The “slow is smooth, smooth is fast” concept likely has its origins in the Military. Practicing at reduced speeds will make you faster when you go full speed. Full speed is often slower than you think, and there is a difference between going full speed and being rushed. When you are rushed you stumble, fumble, and are generally all thumbs (or have two left feet). When you go full speed, you are going as fast as you can comfortably go. Martial artists do this, Soldiers do this, in fact, anyone trying to master a technique would do well to practice slowly and increase speed only when it’s comfortable.
The exact origins of the term are disputed. Doc Holiday said, “Take your time in a hurry” to describe how to win a gunfight. Some also attribute the quote to Sergeant Alvin York (Medal of Honor recipient), others to Wyatt Earp. One thing is for sure, all three were accomplished gunfighters, so the origin of the quote is less important than the truth of it.