($second_beg>=($first_beg-5)) && ($first_beg<=($second_beg+5))

This is the same condition written in two different ways. I think you want something like this:

use List::Util qw(min max);
...;
my $overlap = min($first_end, $second_end) - max($first_beg, $second_b
+eg) + 1;
if($overlap >= 5) { ... };
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