A. Soranzo, E. Epure. Simply Explicitly Invertible Approximations to 4 Decimals of Error Function and Normal Cumulative Distribution Function

Natural Sciences / Mathematics / Statistics

Submitted on: Jul 14, 2012, 11:09:40

Description: We improve the Modified Winitzki's Approximation of the error function $erf(x)cong sqrt{1-e^{-x^2frac{frac{4}{pi}+0.147x^2}{1+0.147x^2}}}$ which has error $|varepsilon (x)| < 1.25 cdot 10^{-4}$ $forall x ge 0$ till reaching 4 decimals of precision with $|varepsilon (x)| < 2.27 cdot 10^{-5}$; also reducing slightly the relative error. Old formula and ours are both explicitly invertible, essentially solving a biquadratic equation, after obvious substitutions. Then we derive approximations to 4 decimals of normal cumulative distribution function $Phi (x)$, of erfc$(x)$ and of the $Q$ function (or cPhi).