Taylor expansion is usually not so good for this, you'll fare better with either Legendre or Chebyshev polynomials. Robin Green has some excellent material on the subject, which I am linking below.

Faster Math Functions: https://basesandframes.files.wordpress.com/2016/05/fast-math... https://basesandframes.files.wordpress.com/2016/05/fast-math...

Even faster math functions GDC 2020: https://gdcvault.com/play/1026734/Math-in-Game-Development-S...