A new algorithm for computing IEEE-compliant rounding is presented, called injection-based rounding. Injection-based rounding is simple and facilitates using the same rounding circuitry for different precisions. We demonstrate the usefulness of injection-based rounding in a design of an IEEE floating-point multiplier capable of performing either a double-precision multiplication or a single-precision multiplication. The multiplier is designed to minimize hardware cost by using only a half-sized multiplication array and by sharing the rounding circuitry for both precisions. The latency of the multiplier is in single-precision two clock cycles and in double precision the latency is three clock cycles, where each pipeline stage contains roughly 15 logic levels.