[wallacoloo]

> You can do a little better with a minimax polynomial that you can derive with the Remez algorithm.

Yup! This was something I meant to mention in a footnote. If I could get equivalent [infinite precision] error bounds using a minimax polynomial of a lower degree, I think this would make a difference. Otherwise, even if the theoretical error bounds are lower for a minimax polynomial of the same degree, it wouldn't do anything to combat the noise introduced by rounding errors, assuming you keep everything else the same - and that noise seems to be by far the limiting factor. But that's just conjecture - no way to know for certain without trying it.

I do still like the Chebyshev polynomial approach because it's the only non-iterative approach I know of. It's something that I could implement without the aid of a CAS program pretty trivially, if I wanted to.

> The best approximation I could find calculated sin(x) and cos(x) for [-pi/4,pi/4] and then combined these to cover all other inputs.

Yes, that also seems to be the way to go (at least for precision). I didn't mention it in the article, but even though the code I showed was rust, the actual implementation is in the form of an AST for a plugin system that only supports f32 addition, multiplication, division, and modulo as operations (no branching, f32<->int conversions, etc). My entire motivation was to create a sine function for this system. It's known that the inputs to the function will be in (-pi, pi), so I don't have to perform reduction over any larger range. I don't see a practical way to reduce down to (-pi/4, pi/4) using the operations available to me, but I could be overlooking something.

[qooiii2]

Under those constraints, Chebyshev approximation will probably outperform minimax. Minimax approximations have non-zero error at the endpoints, so they wouldn't quite hit zero at sin(pi) and your relative error would be horrible.

[psi-squared]

If you want exactly zero at the end points, you could do something like the post does, of approximating sin(x) / x(pi+x)(pi-x), or similar. You can still do that with the Remez algorithm.

Also, a while ago I realized that you can tweak the Remez algorithm to minimize relative error (rather than absolute error) for strictly-positive functions - it's not dissimilar to how this blog post does it for Chebyshev polynomials, in fact. I should really write a blog post about it, but it's definitely doable.

So combining those two, you should be able to get a good "relative minimax" approximation for pi, which might be better than the Chebyshev approximation depending on your goals. Of course, you still need to worry about numerical error, and it looks like a lot of the ideas in the original post on how to deal with that would carry over exactly the same.