// Andrew Taylor - andrewt@unsw.edu.au
//
// The approximate representation of reals can produce unexpected errors.
// If we subtract or divide two values which are very close together
// a large relative error can result.
// This is called cancellation or catastrophic cancellation.
//
// if x is close to 0, cos(x) is close to 1
// and calculating 1 - cos(x) can produce a large error in a calculation
//
// for example below, if x=0.000000011 the correct answer is ~0.5 but we get 0.917540
//
// we can avoid the error by replacing 1 - cos(x) with
// the mathematically equivalent  2 * sin(x/2) * sin(x/2)
//

/*```
$ ./a.out
Enter x: 0.123
(1 - cos(x)) / (x * x) = 0.499370
(2 * sin(x/2) * sin(x/2)) / (x * x) = 0.499370
$ ./a.out
Enter x: 0.000000011
(1 - cos(x)) / (x * x) = 0.917540
(2 * sin(x/2) * sin(x/2)) / (x * x) = 0.500000
```*/

#include <stdio.h>
#include <math.h>

int main(void) {

    double x;
    printf("Enter x: ");
    scanf("%lf", &x);


    printf("(1 - cos(x)) / (x * x) = %lf\n", (1 - cos(x)) / (x * x));
    printf("(2 * sin(x/2) * sin(x/2)) / (x * x) = %lf\n", (2 * sin(x/2) * sin(x/2)) / (x * x));

    return 0;
}