Repitition
My profession is physical and cognitive rehabilitation, specializing in upper extremity work where restoration of fine motor skills after trauma or stroke is a key goal. Learning a new fine motor task, whether whether double tapping or tying shoes, goes like this.
One must repeat an unfamiliar fine motor task accurately 2,000-4,000 times before it has been learned to the point that it can be performed subconsciously.
When we first begin to learn a new physical skill, we think about what we're doing, usually with great concentration. This occurs within the frontal lobe of the brain. Simultaneously, the physical sensation of how to perform this new task is processed within the sensory motor strip of the brain's cerebral cortex. With enough repetition, the memory of how it feels to perform the new activity becomes chemically coded, and that coded molecule is moved out of the front of the brain, and stored within a lower, deeper portion of the brain. This frees the front part of the brain to learn new stuff, and to be available for such things as situational awareness when we are engaged in a complex fine motor activity (driving, typing, tying shoes, etc).
Regarding high speed shooting (double taps), what this means is, do a lot of dry fire practice, drawing, sighting and snapping on an empty chamber. Front sight, press; front sight, press. This isn't bullseye: at pistol gunfighting range you can get very loose with the alignment of the front sight in the rear sight notch, as long as that front sight is superimposed on where you want to hit. Get very precise with grip and stance. Draw and snap, every day, at least twice a day, for a couple of dozen repetitions, or until you begin to lose concentration. Be slow and accurate, resisting the temptation to rush, repeating the same, deliberate correct motion. Enhanced speed will come naturally, and 100 reps a day will give 2,000-4,000 reps fairly quickly. Double tapping will be much more accurate after the body fully understands the "single tap". Live fire practice will go much better after all of the dry firing.
When live firing, you will find that you will begin to anticipate recoil, and will actually be working to pull the gun back down on the target as soon as the first shot has been fired. You'll notice this when you have emptied the gun at the target without realizing it, and you'll experience a sharp dip in the muzzle, kind of like a flinch without being a flinch. This comes from pushing the gun forward to overcome anticipated recoil. I think this is one of the reasons that some people like to use a DA revolver for fast work, because stroking the trigger for the next shot works to pull the muzzle down.
Sorry for the length of this post. My wife tells me that when people ask me for the time I tell them how to build a watch. Hope this helps.