Hyperbolic embeddings achieve excellent performance when embedding hierar- chical data structures like synonym or type hierarchies, but they can be limited by numerical error when ordinary floating-point numbers are used to represent points in hyperbolic space. Standard models such as the Poincaré disk and the Lorentz model have unbounded numerical error as points get far from the origin. To address this, we propose a new model which uses an integer-based tiling to represent any point in hyperbolic space with provably bounded numerical error. This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits. We evaluate our tiling-based model empirically, and show that it can both compress hyperbolic embeddings (down to 2% of a Poincaré embedding on WordNet Nouns) and learn more accurate embeddings on real-world datasets.