$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists?

I am interested in this question, because my matrix $A$ is huge and so is $C$. So computing inverse of $C$ from scratch is not practical, but luckily the matrix $A$ is unitary, so $A^{-1} = A^*$, so I easily have $A^{-1}$, and hence finding ways to use it to get $C^{-1}$.