The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost ever

The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost every time you apply it. A common example is rounding linear programs to integer solutions for the purpose of approximation (or exact solutions, as the case may be). There is almost no way to tell in advance if it will work, and most problems require significantly different versions of this technique. There are many textbooks showcasing the many variations. So you might need days or weeks before you figure out if it will work on your problem, and then it's a "standard calculation" only in retrospect. People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.