Fast computation of incomplete elliptic integral of first kind by half argument transformation

We developed a new method to calculate the incomplete elliptic integral of the first kind, F(j|m){F(\varphi|m)} , by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of j{\varphi} by repeated usage of the formulas while fixing m. The method is sufficiently precise in the sense that the maximum relative error is 3–5 machine epsilons at most. Thanks to the simplicity of the half argument formulas, the new procedure is significantly faster than the existing procedures. For example, it runs 20–60% faster than Bulirsch’ function, el1, and 1.9–2.2 times faster than the method using Carlson’s function, R _F .