Question $1.[20$ marks$]$ Consider IEEE single$-$precision floating$-$point representation with $1$

bit for sign, $8$ bits for biased exponent, and $23$ bits for mantissa.

$(1.$a$)$ marks$]$ Let $x=2^5+2^{-16}+2^{-19}+2^{-21}.$ Find the machine number closest to $x.$

Denote this machine number by $x^{**}.$ What are the absolute and relative errors, respectively,

when converting $x$ to $x^{**}?$

$(1.$b$)[8$ marks$]$ Show that the relative error in converting a real number to a machine number

in this representation is no greater than the machine epsilon $lo{n}_M=2^{-24}.$ Here it is assumed

that the real number lies between the smallest machine number and the largest machine num$-$

ber of this representation.

$(1.$c$)[6$ marks$]$ Consider the function $f(x,y)=x+y$ and assume $x>0$ and $y>0.$ Suppose

that the input values $x$ and $y$ have errors $x$ and $y$ such that their relative errors satisfy

$|x\frac{|}{|}x|$ and $|y\frac{|}{|}y|$ for some constant $>0.$ Show that the relative error of the

output $f(x,y)$ thus caused is also bounded by $$; that is$,$ letting $f$ denotd the error of $f(x,y),$

there is $|f\frac{|}{|}f(x,y)|.$