Xor Becoming Uncanny

We define $f(x)=\begin{cases} f(x \oplus reverse(x))+1 & \text{if } x \neq 0 \\ 0 & \text{otherwise} \end{cases}$ Here, $\oplus$ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR) and $reverse$ is a function that takes a postive integer, reverses its binary representation (without any leading zeros) and returns the resulting number. For example $reverse(2)=