super stable gamma_lccdf

[spinkney]

Summary

This pr closes #3257. The gamma_lccdf experiences numerical instability for alpha approximately greater than 30. It’s due to gamma_q or grad_reg_inc_gamma. This pr rewrites the code to use gamma_p and grad_reg_lower_inc_gamma, which avoids potential instability in a tgamma(alpha) and digamma(alpha) call. It seems to sample a bit faster than just wrapping gamma_lcdf with log1m_exp.

This supercedes and improves upon pr #3264.

Tests

I've implemented new tests in the problematic regime.

Release notes

The gamma_lccdf is made more robust for alpha values. The issue was overflow in the derivatives when alpha was greater than 30.

Checklist

• Copyright holder: Publicis Groupe
  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[syclik]

A few things to change. I'll try to update the branch accordingly; it's nothing about the math itself.