$25$ points$)[$Float Point Design$]$ In a hypothetical $5-$bit floating point format, $3$ leading bits

are used for exponent, E$,$ in two's compliment format and the two remaining bits are for the

fraction F$,$ where the number is determined using x $=1.$F x $2$E$.$ Answer the following

questions:

$($a$)$ What is the decimal value of the five bits set to $01101?$

$($b$)$ What is the decimal value of the five bits set to $10010?$

$($c$)$ What is the range of this format?

$($d$)$ How would a $7$ get represented in these five bits?

$($e$)$ How would $0\mathrm{.}375$ get represented?

$($f$)$ How many distinct numbers can be represented in this format?

$2.(15$ points$)[$Float Point Design$]$ From problem $1,$ if we change how a number is determined

using the following functions, what is the new range, and how many distinct numbers can be

represented?

$($a$)$ x $=$ F$\mathrm{.}0$ x $2$E$.($b$)$ x $=0.$F x $2$E$.($c$)$ x $=$ F$\mathrm{.}1$ x $2$E$.$

$3.(24$ points$)[$IEEE$-754]$ Determine the floating$-$point representation in IEEE$-754$ format of

the following numbers:

$($a$)-101\mathrm{.}4$

$($b$)2\mathrm{.}5$ x $10-3$

$($c$)1\mathrm{.}001$ x $2-142$

$($d$)0\mathrm{.}001$ x $2-146$

$4.(12$ points$)[$IEEE$-754$ Arithmetic$]$ Add the following two IEEE$-754$ formatted numbers

together and report the results in the same format?

$($a$)$ a $=3$f$200000,$ b $=$ be$600000$

$($b$)$ a $=3$f$200000,$ b $=$ ff$800000$

$($c$)$ a $=01100000,$ b $=80$e$00000$

$($d$)$ a $=3$f$200000,$ b $=80$e$00000$

$5.(12$ points$)[$IEEE$-754$ Arithmetic$]$ Multiply the IEEE$-754$ formatted numbers from previous

problem $4$ together and report the results in the same format?

$6.(12$ points$)[$IEEE$-754$ Arithmetic$]$ Divide the IEEE$-754$ formatted numbers from previous

problem $4$ in the form a $/$ b and report the results in the same format?