"Beware of bugs in the above code; I have only proved it correct, not tried it." - Donald Knuth

Tests prove the things you thought of worked, and it often isn't hard to find the likely edge cases such that you have reasonable confidence everything works. You will be wrong from time to time, but not that often. You can prove code correct, but if the proof is wrong (common when a human is doing it), or the spec is wrong (most people have no clue how to write a comprehensive spec) it can still be wrong.