  Let $\Psi(x,y)$ count the number of positive integers $n\le x$ such that
every prime divisor of $n$ is at most $y$. Given inputs $x$ and $y$, what is
the best way to estimate $\Psi(x,y)$? We address this problem in three ways:
with a new algorithm to estimate $\Psi(x,y)$, with a performance improvement to
an established algorithm, and with empirically based advice on how to choose an
algorithm to estimate $\Psi$ for the given inputs.
  Our new algorithm to estimate $\Psi(x,y)$ is based on Ennola's second theorem
[Ennola69], which applies when $y< (\log x)^{3/4-\epsilon}$ for $\epsilon>0$.
It takes $O(y^2/\log y)$ arithmetic operations of precomputation and $O(y\log
y)$ operations per evaluation of $\Psi$.
  We show how to speed up Algorithm HT, which is based on the saddle-point
method of Hildebrand and Tenenbaum [1986], by a factor proportional to
$\log\log x$, by applying Newton's method in a new way.
  And finally we give our empirical advice based on five algorithms to compute
estimates for $\Psi(x,y)$.The challenge here is that the boundaries of the
ranges of applicability, as given in theorems, often include unknown constants
or small values of $\epsilon>0$, for example, that cannot be programmed
directly.
