Recall that a floating$-$point number can be expressed as

where $s,f$ and $e$ are binary numbers. Recall that $($s$)$ is the sign indicator, $($f$)$ the mantissa $($or

fractional part$),$ and $($e$)$ the exponent.

Suppose the sign is $1$ bit long, the mantissa $15$ bits long, and the exponent has $7$ bits. What are the

largest and smallest positive numbers in this computer system?

a$)$ Express the decimal number $1830\mathrm{.}567$ as a binary fraction.

b$)$ Express the binary fraction $10010111\mathrm{.}010001$ as a decimal number.

a$)$ After adding each term, compute the true and the approximate percent relative errors.

b$)$ Continue the iterative process of adding one term at a time until the approximate

percentrelative error falls below an error criterion for $4$ significant figures $($by hand$).$

Evaluate the polynomials. $P(x)=x^3-6x^2+12x-8,Q(x)=((x-6)x+12)x-8,$ and $R(x)=$

$(x-2{)}^3$

a$.$ Using four$-$digit rounding arithmetic compute $P(3\mathrm{.}24),Q(3\mathrm{.}24),$ and $R(3\mathrm{.}24).$

b$.$ Compute the relative error for each, $lo{n}_t=?$ assuming the true value as $1\mathrm{.}906624.$