Unfortunately, you can't simply eliminate reverse. Cody Pendant explained why in the comments in his solution; if the number has an odd number of digits, you'll end doubling the wrong digits. For example, your solution would return true for 548979844, which is not a Luhn number.

You shouldn't eliminate my$s; either. That's not there for strict-ness, but to allow the subroutine to be called more than once. Without it, $s would keep its values between calls, throwing off all the subsequent answers.