I can't think of a single-word adverb precisely having the meaning always correctly. But I can suggest a few adverbs closer to your search.
Flawlessly, seamlessly, accurately or precisely.
flawless (en.oxforddictionaries.com) adjective
Without any imperfections or defects; perfect.
b : perfect, flawless
a seamless performance
accurately (en.oxforddictionaries.com) adverb
In a way that is correct in all details; exactly.
in exact terms; without vagueness.
"the guidelines are precisely defined"
exactly (used to emphasize the complete accuracy or truth of a statement).