Preprint Article Version 1 This version is not peer-reviewed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)

How to cite: Bagul, Y.J.; Chesneau, C. Sigmoid functions for the smooth approximation to $\vert{x}\vert$. Preprints 2019, 2019030140 (doi: 10.20944/preprints201903.0140.v1). Bagul, Y.J.; Chesneau, C. Sigmoid functions for the smooth approximation to $\vert{x}\vert$. Preprints 2019, 2019030140 (doi: 10.20944/preprints201903.0140.v1).

## 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.

## Subject Areas

Sigmoid functions; error function; smooth approximation; Hyperbolic tangent