Mike Day shows a novel attack on the FISR made famous by Quake 3 Arena, offering an optimal solution for all rational powers.

The critical turn is one that doesn't really show up elsewhere: define z = x^a * y^b, which equals 1 exactly when y = x^{-a/b}. Since z = x * y^2 and y_exact^2 = 1/x, we have y / y_exact = sqrt(z). Relative error in y is now expressed in terms of z, and can be resolved with minimax polynomials, provided we know the extrema of z(x).

This means you can define this analytically (which he does, pp. 4-9). It also means you can compare different kinds of polynomial solutions (p. 5, pp. 15-16) for the FRSR, the recirprocal, cube roots, etc.

Octave code for a sketch of the proof here: https://gist.github.com/Protonk/f96a42d1c540c97446afee4309f1... All errors are my own, please read Day's pp. 4-9 for the actual proof.