Does this look right for the value in your example?

$n = unpack 'N', pack 'B32', '01000010001111000000000000000000';;

( $s, $c, $e ) = ( 
    $n & 0x8000_0000, 
    (( $n >> 24 ) & 0x7f) - 64, 
    (($n & 0x00ffffff) / 0xffffff) * 16 
);;

print $s, $c, $e;;
0 2 3.75000022351743

$num = ($s? -1 : 1 ) * $e * 10**$c;;

print $num;;
[download]

375.000022351743

If so, then the following modification of your subroutine may be what you need:

sub parse_E ($) {
    my $data = shift; ## The data comes in as a 4-byte string so...
    $data = unpack 'N', $data; ## We need to treat it as an unsigned i
+nteger
    ## in order to do bitwise math on it

    my ($sign, $characteristic, $fraction);
    $sign = ($data & 0x80000000) ? -1 : 1;
    $characteristic = (($data >> 24) & 0x7f) - 64;
    $fraction = (($data & 0x00ffffff) / 0xffffff) * 16;

## Not the fraction raised to the power of the characteristic
##    my $num = $sign * $fraction ** $characteristic;  

## But rather, the fraction * 10 to the power of the characteristic
    my $num = $sign * $fraction * 10 ** $characteristic; 

    printf("DEBUG: parse_E(%32s)\n\tsign: %d charisteristic: %d ".
        "fraction: %f = %f\n",
        unpack ("B32", $data), $sign, $characteristic, 
        $fraction, $num);
    printf("DEBUG: unpacked characteristic  %s\n", 
       unpack ("B7", ($data >> 24) & 0x7f));
    printf("DEBUG: unpacked fraction        %7s%s\n", " ",
       unpack ("B24",  $data & 0x00ffffff));

    return $num;
}
[download]

Some inferences drawn from here, though it could definitely be more clearly stated.