= Unit roundoff
{title2=$u=\tfrac12\beta^{1-p}$}
For <floating-point arithmetic> with precision $p$ in radix $\beta$ and rounding to nearest, <unit roundoff> is the half-gap scale $u=\beta^{1-p}/2$. If a nonzero exact <real number> $z$ lies in the normal finite range, its nearest rounded value satisfies
$$
\operatorname{fl}(z)=z(1+\delta),\qquad |\delta|\leq u.
$$
Indeed, the absolute distance to the nearest value is at most $\Delta_e/2$, and $|z|\geq\beta^e$ gives the bound. For a correctly rounded <addition>, <subtraction>, <multiplication> or <division>, apply this estimate to the exact result of that one operation on the stored operands. The bound does not include earlier <roundoff error> or automatically bound a whole <algorithm>. Results outside the finite range and tiny <subnormal numbers> require separate treatment. Rounding in a directed mode also changes the bound.https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html[Goldberg, 1991, §§Relative Error and Ulps, Exactly Rounded Operations]{ref}
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