An efficient implementation of online arithmetic

Abstract

We propose the first hardware implementation of standard arithmetic operators – addition, multiplication, and division – that utilises constant compute resource but allows numerical precision to be adjusted arbitrarily at run-time. Traditionally, precision must be set at design-time so that addition and multiplication, which calculate the least significant digit (LSD) of their results first, and division, which calculates the most significant digit (MSD) first, can be chained together. To get around this, we employ online operators, which are always MSD-first, and thus allow successive operations to be pipelined. Even online operators require precision to be fixed at design-time because multiplication and division traditionally involve parallel adders. To avoid this, we propose an architecture, which we have implemented on an FPGA, that reuses a fixed-precision adder and stores residues in on-chip RAM. As such, we can use a single piece of hardware to perform calculations to any precision, limited only by the availability of on-chip RAM. For instance, we obtain an 8x speed-up, compared to the parallel-in-serial-out (PISO) fixedpoint method, when executing 100 iterations of Newton’s method at a precision of 64 digits, while the product of circuit area and latency stays comparable.