Calculating π with the Infinite series formula

Infinite series formula

The infinite series formula for calculating π is the following, according to Leibniz:

π/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 ...

Meaning that if we multiply 4 by all the series it would result π, well, eventually.

π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 ...

If we create an algoritm fast enough to support near infinite terms (it's impossible I know) we will have the best π approximation. The following is a simple approach to calculate π through the inifinte series approach.

function calculatePiByInfiniteSeries(numTerms) {
  let pi = 1;
  let sign = -1;

  for(let odd = 3; odd < numTerms; odd += 2) {
    pi += sign / odd;
    sign *= -1;
  }

  return 4 * pi;
}

A function to measure time execution was also created to measure how much this algorithm take to calcualte a π approximation.

function measureExecSeconds(f) {
  const firstDate = new Date();
  const result = f();
  const secondDate = new Date();
  const diff = secondDate - firstDate;
  const diffSeconds = diff / 1000

  return [`${diffSeconds} seconds`, result];
}

The implementation of these two functions would be the following:

console.log( measureExecSeconds(() => calculatePiByInfiniteSeries(1000000)) )

Results

To generate the results in time and aproximation the following implementation was created:

[
  Math.pow(10, 6), 
  Math.pow(10, 7), 
  Math.pow(10, 8), 
  Math.pow(10, 9), 
  Math.pow(10, 10),
  Math.pow(10, 11),
  Math.pow(10, 12),
].forEach(numTerms => {
  console.log( measureExecSeconds(() => calculatePiByInfiniteSeries(numTerms)) )
});

The following are the results in time and approximation of the algorithm given different powers of 10.

Number of terms	Time	~π
10^6	0.002 seconds	3.141590653589692
10^7	0.009 seconds	3.1415924535897797
10^8	0.075 seconds	3.1415926335902506
10^9	0.62 seconds	3.141592651589258
10^10	6.151 seconds	3.141592653388201
10^11	65.402 seconds	3.1415926535685275
10^12	655.923 seconds	3.1415926535865184

Google calculator π approximation

3.14159265359