------------------------------
-- lgamma: log of absolute value of the gamma function --
---------------------------------------------------------

-- special values
lgam0000 lgamma 0.0 -> inf      divide-by-zero
lgam0001 lgamma -0.0 -> inf     divide-by-zero
lgam0002 lgamma inf -> inf
lgam0003 lgamma -inf -> inf
lgam0004 lgamma nan -> nan

-- negative integers
lgam0010 lgamma -1 -> inf       divide-by-zero
lgam0011 lgamma -2 -> inf       divide-by-zero
lgam0012 lgamma -1e16 -> inf    divide-by-zero
lgam0013 lgamma -1e300 -> inf   divide-by-zero
lgam0014 lgamma -1.79e308 -> inf divide-by-zero

-- small positive integers give factorials
lgam0020 lgamma 1 -> 0.0
lgam0021 lgamma 2 -> 0.0
lgam0022 lgamma 3 -> 0.69314718055994529
lgam0023 lgamma 4 -> 1.791759469228055
lgam0024 lgamma 5 -> 3.1780538303479458
lgam0025 lgamma 6 -> 4.7874917427820458

-- half integers
lgam0030 lgamma 0.5 -> 0.57236494292470008
lgam0031 lgamma 1.5 -> -0.12078223763524522
lgam0032 lgamma 2.5 -> 0.28468287047291918
lgam0033 lgamma 3.5 -> 1.2009736023470743
lgam0034 lgamma -0.5 -> 1.2655121234846454
lgam0035 lgamma -1.5 -> 0.86004701537648098
lgam0036 lgamma -2.5 -> -0.056243716497674054
lgam0037 lgamma -3.5 -> -1.309006684993042

-- values near 0
lgam0040 lgamma 0.1 -> 2.252712651734206
lgam0041 lgamma 0.01 -> 4.5994798780420219
lgam0042 lgamma 1e-8 -> 18.420680738180209
lgam0043 lgamma 1e-16 -> 36.841361487904734
lgam0044 lgamma 1e-30 -> 69.077552789821368
lgam0045 lgamma 1e-160 -> 368.41361487904732
lgam0046 lgamma 1e-308 -> 709.19620864216608
lgam0047 lgamma 5.6e-309 -> 709.77602713741896
lgam0048 lgamma 5.5e-309 -> 709.79404564292167
lgam0049 lgamma 1e-309 -> 711.49879373516012
lgam0050 lgamma 1e-323 -> 743.74692474082133
lgam0051 lgamma 5e-324 -> 744.44007192138122
lgam0060 lgamma -0.1 -> 2.3689613327287886
lgam0061 lgamma -0.01 -> 4.6110249927528013
lgam0062 lgamma -1e-8 -> 18.420680749724522
lgam0063 lgamma -1e-16 -> 36.841361487904734
lgam0064 lgamma -1e-30 -> 69.077552789821368
lgam0065 lgamma -1e-160 -> 368.41361487904732
lgam0066 lgamma -1e-308 -> 709.19620864216608
lgam0067 lgamma -5.6e-309 -> 709.77602713741896
lgam0068 lgamma -5.5e-309 -> 709.79404564292167
lgam0069 lgamma -1e-309 -> 711.49879373516012
lgam0070 lgamma -1e-323 -> 743.74692474082133
lgam0071 lgamma -5e-324 -> 744.44007192138122

-- values near negative integers
lgam0080 lgamma -0.99999999999999989 -> 36.736800569677101
lgam0081 lgamma -1.0000000000000002 -> 36.043653389117154
lgam0082 lgamma -1.9999999999999998 -> 35.350506208557213
lgam0083 lgamma -2.0000000000000004 -> 34.657359027997266
lgam0084 lgamma -100.00000000000001 -> -331.85460524980607
lgam0085 lgamma -99.999999999999986 -> -331.85460524980596

-- large inputs
lgam0100 lgamma 170 -> 701.43726380873704
lgam0101 lgamma 171 -> 706.57306224578736
lgam0102 lgamma 171.624 -> 709.78077443669895
lgam0103 lgamma 171.625 -> 709.78591682948365
lgam0104 lgamma 172 -> 711.71472580228999
lgam0105 lgamma 2000 -> 13198.923448054265
lgam0106 lgamma 2.55998332785163e305 -> 1.7976931348623099e+308
lgam0107 lgamma 2.55998332785164e305 -> inf overflow
lgam0108 lgamma 1.7e308 -> inf overflow

-- inputs for which gamma(x) is tiny
lgam0120 lgamma -100.5 -> -364.90096830942736
lgam0121 lgamma -160.5 -> -656.88005261126432
lgam0122 lgamma -170.5 -> -707.99843314507882
lgam0123 lgamma -171.5 -> -713.14301641168481
lgam0124 lgamma -176.5 -> -738.95247590846486
lgam0125 lgamma -177.5 -> -744.13144651738037
lgam0126 lgamma -178.5 -> -749.3160351186001

lgam0130 lgamma -1000.5 -> -5914.4377011168517
lgam0131 lgamma -30000.5 -> -279278.6629959144
lgam0132 lgamma -4503599627370495.5 -> -1.5782258434492883e+17

