Calculating Pi using for loop

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 <= 1000开发者_运维技巧0000; 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.

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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.

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There are lots of methods for calculating pi. Some converge faster than others.

Also see "Modern Formulae"

the sequence 1 / a converges quartically to pi, giving about 100 digits in three steps and over a trillion digits after 20 steps.

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The problem is that double is not nearly as accurate as you hope. You can't even represent decimal 1.2 with 100% accuracy.

I didn't look at the code closely to see if there are other problems.

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Since the result is wrong after 10 million iterations due to round-up errors, you won't get the correct answer with more loops, only adding more error.