 Outward rounding

Outward rounding replaces an exact lower endpoint by a representable value no larger than it, and an exact upper endpoint by a representable value no smaller than it. This preserves an interval enclosure under floating-point arithmetic. In the binary numeral system, most rational numbers cannot be represented exactly with finitely many digits, so the rounding direction is essential even for elementary calculations. Rump explains directed rounding and its use in interval bounds in §§4 and 5.3 of his survey.^[ref]

To see the issue without long binary digit strings, use an illustrative decimal grid with step $0.001$. Exact interval division gives $[1,1]/[3,3]=[1/3,1/3]$. Rounding both endpoints to the nearest grid value gives $[0.333,0.333]$, which excludes the true answer because $0.333<1/3$. Outward rounding instead givesThis decimal toy illustrates the rule; its displayed endpoints are not the results of binary directed rounding. Real implementations need valid outward rounding throughout the calculation, including input conversion and intermediate interval arithmetic operations; rounding only the final output cannot repair an earlier lost bound. INTLAB's tutorial demonstrates why decimal input strings and previously rounded intermediate values need care.^[ref]