Algorithm 1062: Code Generation for Piecewise Chebyshev Approximation

A Lipschitz continuous function \( f \) that has a high-precision, albeit slow reference implementation can be evaluated more rapidly by piecewise polynomials, obtained by Chebyshev approximation. The partition into subdomains should be made such that the computation of subdomain index and reduced function argument is fast and introduces no rounding errors. A rigorous analysis of truncation and rounding errors shows that the maximum relative error can be chosen arbitrarily close to \(2\epsilon\) , except around zeroes of \( f \) . Alignment of coefficients minimizes cache loading times. Code generation and target algorithms are implemented in the open source project ppapp , maintained at https://jugit.fz-juelich.de/mlz/ppapp , with a snapshot archived in the Collected Algorithms of the ACM. It has been successfully applied to three real-valued functions in the open source complex error function library libcerf .