Optimize Cauchy sampling

[MaximoB]

Adding an implementation to a very fast algorithm for calculating Cauchy random numbers.
My basic testing of plotting some histograms seems to indicate it is about as numerically stable as the current method we are using.
Not sure about how to do paper credits so I am looking for feedback about that too.

[vks]

Could you please provide some before/after benchmarks?

Not sure about how to do paper credits so I am looking for feedback about that too.

For consistency, I would just go with how it was done for the normal distribution.

[dhardy]

1. How accurate is this?
2. Yes, benchmarks please
3. Also include the DOI please

[MaximoB]

1. The paper is not exactly clear on this, but from what I can tell the maximum error in their approximation of the quartile function looks like ±0.002
2. Benchmarks against the current code (current code pulled this morning)

test distr_cauchy_current        ... bench:      28,211 ns/iter (+/- 15,429) = 283 MB/s
test distr_cauchy_new            ... bench:      11,631 ns/iter (+/- 8,514) = 687 MB/s

3. DOI is: 10.1145/50087.50094. Are you saying you would like me to add it to the credit in the documentation?

[dhardy]

Yes, I found the article (but didn't manage to download it yet). But I think it's worth adding to the credits.

Sounds IMO like we should treat this in line with #494; keep the current implementation for now but consider adding this as a fast approximation under distributions::fast or something (still under discussion).

[dhardy]

@vks any suggestion on what we do with this?

[vks]

I would prefer to only maintain one implementation. Can we quantify whether the accuracy of the new implementation is better or worse?

[dhardy]

Agreed. Then we (or someone) needs to look into the accuracy.

[dhardy]

Closing (but referenced from a tracker).

Ideally we should implement #357 before revisiting this.