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      1 -- Testcases for functions in math.
      2 --
      3 -- Each line takes the form:
      4 --
      5 -- <testid> <function> <input_value> -> <output_value> <flags>
      6 --
      7 -- where:
      8 --
      9 --   <testid> is a short name identifying the test,
     10 --
     11 --   <function> is the function to be tested (exp, cos, asinh, ...),
     12 --
     13 --   <input_value> is a string representing a floating-point value
     14 --
     15 --   <output_value> is the expected (ideal) output value, again
     16 --     represented as a string.
     17 --
     18 --   <flags> is a list of the floating-point flags required by C99
     19 --
     20 -- The possible flags are:
     21 --
     22 --   divide-by-zero : raised when a finite input gives a
     23 --     mathematically infinite result.
     24 --
     25 --   overflow : raised when a finite input gives a finite result that
     26 --     is too large to fit in the usual range of an IEEE 754 double.
     27 --
     28 --   invalid : raised for invalid inputs (e.g., sqrt(-1))
     29 --
     30 --   ignore-sign : indicates that the sign of the result is
     31 --     unspecified; e.g., if the result is given as inf,
     32 --     then both -inf and inf should be accepted as correct.
     33 --
     34 -- Flags may appear in any order.
     35 --
     36 -- Lines beginning with '--' (like this one) start a comment, and are
     37 -- ignored.  Blank lines, or lines containing only whitespace, are also
     38 -- ignored.
     39 
     40 -- Many of the values below were computed with the help of
     41 -- version 2.4 of the MPFR library for multiple-precision
     42 -- floating-point computations with correct rounding.  All output
     43 -- values in this file are (modulo yet-to-be-discovered bugs)
     44 -- correctly rounded, provided that each input and output decimal
     45 -- floating-point value below is interpreted as a representation of
     46 -- the corresponding nearest IEEE 754 double-precision value.  See the
     47 -- MPFR homepage at http://www.mpfr.org for more information about the
     48 -- MPFR project.
     49 
     50 
     51 -------------------------
     52 -- erf: error function --
     53 -------------------------
     54 
     55 erf0000 erf 0.0 -> 0.0
     56 erf0001 erf -0.0 -> -0.0
     57 erf0002 erf inf -> 1.0
     58 erf0003 erf -inf -> -1.0
     59 erf0004 erf nan -> nan
     60 
     61 -- tiny values
     62 erf0010 erf 1e-308 -> 1.1283791670955125e-308
     63 erf0011 erf 5e-324 -> 4.9406564584124654e-324
     64 erf0012 erf 1e-10 -> 1.1283791670955126e-10
     65 
     66 -- small integers
     67 erf0020 erf 1 -> 0.84270079294971489
     68 erf0021 erf 2 -> 0.99532226501895271
     69 erf0022 erf 3 -> 0.99997790950300136
     70 erf0023 erf 4 -> 0.99999998458274209
     71 erf0024 erf 5 -> 0.99999999999846256
     72 erf0025 erf 6 -> 1.0
     73 
     74 erf0030 erf -1 -> -0.84270079294971489
     75 erf0031 erf -2 -> -0.99532226501895271
     76 erf0032 erf -3 -> -0.99997790950300136
     77 erf0033 erf -4 -> -0.99999998458274209
     78 erf0034 erf -5 -> -0.99999999999846256
     79 erf0035 erf -6 -> -1.0
     80 
     81 -- huge values should all go to +/-1, depending on sign
     82 erf0040 erf -40 -> -1.0
     83 erf0041 erf 1e16 -> 1.0
     84 erf0042 erf -1e150 -> -1.0
     85 erf0043 erf 1.7e308 -> 1.0
     86 
     87 -- Issue 8986: inputs x with exp(-x*x) near the underflow threshold
     88 -- incorrectly signalled overflow on some platforms.
