Problem $2(20$ points$)$

Consider the representation of numbers in standard floating$-$point form:

$x=(-1{)}^S(1.f)2^{(b-\text{b}\text{i}\text{a}\text{s})}$

Now, consider a hypothetical computer, that represents floating point numbers with $12$ bits, allocating $4$

bits to the exponent, and $7$ bits for the fraction.

$($a$)(10$ points$)$ What is the smallest positive number greater than zero that can be represented on this

computer?

$($b$)(5$ points$)$ What is machine precision, $,$ for this computer?

$($c$)(5$ points$)$ Discuss the tradeoffs between the allocation of bits to the exponent versus the

fraction. In particular, as bits are re$-$allocated from the exponent to the fraction, what happens

to largest finite number, the smallest non$-$zero number, and machine epsilon? Explain briefly