Is there any way to prove that changed implementation does not break system's behavior?

There are two extremities:

• Test the whole system after any change of the code.
• Do not test at all.

Here under 'Testing' I mean running all not automated tests. User Acceptance tests to be more precise.

I'd like to have a solid understanding when it is absolutely safe not to perform manual acceptance tests.

100% code coverage is not sufficient here, I believe.