In this video, I show how to solve the entire cube using variations of "The formulae [RU'L'UR'U'L]" you mentioned. (Except that we include the move U to complete the sequence.) I also explain in the video how/why that sequence works. There are a few instances when you need to conjugate (use premoves/setup moves) before a variation of The Niklas Commutator (the algorithm above --with the added U move at the end --can be rewritten as [R, U' L' U], which is a commutator. It is called the Niklas commutator.) is used to solve some situations. Then those premoves/setup moves are undone afterwards.
If you are interested in learning even more about the scope of commutators, see my post here. More specifically, we can (in theory) solve every position with a single commutator + one move such as R or R2. So if you are actually talking about a "formula" which can represent the form of a solution to any position, then there is the formula. That is, [X,Y] + R or R2 can solve any 3x3x3 Rubik's Cube scramble.
But I am more inclined to think that you were asking about a move sequence. Well the video above demonstrates that direct variations (and conjugations) of one eight move commutator is sufficient to tackle the cube.
As far as answering how people came up with original sequences, this old answer of mine is in agreement with Wood. Furthermore, in the past, I went out of my way to decompose (and/or derive) well-known last layer speedcubing algorithms. See the links I provide in this post. This may "simulate" the type of logic which was used to find the algorithms.
And for what it's worth (expanding on the logic of how they found the algorithms), I don't know if you solved the 4x4x4 yet, but if you have, this guide was probably published in the 80's. One of the last figures on the last page shows the single edge flip case with this caption.
I'm not sure exactly when it was (no later than 1998) that this 25 90 degree rotation algorithm was found to fix that case directly. It was most likely found by a computer, but here is my (human) derivation of it. (Part 1, Part 2)
Well, I took the above challenge seriously (despite that I didn't see this particular guide before I did what I am about to state) and eventually found this 18 90 degree rotation algorithm by hand (in 2016). So in essence, I used a similar logic to what I showed above for 3x3x3 algorithms as I did for finding the above 18 move algorithm. So if there is a human logic base, besides accidentally finding it and/or using a computer, this should put up quite a case that my own logic process is the same that theirs was (since it is very fruitful and is not dependent on computer searches). I mean, you can (I eventually did) conduct a computer search to find the 18 move algorithm I did, but it wasn't how I originally found that 18 move algorithm, for example.