Add Julia native loggamma functionality

[TSGut]

In light of discussions at https://discourse.julialang.org/t/ann-meijerg-jl-a-julia-package-for-calculating-meijer-g-functions/136577 and locally in this repo at #27, here is an adaptation of loggamma based on pre-existing as well as my recent work in SpecialFunctions.jl (JuliaMath/SpecialFunctions.jl#522).

loggamma for reals and bigfloat calls external libraries in SpecialFunctions.jl, so I adapted Julia native versions as part of this. The real bigfloat version just reroutes to the Complex BigFloat one for now but if we care that can be changed for some significant speedups in that route in the future.

I don't know where the best place is to discuss where this goes - the issue, Julia discourse, etc. Or maybe this can restart conversations about a renaming of this package to something more general and thus SEO friendly?

Very happy for feedback or comments.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.09%. Comparing base (4ff4d61) to head (c9bedc4).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #128      +/-   ##
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+ Coverage   95.72%   96.09%   +0.36%     
==========================================
  Files          24       25       +1     
  Lines        2385     2585     +200     
==========================================
+ Hits         2283     2484     +201     
+ Misses        102      101       -1     

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[TSGut]

Alright, tests pass, coverage passes. The only thing failing is invalidations which as far as I can tell has nothing to do with my PR, it didn't add invalidations I think there is some version incompatibility in the action which causes an error?

@heltonmc What's the PR process for this package? I would be happy for a review or suggestions.

Also @juliohm and @oscardssmith since they were part of the chat on Julia Discourse.

Things I intend to leave for the future for now:

1. Dedicated real BigFloat branch rather than using the slower Complex{BigFloat} logic.
2. Export loggammas and gammas? Rebrand? Make Gamma.jl elsewhere? Not sure which if any of this is desirable.

[juliohm]

Thank you for leading this @TSGut ! @oscardssmith is more qualified to help with a review.

Hope this initiative will get the attention it deserves from package authors and maintainers in JuliaMath 🙏🏽

[TSGut]

Benchmarking the implementations with the strategy suggested by the readme I get (compared to ArbNumerics.jl):

Function	Type	Metric	Value
loggamma	Float64	Mean relative error	2.602362769721367e-16
loggamma	Float64	Max relative error	3.674838211509268e-14
loggamma	ComplexF64	Mean relative error	1.3655212256834891e-16
loggamma	ComplexF64	Max relative error	4.869646413689727e-16
logabsgamma	Float64	Mean relative error	2.602362769721367e-16
logabsgamma	Float64	Max relative error	3.674838211509268e-14

And execution time is roughly the same as using SpecialFunctions.jl:

Function	Type	Library	Mean (ns)	Median (ns)	Std (ns)	Speedup
loggamma	Float64	Bessels	8.87	8.84	0.17	1.011×
loggamma	Float64	SpecialFunctions	9.00	8.94	0.16	1.000×
loggamma	ComplexF64	Bessels	28.79	28.78	0.14	1.010×
loggamma	ComplexF64	SpecialFunctions	29.08	29.05	0.22	1.000×
logabsgamma	Float64	Bessels	10.49	10.54	0.23	0.832×
logabsgamma	Float64	SpecialFunctions	8.80	8.77	0.30	1.000×

[TSGut]

@oscardssmith: I think this covers everything in your code review -- anything else left to do before this can be merged?

I leave it to you whether you choose to export these functions but the sooner this can be tagged the sooner I can get started on making the hypergeometric functions be Julia native. 😄 I have had a chat with the maintainer of HypergeometricFunctions.jl and he's happy for me to basically copy and adjust his code as functions within MeijerG.jl, so MeijerG.jl will just contain its own native hypergeometric function logic and probably also export a Julia native pFq, with its only dependency becoming Bessels.jl.

[oscardssmith]

The code looks good to me, but I do think or should probably live in a new Gamma.jl. there are a ton more people that likely will want a good gamma function (e.g. Primes.jl) that probably don't want to pull in all the bessel functions as well.

[TSGut]

I am happy to do that, can you make a Gamma.jl in JuliaMath (we should check that no such named package is currently registered)? While we're at it could I be added to JuliaMath - would make this easier going forward.

I'll move this code into a PR for the new Gamma.jl then.

Also, if this will create a new Julia native Special functions ecosystem then we probably want to have references pointing at the other relevant packages in that ecosystem from the readmes or docs.

[oscardssmith]

https://github.com/JuliaMath/Gamma.jl. If you make the PR, I'll work on getting this to be an actual Julia repo.

[TSGut]

Great, I will close this PR since we are moving development there.