Besides BrowserUk suggestion (that's probably the way to go), some simple modifications on logfact (inlining things and eliminating unnecessary temporaries) makes it twice as fast:

sub hypergeom1 {
    # There are m "bad" and n "good" balls in an urn.
    # Pick N of them. The probability of i or more successful selectio
+ns:
    # (m!n!N!(m+n-N)!)/(i!(n-i)!(m+i-N)!(N-i)!(m+n)!)
    my ($n, $m, $N, $i) = @_;

    my $loghyp1 = logfact1($m) +logfact1($n)+logfact1($N)+logfact1($m+
+$n-$N);
    my $loghyp2 = logfact1($i)+logfact1($n-$i)+logfact1($m+$i-$N)+logf
+act1($N-$i)+logfact1($m+$n);
    return exp($loghyp1 - $loghyp2);
}

sub logfact1 {
    my $x = shift;
    my $ser = (   1.000000000190015
                + 76.18009172947146   / ($x + 2)
                - 86.50532032941677   / ($x + 3)
                + 24.01409824083091   / ($x + 4)
                - 1.231739572450155   / ($x + 5)
                + 0.12086509738661e-2 / ($x + 6)
                - 0.5395239384953e-5  / ($x + 7) );
    my $tmp = $x + 6.5;
    ($x + 1.5) * log($tmp) - $tmp + log(2.5066282746310005 * $ser / ($
+x+1));
}
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