| -- 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 |
| |
| -- Issue 8986: inputs x with exp(-x*x) near the underflow threshold |
| -- incorrectly signalled overflow on some platforms. |
| erf0100 erf 26.2 -> 1.0 |
| erf0101 erf 26.4 -> 1.0 |
| erf0102 erf 26.6 -> 1.0 |
| erf0103 erf 26.8 -> 1.0 |
| erf0104 erf 27.0 -> 1.0 |
| erf0105 erf 27.2 -> 1.0 |
| erf0106 erf 27.4 -> 1.0 |
| erf0107 erf 27.6 -> 1.0 |
| |
| erf0110 erf -26.2 -> -1.0 |
| erf0111 erf -26.4 -> -1.0 |
| erf0112 erf -26.6 -> -1.0 |
| erf0113 erf -26.8 -> -1.0 |
| erf0114 erf -27.0 -> -1.0 |
| erf0115 erf -27.2 -> -1.0 |
| erf0116 erf -27.4 -> -1.0 |
| erf0117 erf -27.6 -> -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 |
| |
| -- Issue 8986: inputs x with exp(-x*x) near the underflow threshold |
| -- incorrectly signalled overflow on some platforms. |
| erfc0100 erfc 26.2 -> 1.6432507924389461e-300 |
| erfc0101 erfc 26.4 -> 4.4017768588035426e-305 |
| erfc0102 erfc 26.6 -> 1.0885125885442269e-309 |
| erfc0103 erfc 26.8 -> 2.4849621571966629e-314 |
| erfc0104 erfc 27.0 -> 5.2370464393526292e-319 |
| erfc0105 erfc 27.2 -> 9.8813129168249309e-324 |
| erfc0106 erfc 27.4 -> 0.0 |
| erfc0107 erfc 27.6 -> 0.0 |
| |
| erfc0110 erfc -26.2 -> 2.0 |
| erfc0111 erfc -26.4 -> 2.0 |
| erfc0112 erfc -26.6 -> 2.0 |
| erfc0113 erfc -26.8 -> 2.0 |
| erfc0114 erfc -27.0 -> 2.0 |
| erfc0115 erfc -27.2 -> 2.0 |
| erfc0116 erfc -27.4 -> 2.0 |
| erfc0117 erfc -27.6 -> 2.0 |
| |
| --------------------------------------------------------- |
| -- 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 |
| -- the next test (expm10302) is disabled because it causes failure on |
| -- OS X 10.4/Intel: apparently all values over 709.78 produce an |
| -- overflow on that platform. See issue #7575. |
| -- 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 |
| |
| -- weaker version of expm10302 |
| expm10307 expm1 709.5 -> 1.3549863193146328e+308 |