     89 erf0100 erf 26.2 -> 1.0
     90 erf0101 erf 26.4 -> 1.0
     91 erf0102 erf 26.6 -> 1.0
     92 erf0103 erf 26.8 -> 1.0
     93 erf0104 erf 27.0 -> 1.0
     94 erf0105 erf 27.2 -> 1.0
     95 erf0106 erf 27.4 -> 1.0
     96 erf0107 erf 27.6 -> 1.0
     97 
     98 erf0110 erf -26.2 -> -1.0
     99 erf0111 erf -26.4 -> -1.0
    100 erf0112 erf -26.6 -> -1.0
    101 erf0113 erf -26.8 -> -1.0
    102 erf0114 erf -27.0 -> -1.0
    103 erf0115 erf -27.2 -> -1.0
    104 erf0116 erf -27.4 -> -1.0
    105 erf0117 erf -27.6 -> -1.0
    106 
    107 ----------------------------------------
    108 -- erfc: complementary error function --
    109 ----------------------------------------
    110 
    111 erfc0000 erfc 0.0 -> 1.0
    112 erfc0001 erfc -0.0 -> 1.0
    113 erfc0002 erfc inf -> 0.0
    114 erfc0003 erfc -inf -> 2.0
    115 erfc0004 erfc nan -> nan
    116 
    117 -- tiny values
    118 erfc0010 erfc 1e-308 -> 1.0
    119 erfc0011 erfc 5e-324 -> 1.0
    120 erfc0012 erfc 1e-10 -> 0.99999999988716204
    121 
    122 -- small integers
    123 erfc0020 erfc 1 -> 0.15729920705028513
    124 erfc0021 erfc 2 -> 0.0046777349810472662
    125 erfc0022 erfc 3 -> 2.2090496998585441e-05
    126 erfc0023 erfc 4 -> 1.541725790028002e-08
    127 erfc0024 erfc 5 -> 1.5374597944280349e-12
    128 erfc0025 erfc 6 -> 2.1519736712498913e-17
    129 
    130 erfc0030 erfc -1 -> 1.8427007929497148
    131 erfc0031 erfc -2 -> 1.9953222650189528
    132 erfc0032 erfc -3 -> 1.9999779095030015
    133 erfc0033 erfc -4 -> 1.9999999845827421
    134 erfc0034 erfc -5 -> 1.9999999999984626
    135 erfc0035 erfc -6 -> 2.0
    136 
    137 -- as x -> infinity, erfc(x) behaves like exp(-x*x)/x/sqrt(pi)
    138 erfc0040 erfc 20 -> 5.3958656116079012e-176
    139 erfc0041 erfc 25 -> 8.3001725711965228e-274
    140 erfc0042 erfc 27 -> 5.2370464393526292e-319
    141 erfc0043 erfc 28 -> 0.0
    142 
    143 -- huge values
    144 erfc0050 erfc -40 -> 2.0
    145 erfc0051 erfc 1e16 -> 0.0
    146 erfc0052 erfc -1e150 -> 2.0
    147 erfc0053 erfc 1.7e308 -> 0.0
    148 
    149 -- Issue 8986: inputs x with exp(-x*x) near the underflow threshold
    150 -- incorrectly signalled overflow on some platforms.
    151 erfc0100 erfc 26.2 -> 1.6432507924389461e-300
    152 erfc0101 erfc 26.4 -> 4.4017768588035426e-305
    153 erfc0102 erfc 26.6 -> 1.0885125885442269e-309
    154 erfc0103 erfc 26.8 -> 2.4849621571966629e-314
    155 erfc0104 erfc 27.0 -> 5.2370464393526292e-319
    156 erfc0105 erfc 27.2 -> 9.8813129168249309e-324
    157 erfc0106 erfc 27.4 -> 0.0
    158 erfc0107 erfc 27.6 -> 0.0
    159 
    160 erfc0110 erfc -26.2 -> 2.0
    161 erfc0111 erfc -26.4 -> 2.0
    162 erfc0112 erfc -26.6 -> 2.0
    163 erfc0113 erfc -26.8 -> 2.0
    164 erfc0114 erfc -27.0 -> 2.0
    165 erfc0115 erfc -27.2 -> 2.0
    166 erfc0116 erfc -27.4 -> 2.0
    167 erfc0117 erfc -27.6 -> 2.0
    168 
    169 ---------------------------------------------------------
    170 -- lgamma: log of absolute value of the gamma function --
    171 ---------------------------------------------------------
    172 
    173 -- special values
    174 lgam0000 lgamma 0.0 -> inf      divide-by-zero
