Indeed, degree-4 rational polynomials are generally enough for JPEG XL. Code for evaluating them using Horner's scheme (just FMAs): https://github.com/libjxl/libjxl/blob/bdde644b94c125a15e532b...

I initially used Chebfun to approximate them, then a teammate implemented the same Caratheodory-Fejer method in C. We subsequently convert from Chebyshev basis to monomials.

Blog sounds good :)