Scheduling algorithms are called “global” if they can migrate tasks between cores. Global scheduling algorithms are the de-facto standard practice for general purpose Operating Systems, to balance the workload between cores. However, the exact schedulability analysis of real-time applications for these algorithms is proven to be weakly NP-hard. Despite such a hardness, the research community keeps investigating the methods for an exact schedulability analysis for its relevance and to tightly estimate the execution requirements of real-time systems. Due to the NP-hardness, the available exact tests are very time and memory demanding even for sets of a few tasks. On another hand, the available sufficient tests are very pessimistic, despite consuming less resources. Motivated by these observations, we propose an exact schedulability test for constrained-deadline sporadic tasks under global multiprocessor fixed-priority scheduling scheduler, which is significantly faster and consumes less memory, compared to any other available exact test. To derive a faster test, we exploit the idea of a state-space pruning, aiming at reducing the number of feasible system states to be examined by the test. The resulted test is multiple orders of magnitude faster with respect to other state-of-the-art exact tests. Our C++ implementation is publicly available.