    175 lgam0001 lgamma -0.0 -> inf     divide-by-zero
    176 lgam0002 lgamma inf -> inf
    177 lgam0003 lgamma -inf -> inf
    178 lgam0004 lgamma nan -> nan
    179 
    180 -- negative integers
    181 lgam0010 lgamma -1 -> inf       divide-by-zero
    182 lgam0011 lgamma -2 -> inf       divide-by-zero
    183 lgam0012 lgamma -1e16 -> inf    divide-by-zero
    184 lgam0013 lgamma -1e300 -> inf   divide-by-zero
    185 lgam0014 lgamma -1.79e308 -> inf divide-by-zero
    186 
    187 -- small positive integers give factorials
    188 lgam0020 lgamma 1 -> 0.0
    189 lgam0021 lgamma 2 -> 0.0
    190 lgam0022 lgamma 3 -> 0.69314718055994529
    191 lgam0023 lgamma 4 -> 1.791759469228055
    192 lgam0024 lgamma 5 -> 3.1780538303479458
    193 lgam0025 lgamma 6 -> 4.7874917427820458
    194 
    195 -- half integers
    196 lgam0030 lgamma 0.5 -> 0.57236494292470008
    197 lgam0031 lgamma 1.5 -> -0.12078223763524522
    198 lgam0032 lgamma 2.5 -> 0.28468287047291918
    199 lgam0033 lgamma 3.5 -> 1.2009736023470743
    200 lgam0034 lgamma -0.5 -> 1.2655121234846454
    201 lgam0035 lgamma -1.5 -> 0.86004701537648098
    202 lgam0036 lgamma -2.5 -> -0.056243716497674054
    203 lgam0037 lgamma -3.5 -> -1.309006684993042
    204 
    205 -- values near 0
    206 lgam0040 lgamma 0.1 -> 2.252712651734206
    207 lgam0041 lgamma 0.01 -> 4.5994798780420219
    208 lgam0042 lgamma 1e-8 -> 18.420680738180209
    209 lgam0043 lgamma 1e-16 -> 36.841361487904734
    210 lgam0044 lgamma 1e-30 -> 69.077552789821368
    211 lgam0045 lgamma 1e-160 -> 368.41361487904732
    212 lgam0046 lgamma 1e-308 -> 709.19620864216608
    213 lgam0047 lgamma 5.6e-309 -> 709.77602713741896
    214 lgam0048 lgamma 5.5e-309 -> 709.79404564292167
    215 lgam0049 lgamma 1e-309 -> 711.49879373516012
    216 lgam0050 lgamma 1e-323 -> 743.74692474082133
    217 lgam0051 lgamma 5e-324 -> 744.44007192138122
    218 lgam0060 lgamma -0.1 -> 2.3689613327287886
    219 lgam0061 lgamma -0.01 -> 4.6110249927528013
    220 lgam0062 lgamma -1e-8 -> 18.420680749724522
    221 lgam0063 lgamma -1e-16 -> 36.841361487904734
    222 lgam0064 lgamma -1e-30 -> 69.077552789821368
    223 lgam0065 lgamma -1e-160 -> 368.41361487904732
    224 lgam0066 lgamma -1e-308 -> 709.19620864216608
    225 lgam0067 lgamma -5.6e-309 -> 709.77602713741896
    226 lgam0068 lgamma -5.5e-309 -> 709.79404564292167
    227 lgam0069 lgamma -1e-309 -> 711.49879373516012
    228 lgam0070 lgamma -1e-323 -> 743.74692474082133
    229 lgam0071 lgamma -5e-324 -> 744.44007192138122
    230 
    231 -- values near negative integers
    232 lgam0080 lgamma -0.99999999999999989 -> 36.736800569677101
    233 lgam0081 lgamma -1.0000000000000002 -> 36.043653389117154
    234 lgam0082 lgamma -1.9999999999999998 -> 35.350506208557213
    235 lgam0083 lgamma -2.0000000000000004 -> 34.657359027997266
    236 lgam0084 lgamma -100.00000000000001 -> -331.85460524980607
    237 lgam0085 lgamma -99.999999999999986 -> -331.85460524980596
    238 
    239 -- large inputs
    240 lgam0100 lgamma 170 -> 701.43726380873704
    241 lgam0101 lgamma 171 -> 706.57306224578736
    242 lgam0102 lgamma 171.624 -> 709.78077443669895
