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Compiling high-level programs to target high-speed packet-processing pipelines is a challenging combinatorial optimization problem. The compiler must configure the pipeline's resources to match the high-level semantics of the program, while packing all of the program's computation into the pipeline's limited resources. State of the art approaches tackle individual aspects of this problem. Yet, they miss opportunities to efficiently produce globally high-quality outcomes. We argue that High-Level Synthesis (HLS), previously applied to ASIC/FPGA design, is the right framework to decompose the compilation problem for pipelines into smaller pieces with modular solutions. We design an HLS-based compiler that works in three phases. Transformation rewrites programs to use more abundant pipeline resources, avoiding scarce ones. Synthesis breaks complex transactional code into configurations of pipelined compute units. Allocation maps the program's compute and memory to the hardware resources. We prototype these ideas in a compiler, CaT, which targets the Tofino pipeline and a cycle-accurate simulator of a Verilog hardware model of an RMT pipeline. CaT can handle programs that existing compilers cannot currently run on pipelines, generating code faster than existing compilers, while using fewer pipeline resources.