Recent algorithmic advances have made equality saturation an appealing technique for program optimization, avoiding the phase-ordering problem by separating the discovery of equivalent expressions from optimal expression selection. Existing work leveraging equality saturation in compilers uses either external equality saturation libraries or custom implementations that are coupled to the specific application. These approaches are inherently limited, as the first discards semantic equivalences when translating back from the external library, while the second restricts equality saturation to a single level of abstraction. We propose an alternative approach that represents an e-graph natively in code, facilitating the application of constructive compiler passes that maintain the e-graph state throughout the compilation flow. We present Tamagoyaki, an implementation of this approach in MLIR and demonstrate its versatility through partial re-implementations of two equality saturation applications. Our software case study improves performance by 1.18$\times$ on average, whilst our hardware case study reduces circuit delay by up to 11% over standard equality saturation. Building on reusable compiler infrastructure, our work expands the scope of equality saturation, making it persistent across and interleavable with other analyses and transformations.

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