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

TL;DR

This paper introduces a new, simple, explicitly invertible approximation of the normal CDF that achieves four-decimal accuracy with minimal complexity, improving upon previous formulas for practical applications.

Contribution

The authors present a novel approximation of the normal CDF that is simpler, explicitly invertible, and more accurate than existing formulas, reaching four decimal places of precision.

Findings

Achieves absolute error less than 4.00×10^{-5}

Reduces absolute and relative errors by about 36% and 28% respectively

Maintains simplicity comparable to Winitzki's erf approximation

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.