-- results close to 0:  positive argument ...
lgam0150 lgamma 0.99999999999999989 -> 6.4083812134800075e-17
lgam0151 lgamma 1.0000000000000002 -> -1.2816762426960008e-16
lgam0152 lgamma 1.9999999999999998 -> -9.3876980655431170e-17
lgam0153 lgamma 2.0000000000000004 -> 1.8775396131086244e-16

-- ... and negative argument
lgam0160 lgamma -2.7476826467 -> -5.2477408147689136e-11
lgam0161 lgamma -2.457024738 -> 3.3464637541912932e-10


---------------------------
-- gamma: Gamma function --
---------------------------

-- special values
gam0000 gamma 0.0 -> inf        divide-by-zero
gam0001 gamma -0.0 -> -inf      divide-by-zero
gam0002 gamma inf -> inf
gam0003 gamma -inf -> nan       invalid
gam0004 gamma nan -> nan

-- negative integers inputs are invalid
gam0010 gamma -1 -> nan         invalid
gam0011 gamma -2 -> nan         invalid
gam0012 gamma -1e16 -> nan      invalid
gam0013 gamma -1e300 -> nan     invalid

-- small positive integers give factorials
gam0020 gamma 1 -> 1
gam0021 gamma 2 -> 1
gam0022 gamma 3 -> 2
gam0023 gamma 4 -> 6
gam0024 gamma 5 -> 24
gam0025 gamma 6 -> 120

-- half integers
gam0030 gamma 0.5 -> 1.7724538509055161
gam0031 gamma 1.5 -> 0.88622692545275805
gam0032 gamma 2.5 -> 1.3293403881791370
gam0033 gamma 3.5 -> 3.3233509704478426
gam0034 gamma -0.5 -> -3.5449077018110322
gam0035 gamma -1.5 -> 2.3632718012073548
gam0036 gamma -2.5 -> -0.94530872048294190
gam0037 gamma -3.5 -> 0.27008820585226911

-- values near 0
gam0040 gamma 0.1 -> 9.5135076986687306
gam0041 gamma 0.01 -> 99.432585119150602
gam0042 gamma 1e-8 -> 99999999.422784343
gam0043 gamma 1e-16 -> 10000000000000000
gam0044 gamma 1e-30 -> 9.9999999999999988e+29
gam0045 gamma 1e-160 -> 1.0000000000000000e+160
gam0046 gamma 1e-308 -> 1.0000000000000000e+308
gam0047 gamma 5.6e-309 -> 1.7857142857142848e+308
gam0048 gamma 5.5e-309 -> inf   overflow
gam0049 gamma 1e-309 -> inf     overflow
gam0050 gamma 1e-323 -> inf     overflow
gam0051 gamma 5e-324 -> inf     overflow
gam0060 gamma -0.1 -> -10.686287021193193
gam0061 gamma -0.01 -> -100.58719796441078
gam0062 gamma -1e-8 -> -100000000.57721567
gam0063 gamma -1e-16 -> -10000000000000000
gam0064 gamma -1e-30 -> -9.9999999999999988e+29
gam0065 gamma -1e-160 -> -1.0000000000000000e+160
gam0066 gamma -1e-308 -> -1.0000000000000000e+308
gam0067 gamma -5.6e-309 -> -1.7857142857142848e+308
gam0068 gamma -5.5e-309 -> -inf overflow
gam0069 gamma -1e-309 -> -inf   overflow
gam0070 gamma -1e-323 -> -inf   overflow
gam0071 gamma -5e-324 -> -inf   overflow

-- values near negative integers
gam0080 gamma -0.99999999999999989 -> -9007199254740992.0
gam0081 gamma -1.0000000000000002 -> 4503599627370495.5
gam0082 gamma -1.9999999999999998 -> 2251799813685248.5
gam0083 gamma -2.0000000000000004 -> -1125899906842623.5
gam0084 gamma -100.00000000000001 -> -7.5400833348831090e-145
gam0085 gamma -99.999999999999986 -> 7.5400833348840962e-145

-- large inputs
gam0100 gamma 170 -> 4.2690680090047051e+304
gam0101 gamma 171 -> 7.2574156153079990e+306
gam0102 gamma 171.624 -> 1.7942117599248104e+308
gam0103 gamma 171.625 -> inf    overflow
gam0104 gamma 172 -> inf        overflow
gam0105 gamma 2000 -> inf       overflow
gam0106 gamma 1.7e308 -> inf    overflow

