I don't know what I was thinking. Clearly, the simplicity of the error code is the best measure of how useful a subroutine is. So a better oddness tester would surely be:

sub odd {
    return 1   if  "1" eq $_[0];
    die "Can't test the oddness of anything but '1'.\n";
}
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But then, neither of these are any more complicated than your first one:

sub odd {
    my $num= shift @_;
    die "Out of range"   if  2**52 <= abs($num);
    return $num % 2;
}

sub odd {
    my $num= shift @_;
    die "Out of range"   if  1 != (1+$num)-$num;
    return $num % 2;
}
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Update: For anyone who cares, I'm done. As is, this is a lot of noise for something pretty trivial and it is going nowhere slow.