Extended-Precision FMA under Parameterized Double-Word Overlap: Tight Error Bounds and Examples

Overview & Synopsis

We study fast fused multiply--add (FMA) kernels for approximating $ab+c$ in extended precision using double-word (DW) representations in floating-point (FP) arithmetic. We focus on how worst-case guarantees depend on the overlap between the high and low parts, modeled by parameterized conditions of the form $|x_\ell|\le k\,\mathrm{ulp}(x_h)$ (and two- and three-parameter variants for multiple DW inputs). We consider the dominance regime $|c|\ge 2|ab|$ and its DW analogues (e.g., $|c_h|\ge 2|ab|$ and $|c_h|\ge 2|ab_h|$), which occur in polynomial evaluation for accurate elementary functions and in cancellation-free constructions. Within this regime, we tighten and extend error analyses for several FMA-based schemes used in recent extended-precision implementations. We analyze FP/DW and DW/DW variants, derive explicit worst-case constants for both the returned low part and the overall relative error, and provide matching worst-case examples. The resulting bounds give a unified and quantitative view of how DW-overlap assumptions affect the accuracy of fast extended-precision FMA building blocks.