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

The error function erf is a special function. It is widely used in statistical computations for instance, where it is also known as the standard normal cumulative probability. The complementary error function is defined as erfc(x)=1−erf(x)erfc(x)=1−erf(x).In this paper, the computation of erf(x)erf(x) and erfc(x)erfc(x) in arbitrary precision is detailed: our algorithms take as input a target precision t′t′ and deliver approximate values of erf(x)erf(x) or erfc(x)erfc(x) with a relative error guaranteed to be bounded by 2−t′2−t′.We study three different algorithms for evaluating erf and erfc. These algorithms are completely detailed. In particular, the determination of the order of truncation, the analysis of roundoff errors and the way of choosing the working precision are presented. The scheme used for implementing erf and erfc and the proofs are expressed in a general setting, so they can directly be reused for the implementation of other functions.We have implemented the three algorithms and studied experimentally what is the best algorithm to use in function of the point x   and the target precision t′t′.