Add expm1(::Float16)

[oscardssmith]

Not sure if this is a regression, but it's definitely bad. Also, fixes a problems I found in expm1(::Float32) with very small inputs.

[jarlebring]

A bit off-topic: Why is this file mixing so much between using evalpoly and exthorner? Isn't horner to prefer over standard poly evaluation for higher degrees? Also, isn't it better to use poly evaluations with less multiplications, such as Paterson-Stockmayer? Or newer poly evaluations methods such as the one by Sastre: https://doi.org/10.1016/j.laa.2017.11.010

[oscardssmith]

evalpoly uses horner's method. exthorner is horner with extended precision by using error free transforms. The link you gave is for evaluating matrix polynomials which is very different from scalar polynomials. There are theoretically more efficient methods for polynomials than horner's method (estirn's for one), but they are only worth-while for polynomials of relatively high degree (12 or more roughly). For small polynomials, you can't do anything more efficient than the series of fma instructions that evalpoly produces.

[jarlebring]

Ok. Thanks.

You have a point that the paper concerns matrix polynomials. I agree that it is more complicated at this low-level, but I thought a rule-of-thumb of computation cost for scalar is the same: Multiplication is much more expensive than addition, and division (which is not present in the paper I referenced) is even more expensive.

For small polynomials, you can't do anything more efficient than the series of fma instructions that evalpoly produces.

Okay. I can accept this directly assuming "small polynomials" means the degree is max (say) 4 or 5. The file has polys with degrees 8 (or maybe higher). Horner eval of poly of degree 8 requires 8 multiplications. With 8 multiplications we can obtain a poly of degree 25 using Paterson-Stockmayer, see figure 1 in http://eprints.maths.manchester.ac.uk/2676/1/fasi18.pdf

I don't understand why a Paterson-Stockmayer approach would be less favorable for fma's.

The estrin scheme also seems to require quite a few multiplications.

[jarlebring]

Ping @mfasi who is an expert on Paterson-Stockmayer. What do you think about the use of PS for scalar polynomial evaluation?

[jarlebring]

Okay. On second thought maybe my comment is less useful for this than I first thought. You have to count also coefficient * argument multiplication as a regular multiplication and the PS is not as competitive.

[jarlebring]

It seems it is possible to get less fma instructions with a PS approach for a polynomial of degree 8. I am happy to discuss it further in a better place: https://discourse.julialang.org/t/fmas-and-poly-eval/61421

[oscardssmith]

Can I get review on this? This fixes a fairly major bug (expm(small_x)), and adds expm(::Float16) which definitely should exist.

[musm]

In general, looks fine to me. I did not verify coefficients, but I trust they're good presumably generated via same method (Remez.jl).

[oscardssmith]

What's buildkite? I haven't seen it before.

[oscardssmith]

Merging in a day or 2 sans objections.