The ability to use computers to brute-force our calculations tends to obscure the fact that very often machine solutions are actually quite crude and inelegant. Even worse, the numerical methods used to digitally derive answers which would be better determined analytically often prove unstable or fail to converge to solutions, or otherwise behave very strangely under certain circumstances. I took a formal course in numerical methods, so I knew the strengths and limitations of numerical solutions. Whenever possible, I actually tried to solve the equations analytically. Before our raw computing power was available, mathematicans often devised clever numerical procedures, approximations and simplifications to ease the calculation load without intoducing too much error.

I'm old enough to have been trained on and used slide rules extensively, (the pocket calculator did not exist back then), and used a big mainframe and fortran to do my heavy-duty number crunching, so I got quite good at using algebra and calculus to simplify and streamline my math prior to coding up my numerical analyses. I took advantage of every significant-digits dodge I could to keep it simple and manageable. This gave me a lot of insight into what was going on computationally, and I was able to detect or avoid numerical artifacts (bogus results arising from accumulated error or built-in imprecisions etc). Nowadays, people tend to be very trusting at the results their software gives them, sometimes with tragic results.