A Highly Accurate Upper Bound for the Cumulative Standard Normal Distribution Function and Related Functions

Abstract

Although there is extensive literature concerning upper bounds for the cumulative standard normal distribution function Φ(x), many existing bounds are not uniformly accurate over the entire domain x ≥ 0. This paper proposes a new explicit upper bound for Φ(x) that is uniformly accurate over x ≥ 0 and improves the classical Polya upper bound and several related bounds available in the literature. The proposed bound is constructed by modifying the classical Polya structure through the introduction of a polynomial correction term inside the exponential function, leading to a flexible parametric form. The coefficients of this correction polynomial are determined using a constrained minimax-type optimization framework combined with local accuracy conditions obtained through Taylor-series matching near the origin, ensuring both global tightness and local stability. It is shown that the maximum absolute error between the proposed bound and the exact function Φ(x) does not exceed 5.785 × 10^(−5) for all x ≥ 0. Numerical comparisons and graphical illustrations demonstrate that the proposed bound is consistently tighter than several well-known alternative bounds, particularly in the small and moderate range of x. The structure of the bound also preserves analytical tractability, making it suitable for theoretical analysis and computational applications. Finally, due to the functional relationships between Φ(x), the Q-function and the error function, the proposed construction can be directly extended to derive corresponding bounds for these related special functions.