Your bicep and tricep are on either side of your upper arm, when you extend to punch your tricep needs to flex and your bicep needs to be relaxed. It works similar to your shoulder. If the muscles are tense on both sides of the joint then theres no movement is there.
If you completely relax the shoulder and arm and throw punches with ease, youll be fast. the more relaxed you are the faster youll get but youll have to supply all your power from the waist. This is all basic technique for Bruce Lees original art, wing chun. The average capable student in my old wing chun class could throw about 6 punches per second, full extension in a 'chain punch' fashion.
I dont train wing chun anymore because its too stylistic and not practical in some waysbut off all classical styles it is the most efficient and progressive actually, imo.