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math_testcases.txt
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-- 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