Unit roundoff ($u=\frac{1}{2}{\beta }^{1-p}$)

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 satisfiesIndeed, the absolute distance to the nearest value is at most ${\Delta }_e/2$, and $|z|\ge {\beta }^e$ 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.^[ref]