University Of Tehran Press Journal of Algorithms and Computation 2476-2776 57 2 2025 12 31 Optimizing Closed-Form Approximations of the Error Function via the Gaussian Combined Arms Metaheuristic 150 169 106212 10.22059/jac.2026.409129.1251 EN Seyed Mohsen Mohammadi Shahid Bahonar University of Kerman Reza Etesami Shahid Bahonar University of Kerman, Kerman, Iran Mohsen Madadi Shahid Bahonar University of Kerman, Kerman, Iran Journal Article 2025 11 01 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. https://jac.ut.ac.ir/article_106212_ed6901f964d16be2c6a4e7c37cfa4d8e.pdf