Practical Explicitly Invertible Approximation to 4 Decimals of Normal Cumulative Distribution Function Modifying Winitzki's Approximation of erf

Abstract

We give a new explicitly invertible approximation of the normal cumulative distribution function: $\Phi (x)\simeq 1/2+1/2\sqrt{1-e^{-x^2\frac{17+x^2}{26.694+2x^2}}}$, $\forall x\ge 0$, with absolute error $<4.00\cdot 10^{-5}$, absolute value of the relative error $<4.53\cdot 10^{-5}$, which, beeing designed essentially for practical use, is much simpler than a previously published formula and, though less precise, still reaches 4 decimals of precision, and has a complexity essentially comparable with that of the approximation of the normal cumulative distribution function $\Phi (x)$ immediatly derived from Winitzki's approximation of erf$(x)$, reducing about 36% the absolute error and about 28% the relative error with respect to that, overcoming the threshold of 4 decimals of precision.

4 pages, 5 figures