I have a smooth enough injective function $f:[a, b]\to \mathbb{R}$ which I want to approximate by something that can be computed quickly, e.g., a Padé approximant of low degree, $$ \frac{\sum_{j=0}^m a_j x^j}{1 + \sum_{k=0}^n b_k x^k}. $$ At the same time, though, I also need the inverse of the approximating function to be computed quickly, so Padé will give me a hard time even with low degrees.
Any ideas?