Branch prediction accuracy is critical for system performance in modern microprocessor architectures. The use of speculative update branch history provides substantial accuracy improvement in branch prediction. However, speculative update branch history is the information about uncommitted branch instruction and thus it may hurts program correctness, in case of miss-speculative execution. Therefore, speculative update branch history requires suitable recovery mechanisms to provide program correctness as well as performance improvement. In this paper, we propose recovery logics for speculative update branch history. The proposed solutions are recovery logics for both global history and local history. In simulation results, our solution provides performance improvement up to 5.64%. In addition, it guarantees the program correctness and almost 90% of additional hardware overhead is reduced, compared to previous works.