Sigmoid functions for the smooth approximation to the absolute value function

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









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.