The correct phrase is to ‘party heartily’. Anyone who says ‘party hearty’ is not doing good. Split infinitives? The rule against them is, as someone who must not be named would say, illogical, Jim. See H.W. Fowler for more on this exciting topic. Finally, mention of Cervantes’ great book reminds me that Don Quixote is correctly pronounced ‘Don Kwixott’. Any alternative from the mouth of a native English-speaker is, like, Beijing, Kebek, Ni-ha-ra-uah and so on, nothing but an affectation.
And, having now condemned affectation, I shall now try to think of something to say in Latin…