Synonyms of 'extemporise'
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If you ad-lib something in a play or a speech, you say something which has not been planned or written beforehand.
He is rather disjointed when he ad-libs.
When performers improvise, they invent music or words as they play, sing, or speak.
Take the story and improvise on it.
Thesaurus for extemporise from the Collins English Thesaurus