Approximations for Standard Normal Distribution Function and Its Invertible

TL;DR

This paper presents highly accurate closed-form approximations for the standard normal distribution function and its inverse, outperforming existing methods in terms of error metrics.

Contribution

Introduces new approximations for the normal distribution and its inverse, demonstrating superior accuracy over existing methods.

Findings

Maximum absolute error of 4.43×10^(-10) for the distribution function

Mean absolute error of 9.62×10^(-11) for the distribution function

Outperforms existing approximations in accuracy measures

Abstract

In this paper, we introduce a new approximation of the cumulative distribution function of the standard normal distribution based on Tocher's approximation. Also, we assess the quality of the new approximation using two criteria namely the maximum absolute error and the mean absolute error. The approximation is expressed in closed form and it produces a maximum absolute error of 4.43*10^(-10) while the mean absolute error is 9.62*10^(-11). In addition, we propose an approximation of the inverse cumulative function of the standard normal distribution based on Polya approximation and compare the accuracy of our findings with some of the existing approximations. The results show that our approximations surpass other existing ones based on the aforementioned accuracy measures.