New Accurate Approximations for the Normal Distribution Function with Application in Engineering

Abstract

This article proposes seven accurate closed-form approximations for the standard normal distribution function and related functions, including the complementary error function, the error function, and the quantile function. The approximations refine Hart’s (1957) [24] approximation by fitting an exponential polynomial correction through nonlinear least squares, and their accuracy is assessed by the maximum absolute error (Max.AE) and the mean absolute error (Mean.AE) over z ∈ [0, 7]. The most accurate proposed approximation attains a maximum absolute error of 8.57 × 10−9; the family is up to 13 times more accurate than the approximations it refines, and among recent methods its most accurate member is surpassed only by a logistic approximation carrying a polynomial of degree 16. We further establish the monotonicity of the approximations, analyse their tail behaviour and computational cost, and benchmark their run time, finding them faster than the built-in normal-CDF routine. The applicability of the approximations is demonstrated on two problems in science and engineering, namely evaluation of the incomplete gamma function and bit-error-rate analysis in digital communications, in both cases with quantitative error assessment.