// Andrew Taylor - andrewt@unsw.edu.au
// 26/2/2019

// Demonstrate approximate representation of reals
// producing error.
// sometimes if we subtract or two approximations which are very close together
// we can can get a large relative error
// correct answer if x == 0.000000011 (1 - cos(x))  / (x * x) is very close to 0.5
// code prints 0.917540 which is wrong by a factor of almost two
// we can avoid the error by replacing 1 - cos(x) with
// the mathematically equivalent  2 * sin(x/2) * sin(x/2)
//

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

int main(void) {

    double x = 0.000000011;
    double y = (1 - cos(x)) / (x * x);

    // correct answer y = ~0.5
    // prints y = 0.917540
    printf("y = %lf\n", y);

    // subtraction of similar approximate value
    // produces large error
    // sometimes called catastrophic cancellation
    // The issue arises because the leading significant digits cancel out, 
    // leaving only the less significant, error-prone bits.
    printf("%.20g\n", cos(x)); 
    return 0;
}