> but there's a reason that all approximations are done using series of polynomials (taylor expansion).

"All" is a tall claim. Have a look at https://perso.ens-lyon.fr/jean-michel.muller/FP5.pdf for example. Jump to slide 18:

> Forget about Taylor series

> Taylor series are local best approximations: they cannot compete on a whole interval.

There is no need to worry about "sh-tt-ng" on their result when there is so much to learn about other approximation techniques.