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The double Weyl semimetal (DWSM) is a newly proposed topological material that hosts Weyl points with chiral charge n=2. The disorder effect in DWSM is investigated by adopting the tight-binding Hamiltonian. Using the transfer matrix method and the noncommutative Kubo formula, we numerically calculate the localization length and the Hall conductivity in the presence of the on-site nonmagnetic disorder or orbital (spin-flip) disorders, and give the corresponding global phase diagrams. For the on-site nonmagnetic disorder, the system undergoes the DWSM-3D quantum anomalous hall (3D QAH) and normal insulator (NI)-DWSM phase transitions, and evolves into the diffusive metal (DM) phase before being localized by strong disorders, which is consistent with the Weyl semimetal. For σ_x orbital disorder, we find that increasing disorder can generate a pair of Weyl nodes at the boundary of the Brillouin zone and induce a 3D QAH-DWSM phase transition. Then we investigate the combined effect of orbital disorders for both disordered 3D QAH phase and DWSM phase. The disorder-induced transitions can be well understood in terms of an effective medium theory based on self-consistent Born approximation.