Regular Expressions (REs) are a computational kernel widely used for finding patterns in data in compute-intensive tasks such as genomic markers research, signature-based detection, and database query. Although flexible on the set of searched REs, software-based solutions cannot fulfill latency or throughput requirements to analyze massive data volumes at a given power budget. For this reason, many approaches exploit hardware accelerators as an offloading engine for REs matching. Indeed, various solutions rely on FPGA reconfigurability to embed automata into the reconfigurable fabric. However, this approach leads to time-consuming updates of the REs to search. This work exploits REs as sequences of basic instructions and builds a Domain-Specific Architecture (DSA), called TiReX, for RE matching on FPGAs. Our approach enables the user to change the desired RE at run-time, providing software programmability, flexibility, and specialized hardware mechanisms. Our DSA delivers performance in line with other state-of-the-art hardware approaches, while providing remarkable flexibility and we underline the importance of energy efficiency for these computations. We compared with multiple state-of-the-art software obtaining remarkable performance while achieving noticeable results with a better energy efficiency that ranges from 3× to 490× with our multi-core.