How can I prove or disprove that the following function is bijection?
Originally, the hash function was defined as follows:

Input: a 64-bit unsigned integer $x$.
Calculate $(9E3779B185EBCB87)_{16} \cdot x = 2^{64} x_h + x_l$, where $x_l,x_h$ are 64-bit unsigned integers.
Output $x_l \oplus \lfloor (x_h + 2x_l)/8 \rfloor$.

Under this definition, there is a collision: both $(2DCE4F983BBEB685)_{16}$ and $(72E04DBED1540FE1)_{16}$ map to $(A5C53FE6E277BD92)_{16}$.
You can find such collisions using a cycle detection algorithm.