    243 lgam0103 lgamma 171.625 -> 709.78591682948365
    244 lgam0104 lgamma 172 -> 711.71472580228999
    245 lgam0105 lgamma 2000 -> 13198.923448054265
    246 lgam0106 lgamma 2.55998332785163e305 -> 1.7976931348623099e+308
    247 lgam0107 lgamma 2.55998332785164e305 -> inf overflow
    248 lgam0108 lgamma 1.7e308 -> inf overflow
    249 
    250 -- inputs for which gamma(x) is tiny
    251 lgam0120 lgamma -100.5 -> -364.90096830942736
    252 lgam0121 lgamma -160.5 -> -656.88005261126432
    253 lgam0122 lgamma -170.5 -> -707.99843314507882
    254 lgam0123 lgamma -171.5 -> -713.14301641168481
    255 lgam0124 lgamma -176.5 -> -738.95247590846486
    256 lgam0125 lgamma -177.5 -> -744.13144651738037
    257 lgam0126 lgamma -178.5 -> -749.3160351186001
    258 
    259 lgam0130 lgamma -1000.5 -> -5914.4377011168517
    260 lgam0131 lgamma -30000.5 -> -279278.6629959144
    261 lgam0132 lgamma -4503599627370495.5 -> -1.5782258434492883e+17
    262 
    263 -- results close to 0:  positive argument ...
    264 lgam0150 lgamma 0.99999999999999989 -> 6.4083812134800075e-17
    265 lgam0151 lgamma 1.0000000000000002 -> -1.2816762426960008e-16
    266 lgam0152 lgamma 1.9999999999999998 -> -9.3876980655431170e-17
    267 lgam0153 lgamma 2.0000000000000004 -> 1.8775396131086244e-16
    268 
    269 -- ... and negative argument
    270 lgam0160 lgamma -2.7476826467 -> -5.2477408147689136e-11
    271 lgam0161 lgamma -2.457024738 -> 3.3464637541912932e-10
    272 
    273 
    274 ---------------------------
    275 -- gamma: Gamma function --
    276 ---------------------------
    277 
    278 -- special values
    279 gam0000 gamma 0.0 -> inf        divide-by-zero
    280 gam0001 gamma -0.0 -> -inf      divide-by-zero
    281 gam0002 gamma inf -> inf
    282 gam0003 gamma -inf -> nan       invalid
    283 gam0004 gamma nan -> nan
    284 
    285 -- negative integers inputs are invalid
    286 gam0010 gamma -1 -> nan         invalid
    287 gam0011 gamma -2 -> nan         invalid
    288 gam0012 gamma -1e16 -> nan      invalid
    289 gam0013 gamma -1e300 -> nan     invalid
    290 
    291 -- small positive integers give factorials
    292 gam0020 gamma 1 -> 1
    293 gam0021 gamma 2 -> 1
    294 gam0022 gamma 3 -> 2
    295 gam0023 gamma 4 -> 6
    296 gam0024 gamma 5 -> 24
    297 gam0025 gamma 6 -> 120
    298 
    299 -- half integers
    300 gam0030 gamma 0.5 -> 1.7724538509055161
    301 gam0031 gamma 1.5 -> 0.88622692545275805
    302 gam0032 gamma 2.5 -> 1.3293403881791370
    303 gam0033 gamma 3.5 -> 3.3233509704478426
    304 gam0034 gamma -0.5 -> -3.5449077018110322
    305 gam0035 gamma -1.5 -> 2.3632718012073548
    306 gam0036 gamma -2.5 -> -0.94530872048294190
    307 gam0037 gamma -3.5 -> 0.27008820585226911
    308 
    309 -- values near 0
    310 gam0040 gamma 0.1 -> 9.5135076986687306
    311 gam0041 gamma 0.01 -> 99.432585119150602
    312 gam0042 gamma 1e-8 -> 99999999.422784343
    313 gam0043 gamma 1e-16 -> 10000000000000000
    314 gam0044 gamma 1e-30 -> 9.9999999999999988e+29
    315 gam0045 gamma 1e-160 -> 1.0000000000000000e+160
    316 gam0046 gamma 1e-308 -> 1.0000000000000000e+308
