Approximating Pi with Polygons

Question

I wrote a program that approximates Pi by using polygons. I used the formulars in the picture beyond. In my code they are called innerPoly (\$c_{2n}\$) and outerPoly (\$C_{2n}\$).

And since you can calculate the circumference of a 2n-polygon with knowing the circumference of a n-polygon you will get the circumferences \$C_8\,ドル \$C_{16}\,ドル \$C_{32}\$ etc., knowing \$C_4\$.

$$ \begin{array}{l} c_{2n} =& 2 \sqrt{2n^2-n\sqrt{(2n)^2-c_n^2}} \qquad&\textrm{for the inner polygon, with}\ c_4=4\sqrt{2} \\ C_{2n} =& \frac{4 n C_n}{2n + \sqrt{(2n)^2 + C_n^2}} &\textrm{for the outer polygon, with}\ C_4=8 \end{array} $$

My thoughts are:

1. Would it make it anyhow better when I have a function called void PiApproximation() that writes my values to stdout already? I mean technically you can always put your full code into the main()-function, but when you have to use the same code-parts over and over again you should make a own function of it and call it, when u need it. So I guess in this case it will make no difference if I use an own function or calculate the circumferences and print it in the main-function.

2. What else can I improve?

pi_approx.c

#include <stdio.h>
#include <math.h>
#define INNER_FOUR 4*sqrt(2); //circumference c_4 of the inner tetragon(square) 
#define OUTER_FOUR 8 //circumference C_4 of the outer tetragon
int power(int n, int p);
int main(void)
{
 int n = 4;
 double innerPoly = INNER_FOUR;
 double outerPoly = OUTER_FOUR;
 printf("PI-APPROXIMATION USING POLYGONS\n");
 printf("===============================\n\n");
 printf(" n I c_n/2 I C_n/2 I\n");
 printf("---------I-----------------I-----------------I\n");
 for (int i=3; n<=8192; n=power(i,2), i++)
 {
 printf(" %4d I %1.8lf I %1.8lf I\n", n, innerPoly / 2, outerPoly / 2);
 innerPoly = 2 * sqrt(2 * n*n - n*sqrt(4 * n*n - innerPoly*innerPoly)); //formular c_2n
 outerPoly = (4 * n * outerPoly) / (2 * n + sqrt(4 * n*n + outerPoly*outerPoly)); //formular C_2n
 }
 return 0;
}
int power(int n, int p)
{
 int pBuffer = 1;
 for (int i=1; i <= n; i++)
 {
 pBuffer *= p; 
 }
 return pBuffer;
}

Answer 1

1. You asked,

   Would it make it better if I had a function called PiApproximation?

   The code in main does just one thing so it would not be an improvement to introduce a new function like this — the question would then be, what's the point of main?

2. There is a dependency between the initial value of n and the initial values of innerPoly and outerPoly. It would make sense to put all the initialization code together.

3. Because n can never be negative it could be unsigned.

4. The loop:

   for (int i=3; n<=8192; n=power(i,2), i++)
   

   is quite hard to follow because the loop variable is i but the termination condition is on a different variable n. Because the loop variable i is not used, it would be easier to follow the logic if you wrote:

   for (; n <= 8192; n *= 2)
   

   (This also avoids the need for the power function.)

5. The l modifier has no effect on the printf format specifier %f, and so should be omitted.

• CAD97: I don't know the C preprocessor very well, but those #define scare me: iirc doing something like double sum = INNER_FOUR + OUTER_FOUR or anything with code after the macro could break in interesting ways. What's the correct way to deal with that?
• Gareth Rees: @CAD97: Parenthesize the macro expansion (see the comp.lang.c FAQ).
• physics: In my case I only use this constant circumferences of a 4-polygon as a start-value of my calculation, i.e. I do not really use them in a formular, where I can immediately see what this constant would mean in a determined context.
• physics: So in my case I could directly write on the variables innerPoly, outerPoly my start-values at the begin of the main()-program. What do you think about that?