A framework for trackable and modular rounding error analysis

arXiv:2610.35420v1 Announce Type: new Abstract: Robust and meaningful rounding error analysis is becoming increasingly challenging as modern numerical computations involve a rapidly evolving variety of arithmetic formats, precisions, and computational kernels. We propose a trackable and modular framework for rounding error analysis, together with a new notation for bounding accumulated rounding errors. The analysis is trackable in the sense that the origin and propagation of individual error terms remain explicit in the final result, and modular in the sense that local error bounds can be replaced according to the arithmetic, precision, or computational kernel without repeating the structural analysis. We also introduce the notation $\gamma_n=(1-{\bm u})^{-n}-1$, which provides an alternative to the classical $\gamma_n^{\mathrm h}=n{\bm u}/(1-n{\bm u})$ notation. A key feature is that our notation does not require the restriction $n{\bm u}