The way that I look at this is pretty mainstream, I think: We aren't training for 99%, we are training for 100%, so tap.
Failures occur in order during the cycle of operation of the pistol. The pistol fails to feed, fails to fire, fails to extract, fails to eject - in that order.
In teaching the masses, versus teaching the basics to folks who are in the pipeline to 'Tier 1' units, on the way to USPSA C class and above, the goal is to have something that will work every time, is easy to remember and is easy to accomplish.
Determining whether the pistol has failed to feed or failed to fire during a string of fire is pretty near impossible, the pistol goes click on both of them. The failure to extract and the failure eject usually feel different to the shooter, the trigger is usually dead, no click. The shooter MAY recognize that a double feed fail to extract feels different, or they may recognize the failure to eject by 'hey that was different, nothing hit my face' but we shouldn't assume anything beyond them knowing the pistol doesn't work for the moment.
What is easiest to remember is gun doesn't go bang we need to do one thing to fix it.
With the goal of every time all the time the only thing that works is ensuring that another round is loaded into the chamber. and in order to be 100% positive that occurs we need to first make sure that another round is available to be fed into the chamber. The only way to endure that is to make sure the magazine is seated. Actually, the first thing we should do is look to make sure a mag is actually there, but in this conversation, we are talking about a quick, reflexive act which gets us back into the fight immediately.
So, we are going to make sure the mag is seated by tapping. Remember we aren't training for 99%, we are training for 100%, so tap.
With that out of the way, the next thing to get the round into the camber is to cycle the action. BUT, on a rare occasion, cycling the action with the pistol upright may not let a case that has failed to eject fall clear. So, in order to ensure the case falls clear we need to roll the pistol so the ejection port is more toward the ground.
So with TAP, RACK AND ROLL, REASSESS/REENGAGE we should be at 100% every time all of the time with three of the four problems that occur during the pistol's cycle of operation.
That leads us to the fail to extract. In looking at ways the fail to extract can present itself, I've seen cases stuck in the chamber with the extractor still engaged, cases in the chamber with the slide forward with the extractor nose resting on the case head, and on some occasions, in the classic double-feed. I can use the fingers of one hand to list the number of times I've seen torn case rims.
If the shooter adopts the strategy of pistol doesn't go bang then immediately tap, rack and roll, reassess, one of three things will happen if there has been a failure to extract: 1) the immediate action will clear the stoppage; 2) the slide will be locked due to the extractor remaining in place on the case rim; 3) the double-feed fail to extract will be set up. Of course, number one is the best outcome.
If the slide moves the first thing the shooter needs to do is lock it to the rear, remove the magazine and, maintaining a firm grip let the slide go forward, preferably using the slide catch/stop.
Let's start with locking the slide to the rear. Some folks teach ripping the mag from the pistol with the slide pressure against the round trying to chamber. It's been my experience that a significant number of folks don't have the strength to do this in a fluid motion. Locking the slide to the rear removes that pressure and makes it easier to remove the magazine.
I prefer to lock the slide to the rear and let the slide go forward under spring pressure rather than immediately 'rack, rack, racking' because I've found that using the slide release allows the recoil spring to impart more impact energy than manually racking the slide.
Hopefully, we get the pistol's extractor to do something it isn't designed to do, ride over the case head in order to grip the case rim. If it doesn't give it two more tries. After that you need to be looking for a range rod.
Which brings us to the question 'seriously, you expect me to do this shit in a gun fight?' Only if you don't have another weapon to transition to, and cannot leave.
If the extractor grabs the rim, the slide may not want to come to the rear. In this case grab the slide in an overhand grip with your support hand, grip as firmly as possibly while taking your strong hand from the pistol and then with master hand extended/open slamming the hand onto the backstrap of the pistol. Slam through the backstrap, your goal is to strip the slide from your support hand grip. Sometimes pistols end up on the ground when doing this maneuver. Hopefully the impact is sufficient to get the case from the chamber. If this doesn't work, you maybe able to use the front corner of your slide against a square edge such as the corner of a brick or board and repeat the process. If you have rear sights such as Heinie Ledges or Dawson Chargers the rear sight can be used in the same manner.
Anyhow, once the chamber is cleared, reload.
TL;DR basic primer on malfunction clearance
Note: I am aware that every time all the time is a redundancy.