-- inputs for which gamma(x) is tiny
gam0120 gamma -100.5 -> -3.3536908198076787e-159
gam0121 gamma -160.5 -> -5.2555464470078293e-286
gam0122 gamma -170.5 -> -3.3127395215386074e-308
gam0123 gamma -171.5 -> 1.9316265431711902e-310
gam0124 gamma -176.5 -> -1.1956388629358166e-321
gam0125 gamma -177.5 -> 4.9406564584124654e-324
gam0126 gamma -178.5 -> -0.0
gam0127 gamma -179.5 -> 0.0
gam0128 gamma -201.0001 -> 0.0
gam0129 gamma -202.9999 -> -0.0
gam0130 gamma -1000.5 -> -0.0
gam0131 gamma -1000000000.3 -> -0.0
gam0132 gamma -4503599627370495.5 -> 0.0

-- inputs that cause problems for the standard reflection formula,
-- thanks to loss of accuracy in 1-x
gam0140 gamma -63.349078729022985 -> 4.1777971677761880e-88
gam0141 gamma -127.45117632943295 -> 1.1831110896236810e-214


-----------------------------------------------------------
-- expm1: exp(x) - 1, without precision loss for small x --
-----------------------------------------------------------

-- special values
expm10000 expm1 0.0 -> 0.0
expm10001 expm1 -0.0 -> -0.0
expm10002 expm1 inf -> inf
expm10003 expm1 -inf -> -1.0
expm10004 expm1 nan -> nan

-- expm1(x) ~ x for tiny x
expm10010 expm1 5e-324 -> 5e-324
expm10011 expm1 1e-320 -> 1e-320
expm10012 expm1 1e-300 -> 1e-300
expm10013 expm1 1e-150 -> 1e-150
expm10014 expm1 1e-20 -> 1e-20

expm10020 expm1 -5e-324 -> -5e-324
expm10021 expm1 -1e-320 -> -1e-320
expm10022 expm1 -1e-300 -> -1e-300
expm10023 expm1 -1e-150 -> -1e-150
expm10024 expm1 -1e-20 -> -1e-20

-- moderate sized values, where direct evaluation runs into trouble
expm10100 expm1 1e-10 -> 1.0000000000500000e-10
expm10101 expm1 -9.9999999999999995e-08 -> -9.9999995000000163e-8
expm10102 expm1 3.0000000000000001e-05 -> 3.0000450004500034e-5
expm10103 expm1 -0.0070000000000000001 -> -0.0069755570667648951
expm10104 expm1 -0.071499208740094633 -> -0.069002985744820250
expm10105 expm1 -0.063296004180116799 -> -0.061334416373633009
expm10106 expm1 0.02390954035597756 -> 0.024197665143819942
expm10107 expm1 0.085637352649044901 -> 0.089411184580357767
expm10108 expm1 0.5966174947411006 -> 0.81596588596501485
expm10109 expm1 0.30247206212075139 -> 0.35319987035848677
expm10110 expm1 0.74574727375889516 -> 1.1080161116737459
expm10111 expm1 0.97767512926555711 -> 1.6582689207372185
expm10112 expm1 0.8450154566787712 -> 1.3280137976535897
expm10113 expm1 -0.13979260323125264 -> -0.13046144381396060
expm10114 expm1 -0.52899322039643271 -> -0.41080213643695923
expm10115 expm1 -0.74083261478900631 -> -0.52328317124797097
expm10116 expm1 -0.93847766984546055 -> -0.60877704724085946
expm10117 expm1 10.0 -> 22025.465794806718
expm10118 expm1 27.0 -> 532048240600.79865
expm10119 expm1 123 -> 2.6195173187490626e+53
expm10120 expm1 -12.0 -> -0.99999385578764666
expm10121 expm1 -35.100000000000001 -> -0.99999999999999944

-- extreme negative values
expm10201 expm1 -37.0 -> -0.99999999999999989
expm10200 expm1 -38.0 -> -1.0
expm10210 expm1 -710.0 -> -1.0
-- the formula expm1(x) = 2 * sinh(x/2) * exp(x/2) doesn't work so
-- well when exp(x/2) is subnormal or underflows to zero; check we're
-- not using it!
expm10211 expm1 -1420.0 -> -1.0
expm10212 expm1 -1450.0 -> -1.0
expm10213 expm1 -1500.0 -> -1.0
expm10214 expm1 -1e50 -> -1.0
expm10215 expm1 -1.79e308 -> -1.0

-- extreme positive values
expm10300 expm1 300 -> 1.9424263952412558e+130
expm10301 expm1 700 -> 1.0142320547350045e+304
expm10302 expm1 709.78271289328393 -> 1.7976931346824240e+308
expm10303 expm1 709.78271289348402 -> inf overflow
expm10304 expm1 1000 -> inf overflow
expm10305 expm1 1e50 -> inf overflow
expm10306 expm1 1.79e308 -> inf overflow