Re: [PATCH] aarch64: Improve SVE sin polynomial

[Wilco Dijkstra <Wilco.Dijkstra@arm.com>]

Hi Paul,

> you can still improve the polynomial with the following one generated by
> Sollya (https://www.sollya.org/) with relative error < 2^-52.454 on
> [-pi/2, pi/2], instead of 2^-51.765 for the one you propose:

So the goal is to reduce the ULP error. Due to the way floating point works,
errors are at their worst when the result is just above a power of 2. For the
wide sin(x) polynomial it happens when sin(x) is close to but larger than 0.5.
The poly is extremely accurate for small x. It's only problematic when x is large
and we cross an exponent boundary.

Szabolcs can explain it better but what we ended up doing is to scale the
sollya function so that errors at the end of the range are and reduced more.

Note also this is not the final polynomial. The ULP is a bit high at 3.3. I had a
more accurate one close to 2ULP but it could give results >1.0. So we need to
teach sollya to estimate the end from below, and not allow it to overshoot.

Cheers,
Wilco