Upgrade to pynac-0.7.16

[Ralf Stephan]

In pynac-0.7.16:

• Python interface / Py3 fixes and improvements (thx Erik Bray, #24561)
• atan2 fixes (#24604)
• beta fixes
• fix atanh/acoth(real) and polylog (#24299)
• immediate atanh/acoth --> log simplifications (#24658)
• legendre_P (#24553)
• chebyshev_T/U (#16812)
• speedup in special polynomial creation
• remove clang-check error
• remove many clang-tidy warnings
• remove unused code
• added CONTRIBUTING.md

​https://github.com/pynac/pynac/releases/download/pynac-0.7.16/pynac-0.7.16.tar.bz2

[Ralf Stephan]

New commits:

​f5b3de4	24668: version/chzksum
​67f9ddc	24668: doctest fixes

[Travis Scrimshaw]

LGTM. Next step: to the buildbots.

[Ralf Stephan]

I'll rebase #24649 on this because of merge conflict.

[Volker Braun]

Some numerical noise:

File "src/sage/symbolic/expression.pyx", line 8331, in sage.symbolic.expression.Expression.arctan2
Failed example:
    SR(1).arctan2(CDF(0,1))
Expected:
    0.7853981633974484 - 19.012501686914433*I
Got:
    0.7853981633974483 - 19.012518200244195*I
**********************************************************************
1 item had failures:
   1 of  35 in sage.symbolic.expression.Expression.arctan2
    [2807 tests, 1 failure, 32.96 s]

[Jeroen Demeyer]

The imaginary part changed a lot. More than just the usual numerical noise.

[Ralf Stephan]

The reason for the relative big error is in the formula that is applied in libs/pynac/pynac.pyx: return -I*py_log((x + I*y)/py_sqrt(x**2 + y**2)) with x=CDF(I), y=1. The denominator sqrt(x**2 + y**2) would be 0, but CDF(I)^2 is -1.0 + 1.2246467991473532e-16*I and so the sqrt denominator becomes 7.82510958117314e-09 + 7.825109581173138e-09*I.

For any other complex type of I as second argument I get an exception so I'm tempted to mark it as known bug.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2568492

24668: fix doctest

[Ralf Stephan]

Actually that numeric result was changed in this ticket. Before, the result was not evaluated. However this pynac version tries to evaluate atan2 even with complex values that have nonzero imag part.