Unless you have a pressing need (like ability to run the script on a different system with a different crypt()) then don't try to do this. Instead just write something like my $shadow = crypt($root_pass, '$6$' . $salt); and you're good to go! The way that crypt() is designed it's equally good for generating or checking a hash.

Oh, but to answer the question you actually asked, the reason is that the "SHA-512" crypt() algorithm isn't SHA-512, it's a key-derivation function using SHA-512. Likewise the "MD5" crypt algorithm isn't MD5, etc. The actual algorithm is specified here as far as I can tell, but you shouldn't worry about implementing it. Let crypt() do that for you.