It's a given that software needs to be tested. The processes around that are a classical "it depends" question.

Most likely any software you deploy will never be "correct" (whatever that means). The quality of that software depends on many variables and it's up to you to try and tweak them while optimizing for things like cost, time etc. Whether it's worthwhile to write the tests ahead of time or after the fact or to do them manually or automatically or any other permutation is just not a question that can be answered in a way that applies to all situations.