Currently the math module uses own implementation of some mathematical functions that are in C99 standard, but not in C89 standard: tgamma, lgamma, erf, erfc. Proposed patch makes it to use functions from standard C library if they are available. They are faster and presumably more accurate.

Here are microbenchmark results (time in microseconds):

                  0.1     1      3     10     30
erf unpatched:   0.506  0.655  0.509  0.548  0.239
erf patched:     0.129  0.252  0.357  0.253  0.253

erfc unpatched:  0.508  0.646  0.532  0.522  0.251
erfc patched:    0.129  0.239  0.373  0.371  0.307

                    0.1    1.5     3     10     10.5
gamma unpatched:   0.369  0.279  0.273  0.274  0.457
gamma patched:     0.24   0.23   0.412  0.741  0.682

lgamma unpatched:  0.351  0.338  0.478  0.627  0.52
lgamma patched:    0.217  0.155  0.37   0.372  0.247

If some libm implementations are pretty bad, they can be disabled by undefining corresponding HAVE_XXX macros.

Faster: maybe. More accurate ... Mark can tell you some funny stories. :)

LGTM. What platform are the benchmarks on?

Is there some way that this can be submitted to the buildbots *before* applying to the 3.6 branch, so that we can get a sense of how widespread (if at all) libm problems are?

The benchmarks were ran on 32-bit Linux with AMD processor.

I'll try to run custom build on buildbots.

I don't know how to run custom build on buildbots with new workflow.

So the GitHub branch fails on my OS X 10.10 machine: there are a number of test failures for gamma and lgamma (but none for erf and erfc).

Some of the gamma and lgamma test failures are accuracy issues; some are more likely to do with errno handling: OS X probably isn't setting errno appropriate on domain or range errors, so any implementation using the libm lgamma/tgamma would need to add code for error handling.

Given this, I'd prefer to leave lgamma and gamma as-is, using their current implementations, at least on OS X; I don't regard the accuracy loss in using the libm functions to be within acceptable limits.

There may still be a case for using the libm versions of erf and erfc.

Tests are failed only on s390x RHEL 3.x.

http://buildbot.python.org/all/builders/s390x%20RHEL%203.x/builds/446/steps/test/logs/stdio
======================================================================
FAIL: test_mtestfile (test.test_math.MathTests)
----------------------------------------------------------------------
Traceback (most recent call last):
  File "/home/dje/cpython-buildarea/3.x.edelsohn-rhel-z/build/Lib/test/test_math.py", line 1287, in test_mtestfile
    '\n  '.join(failures))
AssertionError: Failures in test_mtestfile:
  gam0003: gamma(-inf): expected 'ValueError', got nan (not equal)
  gam0045: gamma(1e-160): expected 1e+160, got 1.0000000000000063e+160 (error = 6.24e+145 (40 ulps); permitted error = 0 or 20 ulps)
  gam0046: gamma(1e-308): expected 1e+308, got 1.0000000000000136e+308 (error = 1.36e+294 (68 ulps); permitted error = 0 or 20 ulps)
  gam0047: gamma(5.6e-309): expected 1.7857142857142848e+308, got 1.785714285714199e+308 (error = 8.58e+294 (430 ulps); permitted error = 0 or 20 ulps)
  gam0065: gamma(-1e-160): expected -1e+160, got -1.0000000000000063e+160 (error = 6.24e+145 (40 ulps); permitted error = 0 or 20 ulps)
  gam0066: gamma(-1e-308): expected -1e+308, got -1.0000000000000136e+308 (error = 1.36e+294 (68 ulps); permitted error = 0 or 20 ulps)
  gam0067: gamma(-5.6e-309): expected -1.7857142857142848e+308, got -1.785714285714199e+308 (error = 8.58e+294 (430 ulps); permitted error = 0 or 20 ulps)
  gam0081: gamma(-1.0000000000000002): expected 4503599627370495.5, got 4503599627370484.5 (error = 11 (22 ulps); permitted error = 0 or 20 ulps)
  gam0084: gamma(-100.00000000000001): expected -7.540083334883109e-145, got -7.540083334882688e-145 (error = 4.21e-158 (296 ulps); permitted error = 0 or 20 ulps)
  gam0085: gamma(-99.99999999999999): expected 7.540083334884096e-145, got 7.540083334884402e-145 (error = 3.06e-158 (215 ulps); permitted error = 0 or 20 ulps)
  gam0100: gamma(170.0): expected 4.269068009004705e+304, got 4.269068009005011e+304 (error = 3.06e+291 (628 ulps); permitted error = 0 or 20 ulps)
  gam0101: gamma(171.0): expected 7.257415615307999e+306, got 7.257415615308882e+306 (error = 8.83e+293 (708 ulps); permitted error = 0 or 20 ulps)
  gam0102: gamma(171.624): expected 1.7942117599248104e+308, got 1.794211759924979e+308 (error = 1.69e+295 (845 ulps); permitted error = 0 or 20 ulps)
  gam0120: gamma(-100.5): expected -3.3536908198076787e-159, got -3.353690819807666e-159 (error = 1.26e-173 (25 ulps); permitted error = 0 or 20 ulps)
  gam0121: gamma(-160.5): expected -5.255546447007829e-286, got -5.255546447008121e-286 (error = 2.92e-299 (313 ulps); permitted error = 0 or 20 ulps)
  gam0122: gamma(-170.5): expected -3.3127395215386074e-308, got -3.312739521538679e-308 (error = 7.16e-322 (145 ulps); permitted error = 0 or 20 ulps)
  gam0140: gamma(-63.349078729022985): expected 4.177797167776188e-88, got 4.177797167776136e-88 (error = 5.19e-102 (93 ulps); permitted error = 0 or 20 ulps)
  gam0141: gamma(-127.45117632943295): expected 1.183111089623681e-214, got 1.183111089623727e-214 (error = 4.6e-228 (223 ulps); permitted error = 0 or 20 ulps)

