What you seem to be asking for is a way to express this operation in conventional mathematical notation. But conventional mathematical notation doesn't have standard symbols for bitwise operations. This isn't because those operations are somehow illegitimate, or because the conventional notation is somehow better. It's just because those operations were studied later than the other ones and are less entrenched in history.
Where the conventional notation does provide a simple expression of a bit operation, as in your example, that is a coincidence.
So it's a bit of a puzzle what you are really looking for. If you need a way to express these operations compactly, you can make up your own compact notation, or you can look at the notation used by computer programming languages, which have solved this problem before, and the conventional notation is unlikely to make anything clearer. If you need an efficient algorithm to calculate the result of this operation, that is a different question, again unrelated to conventional mathematical notation. But the thing you seem to be asking for, a concise expression of this operation in standard number-theoretic symbols, I believe does not exist.