Practical Explicitly Invertible Approximation to 4 Decimals of Normal Cumulative Distribution Function Modifying Winitzki's Approximation of erf

2012/11/27 by Alessandro Soranzo, Soranzo, Alessandro, Emanuela Epure +1
Decision Sciences · Mathematics · #Probabilistic and Robust Engineering Design #Statistical Distribution Estimation and Applications #math.ST #msc:33B20 #msc:33F05 #msc:65D20 #msc:97N50 #stat.TH

paper · pdf · doi:10.48550/arxiv.1211.6403

4 pages, 5 figures

arxiv created 2012/11/27 · arxiv updated 2012/11/28

Abstract

We give a new explicitly invertible approximation of the normal cumulative distribution function: Φ(x) ≃ 1/2 + 1/2 √1-e^-x²\frac17+x²26.694+2x², ∀ x ≥ 0, with absolute error <4.00⋅ 10^-5, absolute value of the relative error <4.53⋅ 10^-5, which, beeing designed essentially for practical use, is much simpler than a previously published formula and, though less precise, still reaches 4 decimals of precision, and has a complexity essentially comparable with that of the approximation of the normal cumulative distribution function Φ(x) immediatly derived from Winitzki's approximation of erf(x), reducing about 36% the absolute error and about 28% the relative error with respect to that, overcoming the threshold of 4 decimals of precision.

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