----------------------------------------------------------------------

PR 632 switch on using libc implementation on Windows. Will see how AppVeyor tests passed.

There are accuracy issues with tgamma() and lgamma() on Windows. But less than on OS X.

======================================================================
FAIL: test_mtestfile (test.test_math.MathTests)
----------------------------------------------------------------------
Traceback (most recent call last):
  File "C:\projects\cpython\lib\test\test_math.py", line 1287, in test_mtestfile
    '\n  '.join(failures))
AssertionError: Failures in test_mtestfile:
  lgam0085: lgamma(-99.99999999999999): expected -331.85460524980596, got -331.86198661322277 (error = 0.00738 (129854318490 ulps); permitted error = 1e-15 or 5 ulps)
  gam0085: gamma(-99.99999999999999): expected 7.540083334884096e-145, got 7.484632144060724e-145 (error = 5.55e-147 (38979707393612 ulps); permitted error = 0 or 20 ulps)
  gam0124: gamma(-176.5): expected -1.196e-321, got 0.0 (error = 1.2e-321 (243 ulps); permitted error = 0 or 20 ulps)
----------------------------------------------------------------------

Does this mean that we should not use libc implementations of tgamma() and lgamma() on Windows?

Please can we revert to using the existing implementations of lgamma and gamma on all platforms? It's clear that there are many issues with OS-provided implementations.

PR 637 revert to using the own implementations of lgamma and gamma on all platforms.

Done.

New changeset 4125e5c60e24ffcff8031817dc60984335917f59 by Serhiy Storchaka in branch 'master':
bpo-26121: Revert to using the own implementations of lgamma and gamma on all platforms. (#637)
https://github.com/python/cpython/commit/4125e5c60e24ffcff8031817dc60984335917f59

New changeset 4dadcd4ed7824c7904add78577e6a05864cfe493 by Serhiy Storchaka in branch 'master':
bpo-26121: Use C library implementation for math functions erf() and erfc() on Windows. (#632)
https://github.com/python/cpython/commit/4dadcd4ed7824c7904add78577e6a05864cfe493

New changeset 97553fdf9daa8231eb05a1ca9933a2b03b0bdad0 by Mark Dickinson (Serhiy Storchaka) in branch 'master':
bpo-26121: Use C library implementation for math functions: (#515)
https://github.com/python/cpython/commit/97553fdf9daa8231eb05a1ca9933a2b03b0bdad0