How would Chomsky categorise this? (Update: Corrected c&p typo per Erez post below.)

#! perl -slw
use strict;

$|++;

my %ops = (
    '+' => sub{ $_[ 0 ] + $_[ 1 ] },
    '-' => sub{ $_[ 0 ] - $_[ 1 ] },
    '*' => sub{ $_[ 0 ] * $_[ 1 ] },
    '/' => sub{ $_[ 0 ] / $_[ 1 ] },
    '**'=> sub{ $_[ 0 ] ** $_[ 1 ] },
);

my @presedence = (
    qr[\*\*],
    qr[\*|/],
    qr[\+|-],
);

my $reVar = qr[[a-z]+];
my $reConst = qr[ [+-]? (?:\d+\.)? \d+ (?: [eE] [+-]? \d+ )? ]x;
my $reArg = qr[$reVar|$reConst];
my $reOps = qr[@{[ join '|', map{ quotemeta } keys %ops ]}];
my $reTokenise = qr[\s*($reArg)(?:\s*($reOps))?];

sub parseEvalExpr {
    my $expr = shift;

    if( $expr =~ m[$reOps \s+ $reArg \s+ $reOps]x ) {
        for my $opset ( @presedence ) {
            return "($expr)" 
                if $expr =~ s[ ( $reArg \s+ $opset \s+ $reArg ) ]{($1)
+}x;
        }
    }
    my @tokens = $expr =~ m[$reTokenise]g;
    pop @tokens unless defined $tokens[ $#tokens ];
    while( @tokens > 1 ) {
        ( my( $arg1, $op, $arg2 ), @tokens ) = @tokens;
        unshift @tokens, $ops{ $op }->( $arg1, $arg2 );
    }
    return $tokens[ 0 ];
}

while( <DATA> ) {
    chomp;
    my $testResult = eval;
    printf "'$_' = ";
    warn "Unbalanced parens '$_'" and next unless tr[(][] == tr[)][];
    while( m[[()\s]] ) {
        s[ \( ( [^()]+ ) \) ]{ parseEvalExpr( $1 ) }xe while m[[()]];
        $_ = parseEvalExpr( $_ );
    }
    print;
    printf STDERR "*** Discrepancy! Eval gets: %s\n", $testResult 
        unless $_ eq $testResult;
}

__DATA__
1 + 2
2 - 1
2 * 1
1 / 2
(((7 + 5) * (9 + 13)) / ((4 + 3) * (17 - 2 + 3)))
23 ** 2
1.1e10 ** -10
1 + 2 * 3
3 + 2 ** 2 ** 2 * 3
[download]

Output

c:\test>expr_parser.pl
'1 + 2' = 3
'2 - 1' = 1
'2 * 1' = 2
'1 / 2' = 0.5
'(((7 + 5) * (9 + 13)) / ((4 + 3) * (17 - 2 + 3)))' = 2.0952380952381
'23 ** 2' = 529
'1.1e10 ** -10' = 3.85543289429532e-101
'1 + 2 * 3' = 7
'3 + 2 ** 2 ** 2 * 3' = 51
[download]