Question

I have just made a program which calculates pi. However, even with 10 million iterations my result is kinda off. I get 3.1415927535897831, whereas already early on is it wrong. It is supposed to be 3.141592653589793238…

So my question is: What is the required amount of iterations to get at least an accurate answer all the way up to 10^-16

Here is my code if anyone is interested:

#include <iostream>
#include <iomanip>
using namespace std;

int main()
{
long double pi = 4.0;
long double tempPi;
for (int i = 1, j = 3; i <= 10000000; i++, j+=2)
{
    tempPi = static_cast<double>(4)/j;
    if (i%2 != 0)
    {
            pi -= tempPi;
    }
    else if (i%2 == 0)
    {
         pi += tempPi;
    }
}        
cout << "Pi has the value of: " << setprecision(16) << fixed << pi << endl;
system("pause");
return 0;

}

Any performance related tips would also be appreciated.

Answer 1

You are using the Leibniz series, which is very, very slow to converge. In an alternating series such as the one you are using, the first omitted term provides a good estimate of the error in the estimate. Your first omitted term is 4/2000005, so you should expect less than six significant digits of precision here.

Note well: Rounding errors, use of double precision numbers has nothing to do with the lack of precision here. The sole factor is that you are using a crappy algorithm.