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TDD is king when refactoring, or proving an algorithm. You have a tests to confirm the output, and near realtime feedback that you assumptions are correct. The rest is obvious. Mission critical component TDD, complicated refactor TDD, algorithm you need to validated TDD. Anything else write the code and get a peer review.

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No amount of tests can prove that an algorithm is correct. At best, they prove that an algorithm works in a particular case.

And generally, I would say that the more interesting sorts of tests (fuzz testing, large-scale system testing) are extremely unpopular with software engineers because "they suddenly fail without reason". Not quite as unpopular as actual proof that an algorithm works, like implementing it in coq for instance, but very unpopular.

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