A fast computation of density of exponentially $S$-numbers

The author \cite{4} proved that, for every set $S$ of positive integers containing 1 (finite or infinite) there exists the density $h=h(E(S))$ of the set $E(S)$ of numbers whose prime factorizations contain exponents only from $S,$ and gave an explicit formula for $h(E(S))\mathrm{.}$ In this paper we give an equivalent polynomial formula for $\log h(E(S))$ which allows to get a fast calculation of $h(E(S))\mathrm{.}$