# $fl(x)=x(1+\delta)$

The floating point representation of a real number $$x$$ in a machine is given by $$fl(x)=x(1+\delta),\: |\delta| = \frac{|x^*-x|}{|x|} \le \epsilon$$.

But I do not find this equation very insightful. Insert $$\delta = \frac{x^*-x}{x}$$ in the equation and you get $$x^*$$. So $$fl(x)$$ is just $$x^*$$. Why write $$x^*$$ in this fancy way: $$fl(x)=x(1+\delta)$$

Does equation have a name by the way?

The kind of guarantee that numerical algorithms can make is: if the input is correct up to a multiplicative error of $$1 \pm \delta$$, then the output will be correct up to a multiplicative error of $$1 \pm \epsilon$$. So $$\delta$$ (in your case, the machine $$\delta$$) measures the relative error involved when expressing real numbers as floating-point numbers, which is the information required to deduce the accuracy of numerical algorithms.