First, it doesn't seem to work correctly.    I tried with the expression (+ 1 2 3 (* 4 3 (- 20 1) (% 5 2)) 3 4) and got the answer 2221 when it should be 241.

sub solvepart($);
...
sub reformat;
...
sub reformat($){
...
sub solvepart($){
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The declaration for reformat doesn't have a prototype but the definition does even though reformat does not accept any arguments    YOU SHOULD NOT USE PROTOTYPES.

    if ($given ~~ /(?:exit)|(?:end)|(?:quit)/i){
...
    if ($VERBOSE or $VERBOSE ~~ /v(?:erbose)?/i){
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Is there any good reason to use the smartmatch operator (~~) instead of the binding operator (=~)?    /(?:exit)|(?:end)|(?:quit)/i could be written more simply as /exit|end|quit/i.

my $num=qr/(-?\d++\.?\d*?)/;
...
    unshift(@stack, [$2,$3,$4]) while $given =~ s/(.*)\((\S++) ++(-?\d
+++\.?\d*?) ++(-?\d++\.?\d*?)?\)/$1/;
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Why define $num if you are not going to use it?

        if ($level[0]eq'+'){
            $level[2]
                ?($value=$level[1]+$level[2])
                :($value+=$level[1]);
        }
        elsif ($level[0]eq'-'){
                        $level[2]
                                ?($value=$level[1]-$level[2])
                                :($value=$level[1]-$value);
                }
        elsif ($level[0]eq'/'){
                        $level[2]
                                ?($value=$level[1]/$level[2])
                                :($value=$level[1]/$value);
                }
        elsif ($level[0]eq'*'){
                        $level[2]
                                ?($value=$level[1]*$level[2])
                                :($value*=$level[1]);
                }
        elsif ($level[0]eq'**'or$level[0]eq'^'){
                        $level[2]
                                ?($value=$level[1]**$level[2])
                                :($value=$level[1]**$value);
                }
        elsif ($level[0]eq'%'){
                        $level[2]
                                ?($value=$level[1] % $level[2])
                                :($value=$level[1] % $value);
                }
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Could be written more simply as:

        if ($level[0]eq'+'){
            $value = $level[1] + $level[2] || $value;
        }
        elsif ($level[0]eq'-'){
                        $value = $level[1] - $level[2] || $value;
                }
        elsif ($level[0]eq'/'){
                        $value = $level[1] / $level[2] || $value;
                }
        elsif ($level[0]eq'*'){
                        $value = $level[1] * $level[2] || $value;
                }
        elsif ($level[0]eq'**'or$level[0]eq'^'){
                        $value = $level[1] ** $level[2] || $value;
                }
        elsif ($level[0]eq'%'){
                        $value = $level[1] % $level[2] || $value;
                }
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Update: this is how I would probably do it:

#!/usr/bin/perl
use warnings;
use strict;

use Scalar::Util 'looks_like_number';

my $MAXITERATIONS = 500;

while ( <> ) {
    my $iterations = 0;
    1 while s/(\([^()]+\))/ die "MAX ITERATIONS ACHIEVED: ABORTING\n" 
+if ++$iterations >= $MAXITERATIONS; solvepart( $1 ) /eg;
    print "\nANSWER: $_\n\n";
    }

sub solvepart {
    my $expression = shift;
    $expression =~ tr/()//d;
    $expression =~ s/\^/**/;
    my ( $operator, @values ) = split ' ', $expression;
    $operator =~ /\*\*|[+\-*\/%]/ && @values >= 2 && @values == grep l
+ooks_like_number( $_ ), @values
        or die "IMPROPER INPUT\n";
    eval join $operator, @values;
    }
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