Cin function

[seadra]

(Disclaimer: first time contributor to a Julia project.)

Added the Cin function, which is the other cosine integral function given by the integral of (1-cost)/t from 0 to x.
Approximation obtained using Mathematica's EconomizedRationalApproximation. Maximum error is 1.1e-10 in the range [0,16] for the input.

I named it cosint2. maybe cin is a better name?

[codecov]

Codecov Report

Merging #397 (017be91) into master (ce58a13) will increase coverage by 1.62%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #397      +/-   ##
==========================================
+ Coverage   91.99%   93.61%   +1.62%     
==========================================
  Files          12       12              
  Lines        2809     2773      -36     
==========================================
+ Hits         2584     2596      +12     
+ Misses        225      177      -48     

Flag	Coverage Δ
unittests	93.61% <100.00%> (+1.62%)	⬆️

Flags with carried forward coverage won't be shown. Click here to find out more.

Impacted Files	Coverage Δ
src/SpecialFunctions.jl	100.00% <ø> (ø)
src/chainrules.jl	100.00% <ø> (ø)
src/sincosint.jl	100.00% <100.00%> (+34.54%)	⬆️
src/beta_inc.jl	92.36% <0.00%> (-0.24%)	⬇️
src/gamma.jl	95.14% <0.00%> (+2.13%)	⬆️
src/gamma_inc.jl	93.46% <0.00%> (+3.45%)	⬆️

---

Continue to review full report at Codecov.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update ce58a13...017be91. Read the comment docs.

[stevengj]

Maximum error is 1.1e-10 in the range [0,16] for the input.

This seems a little large. I would usually aim for a relative error < 1e-13 everywhere.

[seadra]

What are the expectations on relative error when the function value is 0?

[seadra]

I updated the function such that it relies on the existing cosint(x) everywhere except for the region with small |x| in which relative and absolute errors are better than about 1e-15 in my checks

[seadra]

Also, possibly for a different PR but cosint for negative values currently throws domain error, but it can be defined in terms of Cin(x): https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral

[stevengj]

What are the expectations on relative error when the function value is 0?

When the function value is exactly zero, you want to also return 0. And where the function value is near zero, you still want low relative error. Often this means switching to a Taylor expansion in the vicinity of a root.

[seadra]

Often this means switching to a Taylor expansion in the vicinity of a root.

OK, that's what I did exactly. I assume the PR should be fine then.

One might want to tune the y < 0.5 line (the threshold value for |x| beyond which it falls back to existing cosint and log functions), although around |x| = 0.5, the rel and abs errors are still well below the threshold you mentioned.