I worked in Systems Validation at Intel when the 8087 was current. Intel had an engineer dedicated to validating customer bug reports and reproducing them. Day in, day out, that's pretty much all he did. Sooooo many corner cases, and so many opinions on what the 'right' thing to do was when you lost precision[1].

[1] I'd say that over half of the bug reports were people who were annoyed that doing fp instructions in one order got them the right answer but in another order got them the wrong answer.

Mathematicians vs Computer / Systems Engineers. The machine only has so much space, so it's best to imagine every value also has a corresponding error range attached and that managing the growth of that error range so that it remains under the target value is key.