    317 gam0047 gamma 5.6e-309 -> 1.7857142857142848e+308
    318 gam0048 gamma 5.5e-309 -> inf   overflow
    319 gam0049 gamma 1e-309 -> inf     overflow
    320 gam0050 gamma 1e-323 -> inf     overflow
    321 gam0051 gamma 5e-324 -> inf     overflow
    322 gam0060 gamma -0.1 -> -10.686287021193193
    323 gam0061 gamma -0.01 -> -100.58719796441078
    324 gam0062 gamma -1e-8 -> -100000000.57721567
    325 gam0063 gamma -1e-16 -> -10000000000000000
    326 gam0064 gamma -1e-30 -> -9.9999999999999988e+29
    327 gam0065 gamma -1e-160 -> -1.0000000000000000e+160
    328 gam0066 gamma -1e-308 -> -1.0000000000000000e+308
    329 gam0067 gamma -5.6e-309 -> -1.7857142857142848e+308
    330 gam0068 gamma -5.5e-309 -> -inf overflow
    331 gam0069 gamma -1e-309 -> -inf   overflow
    332 gam0070 gamma -1e-323 -> -inf   overflow
    333 gam0071 gamma -5e-324 -> -inf   overflow
    334 
    335 -- values near negative integers
    336 gam0080 gamma -0.99999999999999989 -> -9007199254740992.0
    337 gam0081 gamma -1.0000000000000002 -> 4503599627370495.5
    338 gam0082 gamma -1.9999999999999998 -> 2251799813685248.5
    339 gam0083 gamma -2.0000000000000004 -> -1125899906842623.5
    340 gam0084 gamma -100.00000000000001 -> -7.5400833348831090e-145
    341 gam0085 gamma -99.999999999999986 -> 7.5400833348840962e-145
    342 
    343 -- large inputs
    344 gam0100 gamma 170 -> 4.2690680090047051e+304
    345 gam0101 gamma 171 -> 7.2574156153079990e+306
    346 gam0102 gamma 171.624 -> 1.7942117599248104e+308
    347 gam0103 gamma 171.625 -> inf    overflow
    348 gam0104 gamma 172 -> inf        overflow
    349 gam0105 gamma 2000 -> inf       overflow
    350 gam0106 gamma 1.7e308 -> inf    overflow
    351 
    352 -- inputs for which gamma(x) is tiny
    353 gam0120 gamma -100.5 -> -3.3536908198076787e-159
    354 gam0121 gamma -160.5 -> -5.2555464470078293e-286
    355 gam0122 gamma -170.5 -> -3.3127395215386074e-308
    356 gam0123 gamma -171.5 -> 1.9316265431711902e-310
    357 gam0124 gamma -176.5 -> -1.1956388629358166e-321
    358 gam0125 gamma -177.5 -> 4.9406564584124654e-324
    359 gam0126 gamma -178.5 -> -0.0
    360 gam0127 gamma -179.5 -> 0.0
    361 gam0128 gamma -201.0001 -> 0.0
    362 gam0129 gamma -202.9999 -> -0.0
    363 gam0130 gamma -1000.5 -> -0.0
    364 gam0131 gamma -1000000000.3 -> -0.0
    365 gam0132 gamma -4503599627370495.5 -> 0.0
    366 
    367 -- inputs that cause problems for the standard reflection formula,
    368 -- thanks to loss of accuracy in 1-x
    369 gam0140 gamma -63.349078729022985 -> 4.1777971677761880e-88
    370 gam0141 gamma -127.45117632943295 -> 1.1831110896236810e-214
    371 
    372 
    373 -----------------------------------------------------------
    374 -- expm1: exp(x) - 1, without precision loss for small x --
    375 -----------------------------------------------------------
    376 
    377 -- special values
    378 expm10000 expm1 0.0 -> 0.0
    379 expm10001 expm1 -0.0 -> -0.0
    380 expm10002 expm1 inf -> inf
    381 expm10003 expm1 -inf -> -1.0
    382 expm10004 expm1 nan -> nan
    383 
    384 -- expm1(x) ~ x for tiny x
    385 expm10010 expm1 5e-324 -> 5e-324
