n-th Root Algorithm

Question

What is the fastest way to calculate the n-th root of a number?

I'm aware of the Try and Fail method, but I need a faster algorithm.

Answer 1

The canonical way to do this is Newton's Method. In case you don't know, the derivative of xⁿ is nx^n-1. This will come in handy. 1 is a good first guess. You want to apply it to the function a - xⁿ

IIRC, it's superconvergent on functions of the form a - xⁿ, but either way, it's quite fast. Also, IIRC, the warning in the wiki about it failing to converge would apply to more complex functions that have properties that the 'nice' functions you are interested in lack.