my $digit = '\d'; /$digit/; only compiles a regexp once as well.

Perhaps, but that's not the OP's usage.

my $digit = qr/\d/; foo($digit);  # Compiles once
my $digit = '\\d';  foo($digit);  # Compiles any number of times
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Whether or not passing a pattern as a compile regexp or as string is faster depend on what the subroutine is doing with it.

I said that in the post to which you are replying.