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I've known a lot of mathematicians, some of whom were really good, and five nines (one-in-one hundred thousand) is an implausibly high standard for humans to reach. I'm certainly nowhere near there.

What's worked for me is to develop an intuition about how things 'should' look, so that if I do make a mistake I'll get a 'hey, that's not right' feeling a couple steps later.

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Just to expand on this, when I worked as a grader in college I noted on engineering test keys the professor's answers were always roughly three short lines of handwritten equation. But the student's answer section were often totally darken with pencil they wrote so much. The heuristic was definitely the longer the answer the more wrong it was.

This is something I've taken to heart professionally too. If the equations are getting out of hand it's either wrong or I need some simplifying assumptions (like that cow needs to be spherical). Otherwise one just can't keep track of it and reason about it.

If I need a more precise answer, that's what computer numerics is for.

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