Trinity: Three-Dimensional Tensor Program Optimization via Tile-level Equality Saturation

Modern tensor program optimizers operate at two separate levels: graph-level optimizations (operator fusion, algebraic rewrites) and operator-level scheduling (tiling, parallelization). This separation prevents them from discovering cross-operator, tile-level optimizations that make hand-tuned kernels like FlashAttention effective. We present Trinity, the first tensor program optimizer that achieves scalable joint optimization through tile-level equality saturation. Our key insight is that optimal performance requires simultaneously optimizing three interdependent dimensions -- algebraic equivalence, memory I/O, and compute orchestration. To enable this, Trinity introduces a novel fine-grained IR that exposes all three axes as first-class, rewritable entities and applies equality saturation to perform scalable joint optimization. As a result, Trinity automatically discovers complex optimizations that require coordinated reasoning across all three dimensions. Across diverse Transformer variants, Trinity achieves up to 2.09× speedup over TensorRT and 2.35× over TorchInductor, both state-of-the-art production compilers.