Let $\mathbb{F}(\beta,t,e_{\min},e_{\max})$ be a Floating Point Arithmetic. Let $\text{domain}(\mathbb{F}) = [x_{\min},x_{\max}] \subseteq \mathbb{R}$ for minimal and maximal elements $x_{\min},x_{\max} \in \mathbb{F}$. Let $\text{rd}(x): \text{domain}(\mathbb{F}) \longrightarrow \mathbb{F}$ be a rounding operation in accordance with the IEEE standards for rounding.
Under the above assumptions it is known, that for all $x \in \text{domain}(\mathbb{F})$ a $\delta \in [-u(\mathbb{F}),u(\mathbb{F})]$ exists such that $$rd(x) = x(1+\delta),$$ where $u(\mathbb{F}) := \frac{1}{2}\beta^{1-t}$ is the unit roundoff of $\mathbb{F}$. Now, throughout the literature it is often mentioned that the above expression can equivalently formulated as $$rd(x) = \frac{x}{1+\delta^{\ast}}$$ for some $\delta^{\ast} \in [-u(\mathbb{F}),u(\mathbb{F})]$. While conceptually the alternative expression does make sense to me, I am unable to formalise the equivalence of the expressions. How might one progress in order to so do?
The two representations are not equivalent, but they can be derived using the same basic principles. Let $$x = (1.f_1f_2\dots)_2 \cdot 2^m$$ denote a positive real number in the representational range. Let $$x_- = (1.f_1f_2\dots f_k)_2 \cdot 2^m$$ denote the largest machine number which is less than or equal to $x$ and let $$x_+ = x_1 + 2^{m-k}$$ denote the next floating point number. The floating point representation $\text{fl}(x)$ of $x$ is $x_-$ when $x_-$ is closer to $x$ than $x_+$ and vise versa. Any ties are resolved as dictated by the rounding mode. In any case, we have $$|x- \text{fl}(x)| \leq \frac{1}{2} 2^{m-k} = 2^m u,$$ where $u = 2^{-k-1}$ is the unit roundoff.
This is the point where the analysis branches.
Both representations are useful when analyzing floating point calculations.