Make constants constexpr (where possible)

[andrjohns]

Summary

This PR changes the declaration of constants to use static constexpr where supported.

static constexpr was used over plain constexpr as it compiles to more efficient assembly. Examples provided on godbolt below:

• constexpr: https://godbolt.org/z/9q9cx38z6
• static constexpr: https://godbolt.org/z/1hds8EKKe
• const double: https://godbolt.org/z/MvWj73eY7

Additionally the Euler's Gamma constant was added after a request on the forum

Tests

Added test for egamma, and existing tests for constants all pass

Side Effects

N/A

Release notes

Changed constants to static constexpr for efficiency, added Euler's Gamma constant

Checklist

• Math issue #(issue number)

• Copyright holder: Andrew Johnson

  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

[SteveBronder]

Interesting, it looks like this is failing the sir model in the cmdstan upstream performance tests

https://jenkins.mc-stan.org/job/CmdStan%20Performance%20Tests/job/downstream_tests/2577/consoleFull

FAIL: golds/stat_comp_benchmarks_benchmarks_sir_sir.gold param gamma not within (0.200873675 - 0.199968898) / 0.00232163707201 < 0.3
'stat_comp_benchmarks/benchmarks/sir/sir.stan' had fails '[('gamma', 0.199968898, 0.002321637072015, 0.20087367499999986)]' and errors '[]'

[andrjohns]

@SteveBronder Oddly enough I can't reproduce the performance test failures locally, either on Linux or Windows. Would you mind trying it locally at some point to see if you can reproduce?

[SteveBronder]

Ack sorry I've been so slow on this been making a really steady amount of progress on the new matrix type in the compiler. I can run this on the mac we use for testing tomorrow. My guess is that whatever old clang version is on that mac doesn't follow something in the standard on infinities. But let me run this tmrw before we change anything

[SteveBronder]

Actually I can't login to the mac because I had to wipe my hard drive recently because of a computer crash and think I lost the ssh I needed to get in. I tried replicating this locally to no avail so I think this must be something with the mac

Let me email Ben G and see if I can get access to that mac. The MCSE difference is wide enough that I think it's worth looking into.

[andrjohns]

No wokkas at all, thanks for looking into it! This is much lower priority than the varmat stuff, so no pressure from my end

[SteveBronder]

@serban-nicusor-toptal it looks like after the Jenkins update the little drop down arrows do not work anymore, is it possible to fix that? (Ftr can still see the full log if I go to the Jenkins site but it's nice to have the dropdown)

[serban-nicusor-toptal]

@SteveBronder fixed! I didn't see that one, thanks!

PS: If the mac you want to get into is the one we're using on Jenkins I can help you regain your ssh access, just send me an email.

[SteveBronder]

@serban-nicusor-toptal much appreciated!

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.52	3.52	1.0	0.2% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	1.0	-0.46% slower
eight_schools/eight_schools.stan	0.09	0.09	1.03	2.86% faster
gp_regr/gp_regr.stan	0.14	0.14	0.97	-3.47% slower
irt_2pl/irt_2pl.stan	5.1	5.83	0.88	-14.22% slower
performance.compilation	91.82	90.51	1.01	1.42% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.31	8.15	1.02	2.04% faster
pkpd/one_comp_mm_elim_abs.stan	30.49	31.01	0.98	-1.71% slower
sir/sir.stan	120.98	116.99	1.03	3.3% faster
gp_regr/gen_gp_data.stan	0.03	0.03	0.94	-6.21% slower
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.99	3.0	1.0	-0.36% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.37	0.4	0.94	-6.9% slower
arK/arK.stan	2.79	2.79	1.0	-0.21% slower
arma/arma.stan	0.27	0.25	1.09	8.36% faster
garch/garch.stan	0.65	0.72	0.9	-10.8% slower
Mean result: 0.985763226119

---

Jenkins Console Log
Blue Ocean
Commit hash: f31f43d

---

Machine information

ProductName: Mac OS X ProductVersion: 10.11.6 BuildVersion: 15G22010

CPU:
Intel(R) Xeon(R) CPU E5-1680 v2 @ 3.00GHz

G++:
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr --with-gxx-include-dir=/usr/include/c++/4.2.1
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

Clang:
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

[andrjohns]

@SteveBronder looks like the macOS issues were just temporary, all seems to have completed without issue (phew!)

[syclik]

Awesome. Thanks!