Halide is a domain-specific language for high-performance image processing and tensor computations, widely adopted in industry. Internally, the Halide compiler relies on a term rewriting system to prove properties of code required for efficient and correct compilation. This rewrite system is a collection of handwritten transformation rules that incrementally rewrite expressions into simpler forms; the system requires high performance in both time and memory usage to keep compile times low. In this work, we apply formal techniques to prove the correctness of existing rewrite rules and provide a guarantee of termination. Then, we build an automatic program synthesis system that operates over the undecidable theory of integers in order to craft new, provably correct rules from failure cases where the compiler was unable to prove properties. We identify and fix 5 incorrect rules as well as 8 rules which could give rise to infinite rewriting loops. We demonstrate that the synthesizer can produce better rules than hand-authored ones in five bug fixes, and describe four cases in which it has served as an assistant to a human compiler engineer. We further show that it can proactively improve weaknesses in the compiler by synthesizing a large number of rules without human supervision and showing that the enhanced ruleset lowers peak memory usage of compiled code without appreciably increasing compilation times.