optimize the below code to acheive $4\mathrm{.}5$x speed

#include

#include

#include

#include

#include

#define N $1000000000//$ The upper limit for prime generation

void sieveOfEratosthenesAVX$2($int n$)\{$

char$*$ sieve $=($char$*)$malloc$($n $+1)$;

for $($int i $=0$; i $<=$ n; i$++)\{$

sieve$[$i$]=1$;

$\}$

int sqrtN $=($int$)$sqrt$($n$)$;

for $($int p $=2$; p $<=$ sqrtN; p$++)\{$

if $($sieve$[$p$])\{$

$\_\_$m$256$i avx$\_$p $=\_$mm$256\_$set$1\_$epi$64$x$(($char$)$p$)$;

for $($int i $=$ p $*$ p; i $<=$ n; i $+=$ p$)\{$

if $($i $\%32==0$ && i $+31<=$ n$)\{$

$//$ Use AVX$2$ to unset multiple multiples of p at once

for $($int j $=0$; j $<32$; j$++)\{$

$\_$mm$256\_$storeu$\_$si$256((\_\_$m$256$i$*)$&sieve$[$i $+$ j$],\_$mm$256\_$and$\_$si$256($avx$\_$p$,\_$mm$256\_$setzero$\_$si$256()))$;

$\}$

i $+=31$;

$\}$

else $\{$

sieve$[$i$]=0$;

$\}$

$\}$

$\}$

$\}$

$//$ For measuring time, we only calculate the number of prime numbers $($not printing them$)$

int count $=0$;

for $($int p $=2$; p $<=$ n; p$++)\{$

if $($sieve$[$p$])\{$

count$++$;

$\}$

$\}$

free$($sieve$)$;

$\}$

int main$()\{$

clock$\_$t start$\_$time $=$ clock$()$;

printf$("$Prime numbers up to $\%$d:

$",$ N$)$;

sieveOfEratosthenesAVX$2($N$)$;

clock$\_$t end$\_$time $=$ clock$()$;

double execution$\_$time $=($double$)($end$\_$time $-$ start$\_$time$)/$ CLOCKS$\_$PER$\_$SEC;

printf$("$

Execution time: $\%$f seconds

$",$ execution$\_$time$)$;

return $0$;

$\}$

my original code:

#include

#include

#include

#include

#define N $1000000000//$ The upper limit for prime generation

void sieveOfEratosthenes$($int n$)\{$

char$*$ sieve $=($char$*)$malloc$($n $+1)$;

for $($int i $=0$; i $<=$ n; i$++)\{$

sieve$[$i$]=1$;

$\}$

int sqrtN $=($int$)$sqrt$($n$)$;

for $($int p $=2$; p $<=$ sqrtN; p$++)\{$

if $($sieve$[$p$])\{$

for $($int i $=$ p $*$ p; i $<=$ n; i $+=$ p$)\{$

sieve$[$i$]=0$;

$\}$

$\}$

$\}$

for $($int p $=2$; p $<=$ n; p$++)\{$

if $($sieve$[$p$])\{$

$//$ Print the prime numbers $($comment this out if you only want to measure time$)$

$//$ printf$("\%$d $",$ p$)$;

$\}$

$\}$

free$($sieve$)$;

$\}$

int main$()\{$

clock$\_$t start$\_$time $=$ clock$()$;

printf$("$Prime numbers up to $\%$d:

$",$ N$)$;

sieveOfEratosthenes$($N$)$;

clock$\_$t end$\_$time $=$ clock$()$;

double execution$\_$time $=($double$)($end$\_$time $-$ start$\_$time$)/$ CLOCKS$\_$PER$\_$SEC;

printf$("$

Execution time: $\%$f seconds

$",$ execution$\_$time$)$;

return $0$;

$\}$