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sub is_balanced {
   my $string = shift;
...
   return 0 if $@;
   return 1;
}

my $balanced   = "(1 + 2 * (3 + (5/4) + 8) + 6)";
my $unbalanced1 = "(((1 + 2 * (3 + (5/4) + 8) + 6))";
...
printf("Is $balanced balanced? %s\n", balanced_parens($balanced) ? "ye
+s" : "no");
printf("Is $unbalanced1 balanced? %s\n", balanced_parens($unbalanced1)
+ ? "yes" : "no");
printf("Is $unbalanced2 balanced? %s\n", balanced_parens($unbalanced2)
+ ? "yes" : "no");

Is (1 + 2 * (3 + (5/4) + 8) + 6) balanced? yes
Is (((1 + 2 * (3 + (5/4) + 8) + 6)) balanced? no
Is (((1 + 2 * (3 + (5/4) + 8) + 6)))) balanced? no