The standard relative floating-point model and unit roundoff

A floating-point system. A floating-point system consists of a nonempty finite set $\mathbb{F}\subseteq \mathbb{R}$ of representable numbers with $0\in \mathbb{F}$ and $-\mathbb{F}=\mathbb{F}$, a positive real $N_{\max }\in \mathbb{F}$ (the largest normal magnitude), a positive real $N_{\min }\in \mathbb{F}$ (the smallest normal magnitude), with $N_{\min }\le N_{\max }$, and a rounding map $\mathrm{fl}$ defined on $0$ and on the real numbers $t$ with $N_{\min }\le \mathrm{\mid }t\mathrm{\mid }\le N_{\max }$, taking values in $\mathbb{F}$ and satisfying $\mathrm{fl}(0)=0$.

Unit roundoff. A real $u>0$ is a unit roundoff for the system when every real $t$ with $N_{\min }\le \mathrm{\mid }t\mathrm{\mid }\le N_{\max }$ satisfies

$\mathrm{fl}(t)=t(1+\delta )\text{for some }\delta \text{ with }\mathrm{\mid }\delta \mathrm{\mid }\le u,$

with $\mathrm{\mid }\cdot \mathrm{\mid }$ the absolute value (Absolute value in an ordered field). The model quantifies all subsequent error analysis through the single constant $u$ and says nothing about the internal bit representation beyond the bound $\mathrm{\mid }\delta \mathrm{\mid }\le u$.

The standard relative model. For each operation $\circ \in \{+,-,\times ,÷\}$ and all $x,y\in \mathbb{F}$ such that $x\circ y$ is zero or is a real number in the normal range ($N_{\min }\le \mathrm{\mid }x\circ y\mathrm{\mid }\le N_{\max }$; for $÷$ this includes $y\mathrm{\ne }0$), the computed result is

$\mathrm{fl}(x\circ y)=(x\circ y)(1+\delta )\text{for some }\delta \text{ with }\mathrm{\mid }\delta \mathrm{\mid }\le u\mathrm{.}$

When $x\circ y=0$, this identity is the exact-zero rule $\mathrm{fl}(0)=0$ and one may take $\delta =0$.

The exclusions are part of the model, and are stated explicitly.

• Overflow is excluded: when $\mathrm{\mid }x\circ y\mathrm{\mid }>N_{\max }$ the result is not a real number of the form $(x\circ y)(1+\delta )$ with $\mathrm{\mid }\delta \mathrm{\mid }\le u$.
• Underflow is excluded: when $0<\mathrm{\mid }x\circ y\mathrm{\mid }<N_{\min }$ the result falls into the subnormal or lost range and need not satisfy the relative bound.
• Invalid operations are excluded: when $x\circ y$ is not a real number (for example $0\mathrm{/}0$), there is no real $\delta$ for the displayed identity.

Every item on this page that uses the model therefore carries the hypotheses that the relevant operations lie in the normal range, and statements about overflow, underflow or invalid operations are made only where the exclusion is mentioned.

• The running example. In the IEEE-754 binary64 system the unit roundoff is $u=2^{-53}$, the largest normal magnitude is $N_{\max }=(2-2^{-52})2^{1023}\approx 1.798\cdot {10}^{308}$, and the smallest normal magnitude is $N_{\min }=2^{-1022}$. The counterexample page uses these three numbers to exhibit the excluded events.

• The exclusions are needed, not decorative. Each excluded event genuinely produces a value that is not of the form $(x\circ y)(1+\delta )$ with $\mathrm{\mid }\delta \mathrm{\mid }\le u$: overflow at $((3\mathrm{/}2)2^{1023})\cdot 2$ gives $+\mathrm{\infty }$, underflow at $2^{-1074}\mathrm{/}2$ rounds to $0$, and $0\mathrm{/}0$ gives $\mathrm{NaN}$.

• The model is an assumption, not a theorem about hardware. Statements proved under this model are conditional on the displayed inequalities; they say "if every operation rounds this accurately, then the following error bound holds". That is exactly what makes the bounds of The computed floating-point dot product obeys componentwise and normwise error bounds transportable to any hardware conforming to the model.