Simply Explicitly Invertible Approximations to 4 Decimals of Error Function and Normal Cumulative Distribution Function

2012/01/05 by Alessandro Soranzo, A. Soranzo, Soranzo, A. +3
Computer Science · Mathematics · #33B20 #33F05 #65D20 #97N50 #Computation (stat.CO) #FOS: Computer and information sciences #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms #msc:33B20 #msc:33F05 #msc:65D20 #msc:97N50 #stat.CO

paper · pdf · doi:10.48550/arxiv.1201.1320

3 pages, 1 figure, 1 table

arxiv created 2012/01/05 · openalex publication_date 2012/01/05 · arxiv updated 2012/01/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We improve the Modified Winitzki's Approximation of the error function erf(x)≅ √1-e^-x²\frac\frac4π+0.147x²1+0.147x² which has error |ε (x)| < 1.25 ⋅ 10^-4 ∀ x ≥ 0 till reaching 4 decimals of precision with |ε (x)| < 2.27 ⋅ 10^-5; also reducing slightly the relative error. Old formula and ours are both explicitly invertible, essentially solving a biquadratic equation, after obvious substitutions. Then we derive approximations to 4 decimals of normal cumulative distribution function Φ(x), of erfc(x) and of the Q function (or cPhi).

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