The high computational complexity of the multiple signal classification (MUSIC) algorithm is mainly caused by the subspace decomposition and spectrum search, especially for frequent real-time applications or massive sensors. In this paper, we propose a low-complexity MUSIC algorithm from a finite-precision arithmetic perspective. First, we analyze the computational bottlenecks of the classic low-complexity randomized unitary-based MUSIC (RU-MUSIC), formulating this computational issue as an inner product problem. Then, a mixed-precision method is introduced to address this problem. Specifically, this method partitions summations in inner products into blocks, where intra-block computations use low-precision arithmetic and inter-block sums use high-precision arithmetic. To further improve computational accuracy, we develop an adaptive-precision method that supports adaptive block sizes and multiple precision levels. Finally, simulation results show that the proposed finite-precision MUSIC design achieves direction-of-arrival (DOA) estimation performance similar to that using full-precision arithmetic while reducing more than 50\% computational cost.