Implementation of the log of the lower incomplete gamma function

[JacobHass8]

See #1346 and #1173.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 95.29%. Comparing base (1d670c8) to head (9abdc80).

Additional details and impacted files

@@           Coverage Diff            @@
##           develop    #1349   +/-   ##
========================================
  Coverage    95.29%   95.29%           
========================================
  Files          814      814           
  Lines        67422    67422           
========================================
  Hits         64249    64249           
  Misses        3173     3173           

Files with missing lines	Coverage Δ
include/boost/math/special_functions/gamma.hpp	99.85% <ø> (ø)

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[jzmaddock]

I think your added series is the same as the existing:

math/include/boost/math/special_functions/gamma.hpp

Line 425 in 1d670c8

BOOST_MATH_GPU_ENABLED inline T lower_gamma_series(T a, T z, const Policy& pol, T init_value = 0)

[JacobHass8]

I think your added series is the same as the existing:

math/include/boost/math/special_functions/gamma.hpp

Line 425 in 1d670c8

BOOST_MATH_GPU_ENABLED inline T lower_gamma_series(T a, T z, const Policy& pol, T init_value = 0)

You're absolutely right. Based on the name of the function, I assumed it was a continued fraction implementation. This makes things a lot easier!

[JacobHass8]

I also changed the criteria for when the asymptotic series is used based on code paths in previous functions that used the asymptotic series. With my limited spot testing, this gives accurate results and is ready for testing.

[JacobHass8]

@jzmaddock I think this PR for the lower incomplete gamma function is nearly complete. I'm having some trouble testing long double types on ARM64 architectures where most of the tests are failing. The difference between my computed value and the expected is of order $10^{-18}$ whereas the tolerance is $10^{-30}$. The difference isn't small enough to increase the tolerance.

I'm now thinking the expected values are not quite right. I haven't been able to get Mathematica to output arbitrarily large precision so I've been using the mpmath library for Python. Do you happen to have the Mathematica code you used to calculate values for the upper incomplete gamma function?

[ckormanyos]

I'll bet the platform you are describing uses 10-byte floating-point representations for long double. This is about the precision range where 80-bit long double reaches. But your control values expect the full 128-bits. You can express your tolerances in terms of multiplies of std::numeric_limits<T>::epsilon() and these will actually scale to T, assumed to be the floating-point type in the instantiation of the test(s).