# Machine Floating Point Theorem

Completely stuck on this floating point question.

Let $x \in \mathbb{R}$ have the following floating point representation: $$x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e$$ [Where $\beta$ is the base] Define the floating point round off to $t$ significant figures to be:

$$fl(x) = (-1)^s[0.a_1a_2\dots\tilde{a}_t] , \quad \tilde{a}_t = \begin{cases} a_t &\text{if}\; a_{t + 1} < \beta /2 \\ a_t + 1 &\text{if}\; a_{t + 1} \geq \beta /2 \end{cases}$$ And the following flop computation for $x, y \in \mathbb{R}$ to be: $$x \ominus y := fl(fl(x) - fl(y))$$ Given that the relative error follows: $$\frac{|x - fl(x)|}{|x|} \leq u \left[= \frac{1}{2}\beta^{1-t} \right]$$ Show that [triangle inequality may help]: $$\frac{x \ominus y - (x - y)}{|x - y|} \leq u(2 + u)\frac{|x| + |y|}{|x - y|}$$

Any ideas?

• What is $\beta$, is it the base? – achille hui Oct 19 '14 at 2:02
• yep, just changed it – user150324 Oct 19 '14 at 14:43

## 1 Answer

I am stuck with a question very similar to yours, but I can answer yours.

\begin{equation*} \left\vert \frac{x \ominus y - (x - y)}{x - y} \right\vert \leq \left\vert \frac{x \ominus y - (fl(x) - fl(y))}{x - y} \right\vert + \left\vert \frac{fl(x) - fl(y) - (x - y)}{x - y}\right\vert. \end{equation*}

Let's find an upper bound for the first addend.

\begin{equation*} \left\vert \frac{x \ominus y - (fl(x) - fl(y))}{x - y}\right\vert = \left\vert \frac{fl(fl(x) - fl(y)) - (fl(x) - fl(y))}{x - y}\right\vert = \left\vert \frac{(fl(x) - fl(y))(1 + \delta) - (fl(x) - fl(y))}{x - y}\right\vert = \left\vert \frac{(fl(x) - fl(y))\delta}{x - y}\right\vert \leq \left\vert \frac{(fl(x) - fl(y))}{x - y}\right\vert u \leq \left\vert \frac{fl(x) - x}{x}x + x + \frac{fl(y) - y}{y}y + y\right\vert \frac{u}{\left\vert x - y\right\vert} \leq \left(\left\vert \frac{fl(x) - x}{x}\right\vert \left\vert x\right\vert + \left\vert x\right\vert + \left\vert\frac{fl(y) - y}{y}\right\vert \left\vert y\right\vert + \left\vert y\right\vert\right) \frac{u}{\left\vert x - y\right\vert} \leq (\left\vert x\right\vert + \left\vert y\right\vert)(u + 1)\frac{u}{\left\vert x - y\right\vert}. \end{equation*}

Let's find an upper bound for the second addend. \begin{equation*} \left\vert \frac{fl(x) - fl(y) - (x - y)}{x - y}\right\vert \leq \left\vert \frac{fl(x) - x}{x - y}\right\vert + \left\vert \frac{fl(y) - y}{x - y}\right\vert = \left\vert \frac{fl(x) - x}{x}\frac{x}{x - y}\right\vert + \left\vert \frac{fl(y) - y}{y}\frac{y}{x - y}\right\vert \leq u\frac{\left\vert x\right\vert + \left\vert y\right\vert}{\left\vert x - y\right\vert}. \end{equation*}

Putting all together we find \begin{equation*} \left\vert \frac{x \ominus y - (fl(x) - fl(y))}{x - y} \right\vert + \left\vert \frac{fl(x) - fl(y) - (x - y)}{x - y}\right\vert \leq (\left\vert x\right\vert + \left\vert y\right\vert)(u + 1)\frac{u}{\left\vert x - y\right\vert} + u\frac{\left\vert x\right\vert + \left\vert y\right\vert}{\left\vert x - y\right\vert} = \frac{\left\vert x\right\vert + \left\vert y\right\vert}{\left\vert x - y\right\vert}((u+1)u + u) = \frac{\left\vert x\right\vert + \left\vert y\right\vert}{\left\vert x - y\right\vert}((u+2)u). \end{equation*}

So your problem is solved. This is mine, very linked to yours Relative error of machine summation.