The residue logarithmic number system: theory and implementation

The residue logarithmic number system (RLNS) represents real values as quantized logarithms which, in turn, are represented using the residue number system (RNS). Compared to the conventional logarithmic number system (LNS) in which quantized logarithms are represented as binary integers, RLNS offers faster multiplication and division times. RLNS and LNS use a table lookup involving all bits for addition. The width, dynamic range, precision and naive table size of RLNS (with careful moduli selection) is as good as those for conventional LNS. Conventional LNS can be more efficient than naive addition lookup. First, commutativity allows interchanging arguments. Second, the addition function is often essentially zero, and does not have to be tabulated. In binary, comparisons are easy. In residue, comparisons are slow. Although RLNS inherently demands comparison, this paper shows a novel way comparisons can be performed in parallel to the lookup from a small table. This paper also describes a novel tool that generates synthesizable Verilog, making RLNS viable in practical applications that can benefit from shorter multiply and divide times.