C Programming: How to calculate Pi to 4th decimal place?

Question

Okay, I want to make a C program that calculates pi accurately to 4th decimal place (3.1415...). I thought that double is more accurate than float type... Even with a trillion terms (n=trillion), the program cannot go past 3.1414... Can someone help? Am I using an incorrect data type to store my Pi value or is my loops incorrect?

#include <stdio.h> #include <stdlib.h> int main(int argc, char** argv) { int n; while(1){ printf("Please enter how many terms (n) you wish to add to approximate Pi: "); scanf("%d", &n); if(n>=1) break; } int x; int count =2; double negSum=0; double posSum=0; double pi = 0; for(x=1;x<=n;x++){ do{ if(x%2==1){ posSum += (4.0)/(2.0*x-1.0); count++; } else{ negSum += (-4.0)/(2.0*x-1.0); count++; } pi = negSum + posSum; } while(pi>3.1414999 && pi<3.14160000); } //pi = negSum + posSum; printf("The value of Pi using your approximation is %f, and the iteration was %d", pi, count); return (EXIT_SUCCESS); }

Here is some of my sample input/output:

Please enter how many terms (n) you wish to add to approximate Pi: 98713485 The value of Pi using your approximation is 3.141407, and the iteration was 98713488

Why are you continuing the approximation while(pi>3.1414999 && pi<3.14160000);? Shouldn’t it be the other way around? Also, you don’t need two different running totals. Just add to pi. — Ry-
– Ry- ♦, Commented Sep 13, 2014 at 20:26
Using basic arithmetics for calculating PI has links to various options for calculation. — Richard Chambers
– Richard Chambers, Commented Sep 13, 2014 at 20:40

Jeremy West · Accepted Answer · 2014-09-13 20:38:12Z

The series you are using:

pi = 4(1 - 1/3 + 1/5 - 1/7 + 1/9 ...)

converges REALLY slowly to pi. It is the evaluation of a Taylor series for 4arctan(x) at x=1 and converges conditionally (it is right on edge of the interval of convergence). That's not going to be a very numerically efficient way to compute pi.

Beyond that, I haven't carefully checked your implementation, but some others have pointed out problems in the comments.

Community · Accepted Answer · 2017-05-23 10:27:28Z

To compute Pi to 4th decimal place, you could use Gauss-Legendre algorithm:

#include <math.h> #include <stdio.h> int main(void) { const double PI = acos(-1), SQRT2 = sqrt(2.0); double a = 1, b = 1/SQRT2, t = .25, p = 1; double an, piold, pi = 1, eps = 1e-6; /* use +2 decimal places */ int iteration_count = 0; do { ++iteration_count; an = .5 * (a + b); b = sqrt(a * b); t -= p * (a - an) * (a - an); a = an; p *= 2; piold = pi; pi = (a + b) * (a + b) / (4 * t); } while (fabs(pi - piold) > eps); printf("got pi=%f with rel. err=%.2e in %d iterations\n", pi, (pi - PI) / PI, iteration_count); return 0; }

To run it:

$ gcc *.c -lm && ./a.out

Output

got pi=3.141593 with rel. err=2.83e-16 in 3 iterations

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