A New Accurate Simply Explicitly Invertible Upper Bound for the Q-Function

This paper introduces a new explicitly invertible upper bound for the Gaussian Q-function, optimized for high accuracy in the significant interval [0.45, 4.5], commonly encountered in telecommunications applications. Building on previous research that surveyed and developed explicitly invertible approximations of the Q-function, this work focuses on minimizing the absolute error while maintaining low complexity and explicit invertibility. The proposed bound is a modification of the previously published Q_Soranzo-1 approximation and achieves an absolute error of less than 1.6·10⁻⁵ and a relative error below 2.2·10⁻² in the specified domain. Unlike several earlier bounds requiring the non-elementary Lambert W-function, the new expression is invertible using only elementary functions. This characteristic makes it particularly suitable for practical applications in communications theory where closed-form inversion is critical. Comparative analysis and graphical evaluations demonstrate that this upper bound outperforms all known explicitly invertible bounds in both absolute and relative accuracy within the domain of interest.