Reuse intermediate computations in distributions part 1

[mcol]

Summary

This is the first of a few PRs that aim to clean up the distributions so that we make better use of intermediate computations and composed functions.

This covers the following distributions:

• bernoulli
• bernoulli_logit_glm
• beta
• beta_binomial
• beta_proportion
• binomial
• double_exponential
• exponential
• exp_mod_normal

Tests

None, this is just cleanup.

Side Effects

None.

Checklist

• Math issue: Make use of composed functions and temporaries #1230

• Copyright holder: Marco Colombo

  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	4.83	5.06	0.95	-4.74% slower
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.99	-1.02% slower
eight_schools/eight_schools.stan	0.09	0.09	1.0	0.02% faster
gp_regr/gp_regr.stan	0.22	0.22	0.99	-1.34% slower
irt_2pl/irt_2pl.stan	6.1	6.24	0.98	-2.3% slower
performance.compilation	89.65	86.97	1.03	2.98% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	7.49	7.79	0.96	-3.93% slower
pkpd/one_comp_mm_elim_abs.stan	20.77	20.87	1.0	-0.46% slower
sir/sir.stan	89.13	91.07	0.98	-2.18% slower
gp_regr/gen_gp_data.stan	0.05	0.05	0.98	-1.84% slower
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.97	3.08	0.96	-3.83% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.32	0.32	1.03	2.5% faster
arK/arK.stan	1.72	1.74	0.99	-1.06% slower
arma/arma.stan	0.76	0.77	1.0	-0.41% slower
garch/garch.stan	0.63	0.63	1.0	0.43% faster
Mean result: 0.989115739917

---

Jenkins Console Log
Blue Ocean
Commit hash: 594a56e

---

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

[mcol]

@SteveBronder Is this something you'd like to tackle?

[rok-cesnovar]

If Steve is busy, I can tackle this one tomorrow.

[SteveBronder]

Idt I'll have time today but I will have time tmrw!

Though looking at this I think @bbbales2 or other people better at stats stuff would be a better candidate to review this

[bbbales2]

I'll do this tomorrow. Thanks for ping.

Edit: Also careful with compliments. I'm superstitious on these pull reqs. I don't want a stats ghost to come haunt this and challenge my lowly skillz.

[mcol]

Not much stat going on, only reshuffling of terms. When diffs are nice it's quite straightforward... but that's not often the case! :)

[bbbales2]

Looks good!