Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 4 / c

At a root of size between $1$ and $2$, the spacing of floating-point numbers is $ϵ=2^{-40}$. With ordinary stable function evaluations, a sensible generic absolute position tolerance has scale $2^{-40}\approx 9.09\times 10^{-13}$; sub-spacing progress cannot generally be represented. This is a resolution estimate, not a lower bound excluding an accidentally exactly representable root.

Define one bisection method step as replacing the current bracket by the half retaining the sign change. Starting from $[1,2]$, the width after $N$ such updates is $2^{-N}$. ThereforeThe midpoint of a bracket has an exact-arithmetic error bound of half its width. If tolerance is defined instead as that midpoint error bound, $39$ updates suffice to make it at most $2^{-40}$. Stating the convention avoids an off-by-one difference; once the endpoints are adjacent floating-point numbers, their exact midpoint need not itself be representable.