Improved Error Bounds for Floating-Point Quotients
Florian Bünger · IEEE Transactions on Computers · 2025
Let$x_{0}, y_{1}, \dots, y_{k}$be nonzero floating-point numbers in base$\beta \geq 2$and precision$p \geq 1$. Let$z := x_{0}/ y_{1}/ \dots/ y_{k}$, whereby the divisions are evaluated from left to right, and let$\widehat{z}$be the corresponding floating-point evaluation according to the IEEE 754 standard in rounding to nearest. We prove that, in absence of underflow and overflow,$|\widehat{z}-z| \leq k\text{u}|z|$provided that$k \leq \sqrt{\omega/\beta} \text{u}^{-1/3}$. Here$u := \frac{1}{2} \beta^{1-p}$denotes the relative rounding error unit and$\omega := 2$if$\beta$is even and$\omega := 1$if$\beta$is odd. Thus, the relative rounding error of k consecutive floating-point divisions is bounded byku. This improves on the classical Wilkinson-type bound$\gamma_{k} := k \text{u}/(1 - k \text{u})$.