Arithmetic around the bit heap

A bit heap is a data structure that holds the unevaluated sum of an arbitrary number of bits, each weighted by some power of two. Any multivariate polynomial of binary inputs can be expressed as a bit heap whose bits are simple boolean functions of the input bits. For many large arithmetic designs, viewing them as bit heaps is more relevant than viewing them as a composition of adders and multipliers. It leads to better global optimization at both the algebraic level and the circuit level. However, this notion needs to be supported by tools. This article therefore discusses a generic software framework for the definition, optimization and compression of bit heaps. It is specifically directed towards FPGAs, where complex and application-specific arithmetic circuits must be developped in little time. For this purpose, the textbook notion of a bit array is refined in several ways. Firstly, a bit heap should accept bits arriving at various instants in circuit time, and the bit heap compression process must take this timing into account. Secondly, the DSP blocks of recent FPGAs must be integrated in the bit heap view. Thirdly, the management of signed bit heaps is detailed, and shown to entail no overhead. Finally, a new family of elementary compressors on FPGAs improves upon the state of the art.