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Fully Homomorphic Encryption (FHE) is a cryptographic technique that enables privacy-preserving computation. State-of-the-art Boolean FHE implementations provide a very low-level interface, usually exposing a limited set of Boolean gates that programmers must use to write their FHE applications. This programming model is unnecessarily restrictive: many Boolean FHE schemes supportprogrammable bootstrapping, an operation that allows evaluation of an arbitrary fixed-size lookup table. However, most modern FHE compilers are only capable of reasoning about traditional Boolean circuits, and therefore struggle to take full advantage of programmable bootstrapping. We present COATL, an FHE compiler that makes use of programmable bootstrapping to produce circuits that are smaller and more efficient than their traditional Boolean counterparts. COATL generates circuits usingarithmetic lookup tables, a novel abstraction we introduce for reasoning about computations in Boolean FHE programs. We demonstrate on a variety of benchmarks that COATL can generate circuits that run up to 1.5× faster than those generated by other state-of-the-art compilation strategies.