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.