On the number of impossible differentials of some ARX transformation

The additive differential probabilities of the function $(x \oplus y) \lll r$ are considered, where $x, y \in \mathbb{Z}_2^{n}$ and $1 \leq r < n$. They are interesting in the context of differential cryptanalysis of ciphers whose schemes consist of additions modulo $2^n$, bitwise XORs ($\oplus$) and bit rotations ($\lll r$). We calculate the number of all impossible differentials, i.e. differentials with probability $0$, for all possible $r$ and $n$. The limit of the ratio of this number to the number of all differentials as $r$ and $n-r$ tend to $\infty$ equals $38/245$. We also compare the given numbers and the number of impossible differentials for the function $x \oplus y$.