  An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with
all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of
semismooth integers are utilized in modern integer factoring algorithms, such
as the number field sieve, that incorporate the so-called large prime variant.
Thus, it is useful for factoring practitioners to be able to estimate the value
of $\Psi(x,y,z)$, the number of $(y,z)$-semismooth integers up to $x$, so that
they can better set algorithm parameters and minimize running times, which
could be weeks or months on a cluster supercomputer. In this paper, we explore
several algorithms to approximate $\Psi(x,y,z)$ using a generalization of
Buchstab's identity with numeric integration.
