Lord of the Precision, the Two Floats: Double-Word Arithmetic for Complex Numbers

Abstract

Datacenter GPUs increasingly favour low-precision floating-point formats, offering higher throughput than FP64. Double-word (DW) arithmetic represents numbers as unevaluated sums of two floating-point values, providing higher accuracy. We present the first error analysis of complex DW multiplication and division. Both operations reduce to sums of two DW products, combined using one of three DW addition algorithms: accurate, sloppy, or Madd. Normalising DW products ensures nonoverlapping components, but omitting this step reduces the number of arithmetic operations. We bound the resulting overlap and prove that Madd retains a relative error bound of $3u^2+O(u^3)$ for overlapping inputs, where $u$ is the unit roundoff. For all variants, we establish relative error bounds of the form $(ακ+β)u^2+O(u^3)$, where $κ$ is the condition number of the sum of products. Our analysis shows that the less expensive sloppy addition suffices. We implement these variants in CompleXDW, an open-source C++ library for CPUs and GPUs. In the high-energy physics code MadGraph7, FP32-based DW arithmetic achieves up to 2.9x and 2.1x higher throughput than FP64 on NVIDIA L40S and AMD Radeon Pro W7900 GPUs, respectively, while losing approximately 1.4 correct significant digits.