What term is used when one does something off the top of their head or improvises?

The term "extemporize" specifically refers to performing or speaking without prior preparation, essentially improvising or delivering material spontaneously. This is often seen in situations such as giving a speech, participating in a discussion, or even playing music. The essence of extemporizing lies in its unplanned nature, which involves thinking on one’s feet and adapting to the moment.

In contrast, the other terms have distinct meanings. "Invent" typically implies the creation of something entirely new, often involving a level of planning and development. "Formulate" suggests a methodical approach to developing ideas, often involving systematic arrangements or calculations. "Create" is broader and can refer to bringing something into existence, but it does not inherently convey the same sense of immediacy or spontaneity that "extemporize" does. Thus, "extemporize" accurately captures the essence of improvisational actions taken off the top of one's head.