The functions erf and erfc computed with arbitrary precision and explicit error bounds

摘要

The error function erf is a special function. It is widely used in statistical computationsrnfor instance, where it is also known as the standard normal cumulative probability. Therncomplementary error function is defined as erfc(x) = 1 - erf(x).rnIn this paper, the computation of erf(x) and erfc(x) in arbitrary precision is detailed: ourrnalgorithms take as input a target precision t' and deliver approximate values of erf(x) orrnerfc(x) with a relative error guaranteed to be bounded by 2~(-t').rnWe study three different algorithms for evaluating erf and erfc. These algorithms arerncompletely detailed. In particular, the determination of the order of truncation, the analysisrnof roundoff errors and the way of choosing the working precision are presented. Thernscheme used for implementing erf and erfc and the proofs are expressed in a generalrnsetting, so they can directly be reused for the implementation of other functions.rnWe have implemented the three algorithms and studied experimentally what is the bestrnalgorithm to use in function of the point x and the target precision t'.