extemporize

To perform or speak without prior planning or thought; to act in an impromptu manner; to improvise.

"Will you please tell me whose music you have been playing?" . . . "It's nobody's, miss." "Do you mean you have been extemporizing all this time?"

But while some of his predecessors liked to extemporize, Obama prefers the message to be just so.

To adapt, improvise, or devise action or speech in an impromptu or spontaneous manner.

As the music came fresher on their ears, they danced to its cadence, extemporizing new steps and attitudes.

The small jelly-speck, which we call the amoeba, has no organs save what it can extemporize as occasion arises.