Optimized c++ function for nth prime number execution time - c++

Question

I am implementing a c++ function to get Nth prime number using some predefined indices for time optimization purpose.
my code is :
// file prime.cpp
#include <iostream>
#include <time.h>
using namespace std;
/*
#define primeAt10000 104743
#define primeAt20000 224743
#define primeAt30000 350381
#define primeAt40000 479951
#define primeAt50000 611977
*/
int prime(int n){
int pos = 1,i = 1,temp;
if(n==0)
return 2;
/*
else if(n>50000){
i = primeAt50000;
pos = 50001;
}else if(n>40000){
i = primeAt40000;
pos = 40001;
}else if(n>30000){
i = primeAt30000;
pos = 30001;
}else if(n>20000){
i = primeAt20000;
pos = 20001;
}else if(n>10000){
i = primeAt10000;
pos = 10001;
}*/
while( i+=2 ){
temp = i/2+1;
for(int j = 3 ; j <= temp ; j+=2)
if(i%j == 0)
goto con;
if(pos++ >= n)
return i;
con :;
}
}
int main(int argc, char const *argv[]){
int index;
cin >> index;
clock_t start = clock();
cout << prime(index)<<endl;
cout << (clock()-start)/CLOCKS_PER_SEC<<"sec"<< endl;
return 0;
}
compiled with:
g++ prime.cpp -o prime.exe
I ran this code three times for inputs 9999, 19999 and 29999
1st run : 1sec 6sec 14sec
2nd run : 1sec 7sec 15sec
3rd run : 1sec 7sec 16sec
After enabling commented code again I ran three times with same inputes
1st run : 1sec 5sec 8sec
2nd run : 1sec 5sec 8sec
3rd run : 1sec 5sec 8sec
My question is :
Why this difference in taken time for each execution after second compilation while the loops are running everytime for ~1,25,000 times?
and
Why for input 19999 (~104743 looping times) it is much closer then the first 3 runs after first compilation (~224743 looping times)?

Answer 1

Difference in time for each 9999 interval is different because when we going towards larger numbers to check either it is prime or not it takes more time then smaller ones.
In other words directly We can say that the run-time of for-loop in prime() is increased because of larger value of variable temp.
when we checking for i = 101, the value of temp become 51 and for-loop will run approx 25 times.
while when we check for i = 10001, the value of temp become 5001 and for-loop will run for approx 2500 times.
this difference in run-time of for loop will increase your overall time complexity.

Answer 2

After some discussion with #JonathanLeffler I have further optimized this function to achieve fastest output for larger input values like for index 9999, 19689 and so on...
Now the complexity of my prime function is (N^2)/12 unlike before [it was (N^2)/8].
My new code is :
#include <iostream>
#include <time.h>
using namespace std;
#define primeAt10000 104743-7
#define primeAt20000 224743-7
#define primeAt30000 350381-7
#define primeAt40000 479951-7
#define primeAt50000 611977-7
bool checkPrime(int x){
int temp = x/2+1;
for(int j = 3 ; j <= temp ; j+=2)
if(x%j == 0)
return false;
return true;
}
int prime(int n){
int pos = 2,i = 0;
if(n==0)
return 2;
else if(n==1)
return 3;
else if(n>50000){
i = primeAt50000;
pos = 50000;
}else if(n>40000){
i = primeAt40000;
pos = 40000;
}else if(n>30000){
i = primeAt30000;
pos = 30000;
}else if(n>20000){
i = primeAt20000;
pos = 20000;
}else if(n>10000){
i = primeAt10000;
pos = 10000;
}
while( i+=6 ){
if(checkPrime(i-1))
if(pos++>=n)
return i-1;
if(checkPrime(i+1))
if(pos++>=n)
return i+1;
}
return 0;
}
int main()
{
int index;
cin >> index;
clock_t start = clock();
cout << prime(index)<<endl;
cout << (clock()-start)/(float)CLOCKS_PER_SEC<<"sec";
return 0;
}
Compiled with(as the advice of #NathanOliver && #JonathanLeffler) :
g++ -O3 -Wall -Werror -Wextra prime.cpp -o prime.exe
Now prime.exe taking 1.34, 4.83 and 7.20sec respectivly to inputs 9999, 19999 and 29999.