In the present paper, we propose a new approximation of the inverse Langevin function. This new approximation is based on a two-step modification of the fractional formula introduced by (Cohen 1991). Our proposal is motivated by the minimization of the error between the Cohen formula and the inverse of the Langevin function. It results in two additional terms adopting a remarkable simple power and polynomial forms. The correction provides an excellent agreement with a maximum relative error equal to 0.046 % (against a maximum error of 4.94 % for the Cohen formula).