Although an optimizing compiler often produces a fast program, we cannot expect it to produce an optimal program. Moreover, the amount of headroom between the program we get and the optimally fast program is usually unknown. A superoptimizing compiler—that performs a meaningful search of the program space—can help narrow this gap. We created Minotaur, a superoptimizing LLVM pass that uses program synthesis to improve the quality of its code generation, with a specific focus on integer and floating-point SIMD code. It discovers many profitable optimizations that are missing from LLVM, some of which have been implemented in LLVM as a result of our work. On an Intel Cascade Lake processor, Minotaur achieves an average speedup of 7.3% on the GNU Multiple Precision library (GMP)’s benchmark suite, with a maximum speedup of 13%. On SPEC CPU 2017, Minotaur produces an average speedup of 1.5%, with a maximum speedup of 4.5% for 638.imagick.