Automated Discovery of Numerical Approximation Formulae via Genetic Programming

This paper describes the use of genetic programming to perform automated discovery of numerical approximation formulae. We present results involving rediscovery of known approximations for Harmonic numbers, discovery of rational polynomial approximations for functions of one or more variables, and refinement of existing approximations through both approximation of their error function and incorporation of the approximation as a program tree in the initial GP population. Evolved rational polynomial approximations are compared to Padé approximations obtained through the Maple symbolic mathematics package. We find that approximations evolved by GP can be superior to Padé approximations given certain tradeoffs between approximation cost and accuracy, and that GP is able to evolve approximations in circumstances where the Padé approximation technique cannot be applied. We conclude that genetic programming is a powerful and effective approach that complements but does not replace existing techniques from numerical analysis.