Dyadic floating-point format and directed rounding

An exact dyadic rational is a number $m{2}^q$ with integers $m,q$. Fix integers $p\ge 2$ and $e_{\min }\le e_{\max }$. Put $\eta =2^{e_{\min }-p+1}$ and $N_{\max }=(2^p-1)2^{e_{\max }-p+1}$. The finite representable set $\mathbb{F}$ consists of zero, both signs of every normal number $k{2}^{e-p+1}$ with integers $e,k$ satisfying $e_{\min }\le e\le e_{\max }$ and $2^{p-1}\le k<2^p$, and both signs of every subnormal number $k\eta$ with integer $k$ satisfying $1\le k<2^{p-1}$. The smallest positive subnormal is $\eta$, the smallest positive normal is $2^{e_{\min }}$, and the largest finite magnitude is $N_{\max }$. The extended endpoint set also contains $-\mathrm{\infty }$ and $+\mathrm{\infty }$ as enclosure sentinels; they are not finite machine numbers.

For a real $x$ with $\mathrm{\mid }x\mathrm{\mid }\le N_{\max }$, define $\mathrm{rd}(x)$ as the greatest member of $\mathbb{F}$ not exceeding $x$, and $\mathrm{ru}(x)$ as the least member of $\mathbb{F}$ not less than $x$. Define $\rho (x)=\mathrm{ru}(x)-\mathrm{rd}(x)$. Set ${ϵ}_{\mathrm{a}\mathrm{b}\mathrm{s}}(0)=0$. For nonzero $x$ in the subnormal range, define the absolute-error bound ${ϵ}_{\mathrm{a}\mathrm{b}\mathrm{s}}(x)=\eta$; for $2^e\le \mathrm{\mid }x\mathrm{\mid }<2^{e+1}$ in the normal range, define ${ϵ}_{\mathrm{a}\mathrm{b}\mathrm{s}}(x)=2^{e-p+2}$. For $x>N_{\max }$, set $(\mathrm{rd}(x),\mathrm{ru}(x))=(N_{\max },+\mathrm{\infty })$; for $x<-N_{\max }$, set $(\mathrm{rd}(x),\mathrm{ru}(x))=(-\mathrm{\infty },-N_{\max })$. In either overflow case set ${ϵ}_{\mathrm{a}\mathrm{b}\mathrm{s}}(x)=+\mathrm{\infty }$ and report overflow, even when the selected directed endpoint is finite. The rounding procedure reports finite, subnormal, or overflow, so no relative-error assumption is implicit.

An exact rational intermediate may be converted by comparing integer numerators and denominators after clearing powers of two. A directed interval endpoint is the resulting greatest lower or least upper member of the extended endpoint set $\mathbb{F}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$.