Document Type : Research Paper

Authors

• Seyed Mohsen Mohammadi ¹
• Reza Etesami ²
• Mohsen Madadi ²

¹ Shahid Bahonar University of Kerman

² Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

The error function, erf(x), is crucial in many fields but lacks a closed-form solution. To improve existing closed-form approximations, this paper introduces a global-optimization framework that refines their numerical coefficients without changing their analytical form. The optimization minimizes a composite objective of mean and maximum absolute error (MAE and Max-AE) over selected domains. We solve this using the Gaussian Combined Arms (GCA) metaheuristic. For 16 structural types, the method often reduces error by an order of magnitude while maintaining formula simplicity and low cost. We also present new, highly accurate approximations with closed-form inverses. The framework is a powerful, transferable tool for enhancing approximations of erf(x) and related functions.

Keywords

• closed-form approximation
• coefficient tuning
• numerical refinement
• inverse error function
• function optimization