Good approximations for Gaussian error function - Wang, Pengzhao (Nanyang Technological University, 2023)
The Gaussian error function is a non-fundamental function that is commonly used in probability theory and statistics. In this paper, we present a new method for approximating the error function. The expression consists of two parts: one part is a function that decreases at a rate similar to the error function, while the other part is a polynomial function with several parameters that assists the approximation in approaching the exact error function. Two criteria, absolute error and relative error, are utilized to analyze the error of the approximation, as they have distinct impacts. Absolute error aims to minimize the error at the start of the interval because the error function declines exponentially, causing the error at the right end of the interval to be much larger. Conversely, all errors throughout the argument region have the same importance when computing the relative error. As a result, approximations based on relative error exhibit more exceptional performance as the variable's magnitude increases. By considering these two types of errors, we can merge them together, with the first half utilizing an approximation determined by absolute error and the second half using relative error. Moreover, the number of terms in the polynomial function also affects the performance of the approximations. After comparing the errors, we select the third-order function. The novel approximation we presented surpasses previous approximations in terms of performance. The results of the calculations indicate that this new method has a tenfold reduction in error and requires only half the computation time compared to Karagiannidis' approximation. Consequently, the proposed approximation in this paper represents a remarkable advancement.