Sometimes you need a fairly high degree polynomial to get reasonable accuracy, but can achieve a far greater accuracy with a much lower degree using rational functions.

Theorem: An irreducible function \(R^{\ast}=P/Q\) is the minimax rational approximation to \(f\) on \([a,b]\) among the rational functions belonging to \(\Rn_{n,m}\) if and only if there exist at least \(k=2+\max\{m+\degree(P),n+\degree(Q)\}\) values \(a\le x_{0}<\cdots<x_{k-1}\le b\) such that:

There is an analogue of Remez’s algorithm that will yield the minimax rational approximation. However, the problem is ill posed, and the accuracy of the coefficients can be poor. Yet, the result of the algorithm is still usually a very good approximation.

Pade approximations are like Taylor approximations - they yield good
*local* behavior, but are poor in general.

In general, it is hard to determine whether a rational approximation will work better than a polynomial one. Another factor to take into account is that even if the degree is lower in a rational approximation, division is expensive, and can offset some or all of that gain.