Precise and fast computation of inverse Fermi–Dirac integral of order 1/2 by minimax rational function approximation

The single and double precision procedures are developed for the inverse Fermi–Dirac integral of order 1/2 by the minimax rational function approximation. The maximum error of the new approximations is one and 7 machine epsilons in the single and double precision computations, respectively. Meanwhile, the CPU time of the new approximations is so small as to be comparable to that of elementary functions. As a result, the new double precision approximation achieves the 15 digit accuracy and runs 30–84% faster than Antia’s 28 bit precision approximation (Antia, 1993). Also, the new single precision approximation is of the 24 bit accuracy and runs 10–86% faster than Antia’s 15 bit precision approximation.