  The inverse Langevin function is a fundamental part of the statistical chain
models used to describe the behavior of polymeric-like materials, appearing
also in other fields such as magnetism, molecular dynamics and even
biomechanics. In the last four years, several approximants of the inverse
Langevin function have been proposed. In most of them, optimization techniques
are used to reduce the relative error of previously published approximants to
reach orders of magnitude of O(10^-2, 10^-3). In this paper a new simple and
efficient numerical approach to calculate the inverse Langevin function is
proposed. Its main feature is the reduction of the relative errors in all the
domain x=[0,1) to near machine precision, maintaining function evaluation CPU
times similar to those of the most efficient approximants. The method consists
in the discretization of the Langevin function, the calculation of the inverse
of these discretization points and their interpolation by cubic splines. In
order to reproduce the asymptotic behavior of the inverse Langevin function, a
rational function is considered only in the asymptotic zone keeping C^1
continuity with the cubic splines. We include customizable Matlab codes to
create the spline coefficients, to evaluate the function, and to compare
accuracy and efficiency with other published proposals.
