Computers internally use moderate-degree taylor approximations of most analytic functions, integration is really simple on polynomial series, and series reversion is a well-defined problem. Therefore, I wonder if it would be reasonable to---say at some compile time step---figure out the high order terms of the taylor expansion of P^{-1} and use this to be able to do inversion sampling on analytic functions very quickly.
Computers internally use moderate-degree taylor approximations of most analytic functions, integration is really simple on polynomial series, and series reversion is a well-defined problem. Therefore, I wonder if it would be reasonable to---say at some compile time step---figure out the high order terms of the taylor expansion of P^{-1} and use this to be able to do inversion sampling on analytic functions very quickly.