Efficient Table-based Function Approximation on FPGAs Using Interval Splitting and BRAM Instantiation

Abstract

This article proposes a novel approach for the generation of memory-efficient table-based function approximation circuits for edge devices in general and FPGAs in particular. Given a function f(x) to be approximated in a given interval [ x 0 , x 0+a ) and a maximum approximation error E a , the goal is to determine a function table implementation with a minimized memory footprint, i.e., number of entries that need to be stored. Rather than state-of-the-art work performing an equidistant sampling of the given interval by so-called breakpoints and using linear interpolation between two adjacent breakpoints to determine f(x) at the maximum error bound, we propose and compare three algorithms for splitting the given interval into sub-intervals to reduce the required memory footprint drastically based on the observation that in sub-intervals of low gradient, a coarser sampling grid may be assumed while guaranteeing the maximum interpolation error bound E a . Experiments on elementary mathematical functions show that a large fraction in memory footprint may be saved. Second, a hardware architecture implementing the sub-interval selection, breakpoint lookup, and interpolation at a latency of just 9 clock cycles is introduced. Third, for each generated circuit design, BRAMs are automatically instantiated rather than synthesizing the reduced footprint function table using LUT primitives, providing an additional degree of resource efficiency. The approach presented here for FPGAs can equally be applied to other circuit technologies for fast and, at the same time, memory-optimized function approximation at the edge.