Combining Deep and Shallow Embedding for EDSL

When compiling embedded languages it is natural to use an abstract syntax tree to represent programs. This is known as a deep embedding and it is a rather cumbersome technique compared to other forms of embedding, typically leading to more code and being harder to extend. In shallow embeddings, language constructs are mapped directly to their semantics which yields more flexible and succinct implementations. But shallow embeddings are not well-suited for compiling embedded languages. We present a technique to combine deep and shallow embedding in the context of compiling embedded languages in order to provide the benefits of both techniques. In particular it helps keeping the deep embedding small and it makes extending the embedded language much easier. Our technique also has some unexpected but welcome knock-on effects. It provides fusion of functions to remove intermediate results for free without any additional effort. It also helps to give the embedded language a more natural programming interface