holy shit, now that's a post. very well written, easy to follow. nice.
unfortunately, what you've described is basically what i've been doing, which is way more steps than i tend to need with the others. i don't know if it exists in other places but we had an entire module in middle school on "överslagsräkning", which basically translate to "rough calculations". it was about using iterative methods like this to land at a plausible solution to a problem.
unfortunately, i mostly need actual solutions. as an example i'll tell you what i did today.
today, i tried to refine an algorithm that, given a set of points (represented by a c array) and a "fractional index" (a positive non-integer value) can give result for a number between two defined points. i had previously done this through simple linear interpolation, but since we sample data from actual physical objects the points tend to represent a curve, and the common way to solve that is with a lagrange polynomial. so, i found an algorithm that, given a set of n + 1 points, evaluates the unique lagrangian of at most degree n that intersects all the points at the given index. i split the terms out to an (a + b) / c form and then i realised that i have no fucking idea what i'm doing.
like yes, i can run the algorithm to see if the numbers line up, but i could never actually do the math. i'm not talking about the algebra part, i'm talking about the arithmetic. if i plug some numbers in and do the parts i can do, i end up with something like 0.23 / -7 and i don't even know where to start. i have no idea where that would even land. so, while i can verify that my test data works, i have no way to quickly extrapolate numbers. which in turn means i can't come up with plausible input. i'm back to using a calculator for that, which is so damn annoying.