I love these things! Approximation is art.

I have a similar example in my book Geometry for Programmers (https://www.manning.com/books/geometry-for-programmers). There I make a polynomial approximation for sine that is: 1) odd (meaning antisymmetric); 2) precise at 0 and Pi/2 (and -Pi/2 automatically); 3) have precise derivatives and 0 and Pi/2 (and -Pi/2); 4) have better overall precision than the equivalent approximation with Taylor series; and it only consists of 4 members! Adding more members makes it impractical in the modern world since we have hardware implementations for sine on CPUs and GPUs.

Spoiler alert, the secret sauce is the integral equation ;-)