  Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ having rank
$l$ and let $V=L(\lambda)$ be an irreducible finite-dimensional
$\mathfrak{g}$-module having highest weight $\lambda.$ Computations of weight
multiplicities in $V,$ usually based on Freudenthal's formula, are in general
difficult to carry out in large ranks or for $\lambda$ with large coefficients
(in terms of the fundamental weights). In this paper, we first show that in
some situations, these coefficients can be "lowered" in order to simplify the
calculations. We then investigate how this can be used to improve the
aforementioned formula of Freudenthal, leading to a more efficient version of
the latter in terms of complexity as well as a way of dealing with certain
computations in unbounded ranks. We conclude by illustrating the last assertion
with a concrete example.
