Accurately Computing log(1− exp(− |a|)) Assessed by the Rmpfr package

In this note, we explain how f(a) = log(1− e−a) = log(1− exp(−a)) can be computed accurately, in a simple and optimal manner, building on the two related auxiliary functions log1p(x) (= log(1 + x)) and expm1(x) (= exp(x)− 1 = ex − 1). The cutoff, a0, in use in R since 2004, is shown to be optimal both theoretically and empirically, using Rmpfr high precision arithmetic. As an aside, we also show how to compute log (