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A Sharp Polya-Based Approximation to the Normal CDF

We introduce a closed form approximation to the cumulative distribution function of the standard normal variable involving only five explicit constants with an approximation error of 5.79 x 10^{-6} across the entire range of real numbers. With its simple form and applicability for all real numbers, our approximation surpasses either in computational efficiency or in accuracy, and most often in both, other approximation formulas based on numerical algorithms or ad-hoc approximations. An extensive overview and classification of the existing approximations from the literature is included.