Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersom

Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersome (and, importantly, not better). My argument here is not that applied mathematics is inferior (it’s what my degree is in), just that it’s generally not a good idea to carry applied techniques back into theory. You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details. Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.