Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Sigmoid functions for the smooth approximation to $\vert{x}\vert$

Version 1 : Received: 11 March 2019 / Approved: 13 March 2019 / Online: 13 March 2019 (09:12:38 CET)

A peer-reviewed article of this Preprint also exists.

Journal reference: Moroccan Journal of pure and applied analysis 2020, 7, 12-19
DOI: 10.2478/mjpaa-2021-0002

## Abstract

We present smooth approximations to $\vert{x} \vert$ using sigmoid functions. In particular $x\,erf(x/\mu)$ is proved to be better smooth approximation for $\vert{x} \vert$ than $x\,tanh(x/\mu)$ and $\sqrt{x^2 + \mu}$ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.

## Keywords

Sigmoid functions; error function; smooth approximation; Hyperbolic tangent

## Subject

MATHEMATICS & COMPUTER SCIENCE, Analysis

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