Towards the Multiple Constant Multiplication at Minimal Hardware Cost

Abstract: Multiple Constant Multiplication (MCM) over integers is a frequent operation arising in embedded systems that require highly optimized hardware. An efficient way is to replace costly generic multiplication by bit-shifts and additions, i.e. a multiplierless circuit. In this work, we improve the state-of-the-art optimal approach for MCM, based on Integer Linear Programming (ILP). We introduce a new lower-level hardware cost, based on counting the number of one-bit adders and demonstrate that it is strongly correlated with the LUT count. This new model for the multiplierless MCM circuits permitted us to consider intermediate truncations that permit to significantly save resources when a full output precision is not required. We incorporate the error propagation rules into our ILP model to guarantee a user-given error bound on the MCM results. The proposed ILP models for multiple flavors of MCM are implemented as an open-source tool and, combined with the FloPoCo code generator, provide a complete coefficient-to-VHDL flow. We evaluate our models in extensive experiments, and propose an in-depth analysis of the impact that design metrics have on actually synthesized hardware.