This is an easy mistake to make. I've made it myself. Notice that in $bit, you have parentheses '()' and a plus '+'. What your regex is doing is trying to find one or more fours followed by one five. It then, if successful, would capture this to $1.

The easiest way to deal with the is to use the "quote" metacharacters (\Q and \E) in the regex.

$formula = $bit = '(4+5)'; 
if( $formula =~ /\Q$bit\E/ ) { 
    print "yeah\n"; 
} else { 
    print "Oh no!\n"; 
}
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You cannot include a literal $ or @ within a \Q sequence. An unescaped $ or @ interpolates the corresponding variable, while escaping will cause the literal string \$ to be inserted. You'll need to write something like m/\Quser\E\@\Qhost/.

You could also use quotemeta(), which is how the \Q is implemented. Ie.,

$formula = '(4+5)';
$bit = quotemeta $formula;
if ( $formula =~ /$bit/ ) {
  print "yeah\n"; 
} else { 
  print "Oh no!\n"; 
}