Prime Density: A Closed-Form Logistic Approximation to the Normal Cumulative Distribution Function

Prime Density: A Closed-Form Logistic Approximation to the Normal Cumulative Distribution Function | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 31 March 2025 V1 Latest version Share on Prime Density: A Closed-Form Logistic Approximation to the Normal Cumulative Distribution Function Authors : Michael Froelich 0000-0001-5779-1331 [email protected] and Michael A Frölich Authors Info & Affiliations https://doi.org/10.22541/au.174345555.51224749/v1 241 views 109 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We introduce the Prime Density function, a novel logistic-cubic closed-form approximation to the standard normal cumulative distribution function (CDF). Existing approximations either lack analytic simplicity and invertibility or compromise accuracy, particularly in distribution tails. To overcome these limitations, the Prime Density function employs a logistic sigmoid function with a cubic polynomial argument. Parameters were rigorously optimized through a hybrid global-local procedure combining Differential Evolution and Nelder-Mead methods, minimizing maximum absolute error and root-mean-square deviation across the real line. Our optimized approximation achieves a maximum absolute error below 1.7 × 10 −4, surpassing classical logistic and rivaling complex rational approximations. The function maintains analytical invertibility, differentiability, and symbolic simplicity, providing distinct computational advantages for real-time analytics, symbolic computation, and resource-limited hardware. Empirical evaluations across diverse datasets-including environmental pollutant indices (PM2.5 AQI), financial returns (S&P 500), and biomedical markers (glucose and triglycerides)-demonstrate the Prime Density function's superior empirical flexibility and precision compared to traditional approximations. The results position the Prime Density function as a practical, rigorously validated, and computationally efficient alternative, effectively bridging analytic simplicity and high-performance demands. Supplementary Material File (outline.pdf) Download 474.62 KB Information & Authors Information Version history V1 Version 1 31 March 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords closed-form inverse cumulative distribution function logistic approximation normal distribution numerical approximation symbolic computation Authors Affiliations Michael Froelich 0000-0001-5779-1331 [email protected] View all articles by this author Michael A Frölich Department of Anesthesiology & Perioperative Medicine, University of Alabama at Birmingham View all articles by this author Metrics & Citations Metrics Article Usage 241 views 109 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Michael Froelich, Michael A Frölich. Prime Density: A Closed-Form Logistic Approximation to the Normal Cumulative Distribution Function. Authorea . 31 March 2025. DOI: https://doi.org/10.22541/au.174345555.51224749/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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