{"paper":{"title":"Simply Explicitly Invertible Approximations to 4 Decimals of Error Function and Normal Cumulative Distribution Function","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"stat.CO","authors_text":"A. Soranzo, E. Epure","submitted_at":"2012-01-05T22:43:16Z","abstract_excerpt":"We improve the Modified Winitzki's Approximation of the error function $erf(x)\\cong \\sqrt{1-e^{-x^2\\frac{\\frac{4}{\\pi}+0.147x^2}{1+0.147x^2}}}$ which has error $|\\varepsilon (x)| < 1.25 \\cdot 10^{-4}$ $\\forall x \\ge 0$ till reaching 4 decimals of precision with $|\\varepsilon (x)| < 2.27 \\cdot 10^{-5}$; 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 $\\Phi (x)$, of erfc$(x)$ and of the $Q$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1320","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}