New Mathematical Approximations for the Cumulative Normal Distribution Function

Abstract

The normal cumulative distribution function and its derivatives, such as the error function, the Q-function, and the Mills ratio, are widely used in engineering, mathematics, statistics, computer science, diffusion theory, communication theory, physics, and chemistry. However, their non-closed form nature has led to the development of new approximations with varying levels of accuracy and complexity. These new approximations are often more accurate; nevertheless, they can also be more complex, which may limit their practical utility. In this article, a new approach for approximating is proposed. which combines Taylor series expansion and logistic function to create an initial approximation, to enhance the accuracy of the initial approximation, the Hunter-Prey Optimization algorithm is utilized to minimize both the maximum absolute error and the mean absolute error, leading to a significantly more precise approximation. Furthermore, this algorithm is employed to enhance the accuracy of other existing approximations introduced by researchers. The results showed that the improved approximations have much higher accuracy. To show the effectiveness of the new findings of this article, two case studies with applications are presented.