Approximating the number of integers without large prime factors

$\Psi(x,y)$ denotes the number of positive integers $\leq x$ and free of prime factors $>y$. Hildebrand and Tenenbaum gave a smooth approximation formula for $\Psi(x,y)$ in the range $(\log x)^{1+\epsilon}< y \leq x$, where $\epsilon$ is a fixed positive number $\leq 1/2$. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate $\Psi(x,y)$. The computational complexity of this algorithm is $O(\sqrt{(\log x)(\log y)})$. We give numerical results which show that this algorithm provides accurate estimates for $\Psi(x,y)$ and is faster than conventional methods such as algorithms exploiting Dickman's function.