The formula of maximal possible rank of commutator subgroups of finite $p$-groups

All groups in the abstract are finite. We define rank $d(G)$ of a $p$-group $G$ as the minimal number of generators of $G$. In this paper, we obtain a compact formula for the strict upper bound of the ranks of commutator subgroups of finite $p$-groups generated by elements of given orders. This bound was described in a recent article of the author. But the corresponding formula was very complicated although containing some useful information. The new formula is much more simple and clear.