Refactoring code is the same as refactoring a math equation. Make sure you show every step so you don't screw up. If you skip steps you will screw it up.

If it's risky, you're doing it wrong. (Or you're using an unchecked language like Python (which is incidentally why Python is not suited for large projects IMO, because you can't safely refactor them).)

P.S. the steps are covered in Martin Fowler's book, but they are equivalent to refactoring functions in math, naturally.