Sigmoid functions for the smooth approximation to the absolute value function

Bagul, Yogesh J.; Chesneau, Christophe

doi:10.2478/mjpaa-2021-0002

Abstract

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

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Sigmoid functions for the smooth approximation to the absolute value function

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