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