Summary

1. Removes the log gamma q and Q eval structs and instead uses std::pair
2. Decreases precision default to sqrt(machine.error)
3. Rewrote parts of log_q_gamma_cf to remove some state variables for in loop variables
4. Used auto in functions where there are unrestricted template types. It was unclear to me if log_gamma_q_dgamma should ever take in autodiff variables. If so then the auto makes sense to me. But if they are only ever taking in or working with doubles then we should remove the templates entirely
5. Uses std::optional<std::pair> to replace the bool ok{false}; in Q_eval.
6. Replace *_dbl with *_val in functions with template parameters. If these functions accept autodiff types then *_dbl would imply the wrong type.
7. Functions that can return back std::nullopt (empty std::optional) now early exit when the function can return std::nullopt

Checklist

• Copyright holder: Simons Foundation

  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