Add 4 new OpenCL device functions

[t4c1]

Summary

Adds new device functions: lgamma_stirling, lgamma_stirling_diff, lbeta and binomial_coefficient_log. All these are based on existing C++ implementations in Stan Math.

Tests

New device functions are tested.

Side Effects

None.

Release notes

Added 4 new OpenCL device functions: lgamma_stirling, lgamma_stirling_diff, lbeta and binomial_coefficient_log.

Checklist

• Math issue Add OpenCL distributions #2152

• Copyright holder: Tadej Ciglarič

  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

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.11	3.09	1.01	0.65% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.91	-9.7% slower
eight_schools/eight_schools.stan	0.12	0.12	1.0	0.36% faster
gp_regr/gp_regr.stan	0.17	0.17	1.0	-0.17% slower
irt_2pl/irt_2pl.stan	5.73	5.68	1.01	0.89% faster
performance.compilation	90.9	88.2	1.03	2.97% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.47	8.46	1.0	0.03% faster
pkpd/one_comp_mm_elim_abs.stan	28.64	30.24	0.95	-5.61% slower
sir/sir.stan	131.99	127.22	1.04	3.62% faster
gp_regr/gen_gp_data.stan	0.05	0.04	1.04	3.74% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.96	2.98	0.99	-0.67% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.39	0.38	1.0	0.09% faster
arK/arK.stan	1.82	1.8	1.01	1.35% faster
arma/arma.stan	0.7	0.7	0.99	-1.09% slower
garch/garch.stan	0.68	0.67	1.0	0.25% faster
Mean result: 0.998858454267

---

Jenkins Console Log
Blue Ocean
Commit hash: b3c0307

---

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

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.13	3.1	1.01	0.75% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.98	-2.21% slower
eight_schools/eight_schools.stan	0.12	0.12	1.01	0.69% faster
gp_regr/gp_regr.stan	0.17	0.17	0.99	-1.21% slower
irt_2pl/irt_2pl.stan	5.69	5.62	1.01	1.3% faster
performance.compilation	91.19	88.26	1.03	3.22% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.49	8.46	1.0	0.32% faster
pkpd/one_comp_mm_elim_abs.stan	29.57	29.67	1.0	-0.32% slower
sir/sir.stan	143.95	144.3	1.0	-0.25% slower
gp_regr/gen_gp_data.stan	0.05	0.04	1.02	2.39% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.98	2.97	1.0	0.31% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.38	0.39	0.97	-3.52% slower
arK/arK.stan	1.79	1.81	0.99	-1.4% slower
arma/arma.stan	0.61	0.6	1.01	1.18% faster
garch/garch.stan	0.75	0.75	0.99	-0.72% slower
Mean result: 1.00062750827

---

Jenkins Console Log
Blue Ocean
Commit hash: 8f92188

---

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

[rok-cesnovar]

Need to uncomment Jenkinsfile changes and then this is good to go.

[rok-cesnovar]

This needs to be uncommented back.

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.04	3.07	0.99	-0.95% slower
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.95	-4.9% slower
eight_schools/eight_schools.stan	0.12	0.11	1.02	2.27% faster
gp_regr/gp_regr.stan	0.17	0.17	1.0	0.48% faster
irt_2pl/irt_2pl.stan	5.69	5.76	0.99	-1.19% slower
performance.compilation	87.75	85.17	1.03	2.95% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.47	8.5	1.0	-0.32% slower
pkpd/one_comp_mm_elim_abs.stan	30.25	29.66	1.02	1.95% faster
sir/sir.stan	135.86	137.53	0.99	-1.23% slower
gp_regr/gen_gp_data.stan	0.04	0.04	1.0	0.14% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.97	2.98	1.0	-0.19% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.37	0.38	0.98	-1.86% slower
arK/arK.stan	1.77	1.76	1.0	0.15% faster
arma/arma.stan	0.6	0.61	0.99	-0.51% slower
garch/garch.stan	0.74	0.75	0.98	-1.53% slower
Mean result: 0.997181407867

---

Jenkins Console Log
Blue Ocean
Commit hash: 7b4a83b

---

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