Many ways of doing this, for instance, this is rather inelegant but shows the logic clearly:

my $count = 0;
while (<>) {
    if (/formulae/) {
        $count++;
        }
    if ($count == 2) {
        print "Found 2nd\n";
        }
    }
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Or you can just change your do-loop a bit:

$count = 0;
do {$count++; $_ = <> } until ((/formulae/) && ($count == 2))
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