Cachinnate . . .
has been whooping it up in English since the 19th century.
The word derives from the Latin verb cachinnare, meaning "to laugh loudly," and "cachinnare" was probably coined in imitation of a loud laugh. As such, "cachinnare" is much like the Old English "ceahhetan," the Old High German kachazzen, and the Greek kachazein - all words of imitative origin that essentially meant "to laugh loudly."
Our word cackle has a different ancestor than any of these words (the Middle English "cakelen"), but this word, too, is believed to have been modeled after the sound of laughter.