To explain further: use bigint
only causes numeric literals to become bigints. It does not change the meaning of operators (like -), no matter what the perldoc might lead you to believe. 0 is a literal, and becomes a bigint. $a and $b are strings, and do not become bigints unless you add 0 to them. Just saying "$a-$b" does not trigger bigint conversion.
Edit: Here's a quick demonstration:
> perl -le 'use bigint; print 22/7; print "22"/"7"'
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