The absolute value function $x \mapsto |x|$ is frequently used as a loss function in regression methods in place of the square function $x \mapsto x^2$ because it is less sensitive to outliers. Unfortunately it is not differentiable, so for applying e.g. Newton methods it is often approximated by a "smoothed version"
$$ x \mapsto \sqrt{x^2 + \epsilon }$$
for some small $\epsilon >0$.
Does this approximation have a name? Is there a known person who first introduced it?