New highly accurate improvements for single-term approximations of the standard normal distribution function

Abstract

This paper proposed new, highly accurate, single-term, and explicitly invertible approximations for the standard normal distribution function and its related functions, such as the error function and the quantile function. The proposed approximations are built based on some existing approximations, however, the proposed ones are much more accurate. The accuracy of the proposed approximations is measured via maximum absolute error and mean absolute error. Some of the proposed approximations are at least five times more accurate than the original ones and two of them have maximum absolute error lower than 1.8×10^-4, which is quite sufficient for most of real-world applications. Two real applications are studied to show the applicability of the proposed improvements. These applications showed the superiority of one of the proposed approximations over some of the available single-term approximations even though the latter have smaller maximum absolute error.