write a more efficient version of the function called poly$\_$opt that takes the same parameters in the same order and returns the same result. double poly$\_$opt$($double a$[],$double x$,$ long degree$)$

$\{$

$\}$ Practice Problem $5\mathrm{.}5($solution page $611)$

Suppose we wish to write a function to evaluate a polynomial, where a polynomial of degree $\backslash ($ n $\backslash )$ is defined to have a set of coefficients $\backslash ($ a$\_\{0\},$ a$\_\{1\},$ a$\_\{2\},\backslash$ldots$,$ a$\_\{$n$\}\backslash ).$ For a value $\backslash ($ x $\backslash ),$ we evaluate the polynomial by computing

$\backslash [$

a$\_\{0\}+$a$\_\{1\}$ x$+$a$\_\{2\}$ x$^\{2\}+\backslash$cdots$+$a$\_\{$n$\}$ x$^\{$n$\}$

$\backslash ]$

This evaluation can be implemented by the following function, having as arguments an array of coefficients a$,$ a value x $,$ and the polynomial degree degree $($the value $\backslash ($ n $\backslash )$ in Equation $5\mathrm{.}2).$ In this function, we compute both the successive terms of the equation and the successive powers of $\backslash ($ x $\backslash )$ within a single loop:

$```$

double poly$($double a$[],$ double x$,$ long degree$)$

$\{$

long i;

double result $=$ a$[0]$;

double xpwr $=$ x; $/*$ Equals x$^$i at start of loop $*/$

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

result $+=$ a$[$i$]*$ xpwr;

xpwr $=$ x $*$ xpwr;

$\}$

return result;

$\}$

$```$

A$.$ For degree $\backslash ($ n $\backslash ),$ how many additions and how many multiplications does this code perform?

B$.$ On our reference machine, with arithmetic operations having the latencies shown in Figure $5\mathrm{.}12,$ we measure the CPE for this function to be $5\mathrm{.}00.$ Explain how this CPE arises based on the data dependencies formed between iterations due to the operations implementing lines $\backslash (7-8\backslash )$ of the function. Practice Problem $5\mathrm{.}6($solution page $611)$

Let us continue exploring ways to evaluate polynomials, as described in Practice Problem $5\mathrm{.}5.$ We can reduce the number of multiplications in evaluating a polynomial by applying Horner's method, named after British mathematician William G$.$ Horner $(1786-1837).$ The idea is to repeatedly factor out the powers of $\backslash ($ x $\backslash )$ to get the following evaluation:

$\backslash [$

a$\_\{0\}+$x$\backslash$left$($a$\_\{1\}+$x$\backslash$left$($a$\_\{2\}+\backslash$cdots$+$x$\backslash$left$($a$\_\{$n$-1\}+$x a$\_\{$n$\}\backslash$right$)\backslash$cdots$\backslash$right$)\backslash$right$)$

$\backslash ]$

Using Horner's method, we can implement polynomial evaluation using the following code:

$```$

$/*$ Apply Horner's method $*/$

double polyh$($double a$[],$ double x$,$ long degree$)$

$\{$

$```$

$```$

long i;

double result $=$ a$[$degree$]$;

for $($i $=$ degree$-1$; i $>=0$; i$--)$

result $=$ a$[$i$]+$ x$*$result;

return result;

$\}$

$```$

A$.$ For degree $\backslash ($ n $\backslash ),$ how many additions and how many multiplications does this code perform?

B$.$ On our reference machine, with the arithmetic operations having the latencies shown in Figure $5\mathrm{.}12,$ we measure the CPE for this function to be $8\mathrm{.}00.$ Explain how this CPE arises based on the data dependencies formed between iterations due to the operations implementing line $7$ of the function.

C$.$ Explain how the function shown in Practice Problem $5\mathrm{.}5$ can run faster, even though it requires more operations. Practice Problem $5\mathrm{.}5($solution page $611)$

Suppose we wish to write a function to evaluate a polynomial, where a polynomial of degree $\backslash ($ n $\backslash )$ is defined to have a set of coefficients $\backslash ($ a$\_\{0\},$ a$\_\{1\},$ a$\_\{2\},\backslash$ldots$,$ a$\_\{$n$\}\backslash ).$ For a value $\backslash ($ x $\backslash ),$ we evaluate the polynomial by computing

$\backslash [$

a$\_\{0\}+$a$\_\{1\}$ x$+$a$\_\{2\}$ x$^\{2\}+\backslash$cdots$+$a$\_\{$n$\}$ x$^\{$n$\}$

$\backslash ]$

This evaluation can be implemented by the following function, having as arguments an array of coefficients a$,$ a value x $,$ and the polynomial degree degree $($the value $\backslash ($ n $\backslash )$ in Equation $5\mathrm{.}2).$ In this function, we compute both the successive terms of the equation and the successive powers of $\backslash ($ x $\backslash )$ within a single loop:

$```$

double poly$($double a$[],$ double x$,$ long degree$)$

$\{$

long i;

double result $=$ a$[0]$;

double xpwr $=$ x; $/*$ Equals x$^$i at start of loop $*/$

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

result $+=$ a$[$i$]*$ xpwr;

xpwr $=$ x $*$ xpwr;

$\}$

return result;

$\}$

$```$

A$.$ For degree $\backslash ($ n $\backslash ),$ how many additions and how many multiplications does this code perform?

B$.$ On our reference machine, with arithmetic operations having the latencies shown in Figure $5\mathrm{.}12,$ we measure the CPE for this function to be $5\mathrm{.}00.$ Explain how this CPE arises based on the data dependencies formed between iterations due to the operations implementing lines $\backslash (7-8\backslash )$ of the function. Practice Problem $5\mathrm{.}6($solution page $611)$

Let us continue exploring ways to evaluate polynomials, as described in Practice Problem $5\mathrm{.}5.$ We can reduce the number of multiplications in evaluating a polynomial by applying Horner's method, named after British mathematician William G$.$ Horner $(1786-1837).$ The idea is to repeatedly factor out the powers of $\backslash ($ x $\backslash )$ to get the following evaluation:

$\backslash [$

a$\_\{0\}+$x$\backslash$left$($a$\_\{1\}+$x$\backslash$left$($a$\_\{2\}+\backslash$cdots$+$x$\backslash$left$($a$\_\{$n$-1\}+$x a$\_\{$n$\}\backslash$right$)\backslash$cdots$\backslash$right$)\backslash$right$)$

$\backslash ]$

Using Horner's method, we can implement polynomial evaluation using the following code:

$```$

$/*$ Apply Horner's method $*/$

double polyh$($double a$[],$ double x$,$ long degree$)$

$\{$

$```$

$```$

long i;

double result $=$ a$[$degree$]$;

for $($i $=$ degree$-1$; i $>=0$; i$--)$

result $=$ a$[$i$]+$ x$*$result;

return result;

$\}$

$```$

A$.$ For degree $\backslash ($ n $\backslash ),$ how many additions and how many multiplications does this code perform?

B$.$ On our reference machine, with the arithmetic operations having the lat