[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.