    386 expm10011 expm1 1e-320 -> 1e-320
    387 expm10012 expm1 1e-300 -> 1e-300
    388 expm10013 expm1 1e-150 -> 1e-150
    389 expm10014 expm1 1e-20 -> 1e-20
    390 
    391 expm10020 expm1 -5e-324 -> -5e-324
    392 expm10021 expm1 -1e-320 -> -1e-320
    393 expm10022 expm1 -1e-300 -> -1e-300
    394 expm10023 expm1 -1e-150 -> -1e-150
    395 expm10024 expm1 -1e-20 -> -1e-20
    396 
    397 -- moderate sized values, where direct evaluation runs into trouble
    398 expm10100 expm1 1e-10 -> 1.0000000000500000e-10
    399 expm10101 expm1 -9.9999999999999995e-08 -> -9.9999995000000163e-8
    400 expm10102 expm1 3.0000000000000001e-05 -> 3.0000450004500034e-5
    401 expm10103 expm1 -0.0070000000000000001 -> -0.0069755570667648951
    402 expm10104 expm1 -0.071499208740094633 -> -0.069002985744820250
    403 expm10105 expm1 -0.063296004180116799 -> -0.061334416373633009
    404 expm10106 expm1 0.02390954035597756 -> 0.024197665143819942
    405 expm10107 expm1 0.085637352649044901 -> 0.089411184580357767
    406 expm10108 expm1 0.5966174947411006 -> 0.81596588596501485
    407 expm10109 expm1 0.30247206212075139 -> 0.35319987035848677
    408 expm10110 expm1 0.74574727375889516 -> 1.1080161116737459
    409 expm10111 expm1 0.97767512926555711 -> 1.6582689207372185
    410 expm10112 expm1 0.8450154566787712 -> 1.3280137976535897
    411 expm10113 expm1 -0.13979260323125264 -> -0.13046144381396060
    412 expm10114 expm1 -0.52899322039643271 -> -0.41080213643695923
    413 expm10115 expm1 -0.74083261478900631 -> -0.52328317124797097
    414 expm10116 expm1 -0.93847766984546055 -> -0.60877704724085946
    415 expm10117 expm1 10.0 -> 22025.465794806718
    416 expm10118 expm1 27.0 -> 532048240600.79865
    417 expm10119 expm1 123 -> 2.6195173187490626e+53
    418 expm10120 expm1 -12.0 -> -0.99999385578764666
    419 expm10121 expm1 -35.100000000000001 -> -0.99999999999999944
    420 
    421 -- extreme negative values
    422 expm10201 expm1 -37.0 -> -0.99999999999999989
    423 expm10200 expm1 -38.0 -> -1.0
    424 expm10210 expm1 -710.0 -> -1.0
    425 -- the formula expm1(x) = 2 * sinh(x/2) * exp(x/2) doesn't work so
    426 -- well when exp(x/2) is subnormal or underflows to zero; check we're
    427 -- not using it!
    428 expm10211 expm1 -1420.0 -> -1.0
    429 expm10212 expm1 -1450.0 -> -1.0
    430 expm10213 expm1 -1500.0 -> -1.0
    431 expm10214 expm1 -1e50 -> -1.0
    432 expm10215 expm1 -1.79e308 -> -1.0
    433 
    434 -- extreme positive values
    435 expm10300 expm1 300 -> 1.9424263952412558e+130
    436 expm10301 expm1 700 -> 1.0142320547350045e+304
    437 -- the next test (expm10302) is disabled because it causes failure on
    438 -- OS X 10.4/Intel: apparently all values over 709.78 produce an
    439 -- overflow on that platform.  See issue #7575.
    440 -- expm10302 expm1 709.78271289328393 -> 1.7976931346824240e+308
    441 expm10303 expm1 709.78271289348402 -> inf overflow
    442 expm10304 expm1 1000 -> inf overflow
    443 expm10305 expm1 1e50 -> inf overflow
    444 expm10306 expm1 1.79e308 -> inf overflow
    445 
    446 -- weaker version of expm10302
    447 expm10307 expm1 709.5 -> 1.3549863193146328e+308
    448