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

Use the standard spacing convention for machine epsilon: the next representable floating-point number above $1$ is $1+2^{-40}$. In a normalized binary representation this corresponds to $41$ significant binary digits, including the leading digit. The spacing throughout $[1,2)$ is therefore $2^{-40}$.

With rounding to nearest, the associated unit roundoff is half that spacing, $u=2^{-41}$; for a normal result the relative round-off error is bounded by $u$. Some texts instead call $u$ machine epsilon. Here $2^{-40}$ denotes the spacing scale, and the iteration counts below use that convention. It describes representation precision, not a guarantee that an arbitrarily evaluated function has a relative error this small.