Unit testing would let you validate the correctness of the code.
As far as improving performance characteristics, the argument is that unit testing helps indirectly by improving awareness of the behavior of your system. Having a good idea of where there aren't bugs will let you be more aggressive in refactoring to improve things like performance characteristics.
Sun Tzu:
If you know your enemies and know yourself, you will not be imperiled in a hundred battles... if you do not know your enemies nor yourself, you will be imperiled in every single battle.
...if it's not clear, you are your code and bugs are enemies. It's all about mitigating risk through better understanding of your code base.
As far as improving performance characteristics, the argument is that unit testing helps indirectly by improving awareness of the behavior of your system. Having a good idea of where there aren't bugs will let you be more aggressive in refactoring to improve things like performance characteristics.
Sun Tzu:
If you know your enemies and know yourself, you will not be imperiled in a hundred battles... if you do not know your enemies nor yourself, you will be imperiled in every single battle.
...if it's not clear, you are your code and bugs are enemies. It's all about mitigating risk through better understanding of your code base.