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Optimizing compilers rely on peephole optimizations to simplify combinations of instructions and remove redundant instructions. Typically, a new peephole optimization is added when a compiler developer notices an optimization opportunity---a collection of dependent instructions that can be improved---and manually derives a more general rewrite rule that optimizes not only the original code, but also other, similar collections of instructions. In this paper, we present Hydra, a tool that automates the process of generalizing peephole optimizations using a collection of techniques centered on program synthesis. One of the most important problems we have solved is finding a version of each optimization that is independent of the bitwidths of the optimization's inputs (when this version exists). We show that Hydra can generalize 75% of the ungeneralized missed peephole optimizations that LLVM developers have posted to the LLVM project's issue tracker. All of Hydra's generalized peephole optimizations have been formally verified, and furthermore we can automatically turn them into C++ code that is suitable for inclusion in an LLVM pass.