Consider that floating$-$point numbers are represented in a $16$ bit format as shown below.

where

$-$ The scale factor $($exponent$)$ is an implied base of $2$ and a $7-$bit, excess $63$ exponent, with the two end values of $0$ and $127$ use to signify exact $0$ and infinity, respectively.

$-$ The $8-$bit mantissa is normalized, with an implied $1$ to the left of the binary point.

$-$ The sign bit is $1$ bit.

Answer the following questions

$-$ Represent the numbers $2\mathrm{.}125$ and $-1\mathrm{.}012$ in this format.

$-$ Describe two steps to add floating point numbers $0\mathrm{.}0125$ and $20\mathrm{.}0.$

$-$ The following are three solutions to add $\backslash (0\mathrm{.}01,0\mathrm{.}02,0\mathrm{.}03,\backslash$ldots$,1\mathrm{.}00\backslash ),$ in total $100$ floating point numbers with increment $0\mathrm{.}01.$ Consider how we add two floating point numbers and explain which one would yield most accurate result and what the main problems are with the other two.

Listing $1$: snippet $1$

$```$

double sum $=0\mathrm{.}0$;

for $($double f$=0\mathrm{.}0$;f;f$+=0\mathrm{.}01)\{$

sum $+=$ f;

$\}$

$```$

Listing $2$: snippet $2$

$```$

double sum $=0\mathrm{.}0$;

for $($int i$=1$; i; i$++)\{$

sum $+=0\mathrm{.}01*$ ; ;

$\}$

$```$

Listing $3$: snippet $3$

$```$

double sum $=0\mathrm{.}0$;

for $($int i $=100$; i$>=1$; i $--)\{$

sum $+=0\mathrm{.}01*$ i;

$\}$

$```$