  In this work, we propose an extensive numerical study on approximating the
absolute value function. The methods presented in this paper compute
approximants in the form of rational functions and have been proposed
relatively recently, e.g., the Loewner framework and the AAA algorithm.
Moreover, these methods are based on data, i.e., measurements of the original
function (hence data-driven nature). We compare numerical results for these two
methods with those of a recently-proposed best rational approximation method
(the minimax algorithm) and with various classical bounds from the previous
century. Finally, we propose an iterative extension of the Loewner framework
that can be used to increase the approximation quality of the rational
approximant.
