Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation

Piecewise minimax rational function approximations with the single and double precision accuracies are developed for (i) K(m)K(m) and E(m)E(m), the complete elliptic integral of the first and second kind, respectively, and (ii) B(m)≡(E(m)−(1−m)K(m))/mB(m)≡(E(m)−(1−m)K(m))/m and D(m)≡(K(m)−E(m))/mD(m)≡(K(m)−E(m))/m, two associate complete elliptic integrals of the second kind. The maximum relative error is one and 5 machine epsilons in the single and double precision computations, respectively. The new approximations run faster than the exponential function. When compared with the previous methods (Fukushima, 2009; Fukushima, 2011), which have been the fastest among the existing double precision procedures, the new method requires around a half of the memory and runs 1.7–2.2 times faster.