How can I optimize my program? - c++

Question

I wrote program which realises this formula:
Pi = 1/n * summ( 4 / ( 1 + ((i-0.5) /n)^2)
Program code:
#include <iostream>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <sys/time.h>
using namespace std;
const long double PI = double(M_PI);
int main(int argc, char* argv[])
{
typedef struct timeval tm;
tm start, end;
int timer = 0;
int n;
if (argc == 2) n = atoi(argv[1]);
else n = 8000;
long double pi1 = 0;
gettimeofday ( &start, NULL );
for(int i = 1; i <= n; i++) {
pi1 += 4 / ( 1 + (i-0.5) * (i-0.5) / (n*n) );
}
pi1/=n;
gettimeofday ( &end, NULL );
timer = ( end.tv_usec - start.tv_usec );
long double delta = pi1 - PI;
printf("pi = %.12Lf\n",pi1);
printf("delta = %.12Lf\n", delta);
cout << "time = " << timer << endl;
return 0;
}
How to present it in an optimal way? when there will be less floating-point operations in this part:
for(int i = 1; i <= n; i++) {
pi1 += 4 / ( 1 + (i-0.5) * (i-0.5) / (n*n) );
}
Thanks

Answer 1

one idea will be:
double nn = n*n;
for(double i = 0.5; i < n; i += 1) {
pi1 += 4 / ( 1 + i * i / nn );
}
but you need to test if it is any difference with current code.

Answer 2

I suggest you read this excellent document:
Software Optimization Guide for AMD64 Processors
Which is also great when you do not have an AMD processor.
But if I were you, I would replace the whole calculation loop with just
pi1 = M_PI;
Which will probably be the fastest... If you are actually interested in a faster algorithm for Pi calculations, look at the Wikipedia article: Category:Pi algorithm
If you just want to microoptimize your code, read the above mentioned software optimization guide.

Answer 3

Examples of simple optimization:
compute double one_per_n = 1/n; outside the for loop reducing the cost of dividing by non each iteration
compute double j = (i-0.5) * one_per_n inside the loop
pi1 += 4 / (1 + j*j);
This should be faster and also avoid the integer overflow you have for greater values of n. For even more optimized code you will have to look at the generated code and use a profiler to make appropriate changes. The optimized code this way might behave differently on machines with a different CPU or cache.... Avoiding divisions is something that is always good to do to save computation time.

Answer 4

#include <iostream>
#include <cmath>
#include <chrono>
#ifndef M_PI //M_PI is non standard make you sure catch this case
#define M_PI 3.14159265358979323846
#endif
typdef long double float_t;
const float_t PI = double(M_PI);
int main(int argc, char* argv[])
{
int n = argc == 2 ? atoi(argv[1]) : 8000;
float_t pi1=0.0;
//if you can using auto here is a no brainer
std::chrono::time_point start
=std::chrono::system_clock::now();
unsigned n2=n*n;
for(unsigned i = 1; i <= n; i++)
{
pi1 += 4.0 / ( 1.0 + (i-0.5) * (i-0.5) / n2 );
}
pi1/=n;
std::chrono::duration<double> time
=std::chrono::system_clock::now()-start;
float_t delta = pi1 - PI;
std::cout << "pi = " << std::setprecision(12) << pi1
<< "delta = " << std::setprecision(12) << delta
<< "\ntime = " << time.count() << std::endl;
return 0;
}