Consider a $($binary$)$ floating point system of the form

$(1.s_1{s}_2{s}_3{)}_2{2}^m$ where min$[-31,32].$ Calculate the relative

error, with respect to $2-$norm if we convert the vector $n{R}^3$ to

the given floating point system. When converting to floating point,

first convert to a binary number then truncate any additional bits.

$x=\left[\begin{array}{c}1\mathrm{.}4375000\\ 7\mathrm{.}1250000\\ 7\mathrm{.}0000000\end{array}\right]$

Note: We are interested in the representation accuracy of floating

point. So assume that all operations performed $($addition$,$

subtraction, etc.$)$ do not increase the error.

relative error$(x)=$

number rtol $=0\mathrm{.}01,$ atol $=0\mathrm{.}001$