Floating-point bitwidth analysis via automatic differentiation

Automatic bitwidth analysis is a key ingredient for highlevel programming of FPGAs and high-level synthesis of VLSI circuits. The objective is to find the minimal number of bits to represent a value in order to minimise the circuit area and to improve efficiency of the respective arithmetic operations, while satisfying user-defined numerical constraints. We present a novel approach to bitwidth- or precision-analysis for floating-point designs. The approach involves analysing the dataflow graph representation of a design to see how sensitive the output of a node is to changes in the outputs of other nodes: higher sensitivity requires higher precision and hence more output bits. We automate such sensitivity analysis by a mathematical method called automatic differentiation, which involves differentiating variables in a design with respect to other variables. We illustrate our approach by optimising the bitwidth for two examples, a discrete Fourier transform (DFT) implementation and a Finite Impulse Response (FIR) filter implementation.