How can I change my C program to give correct output for differentiation calculation?

Question

I am trying to calculate the absolute error in the calculation of f'(1) (where f(x)=sin(x)) caused, due to direct calculation ie(cos(1)), and that done with the formula

f'(x) = (f(x+h)-f(x))/h  

ie Error function = cos(1) - ((sin(1+h)-sin(1))/h), for a small value of h.

But for a certain value of 'h' (say 10^{-8}), when I calculated through a very precise calculator( Kelsan Calculator ), the absolute error came to be 4.207355e-9 but computing the same thing using following program:

#include <stdio.h>
#include <math.h>
int main() {
  double h = 1e-8;
  double a = (sin(1 + h) - sin(1)) / h;
  printf("%10e", cos(1) - a);
}

I got 2.969885e-09. Is it possible to calculate this correctly on C?

Answer 1

Once you round two values which are extremely close to begin with and subtract the rounded values, the results are going to be all over the place.

Using a little trigonometry, you can represent the difference of two sines much more accurately.

double a = 2 * cos(1+h/2) * sin(h/2) / h;

This change results in printing exactly 4.207355e-09.