except by the definition of dual numbers eps^2 is 0, as is eps^3. So the slope you computed is 0, which is rather wrong for (a complicated representation of) y=x.
> a complicated representation of) y=x.
The problem started as that. So no surprise.
Sure, we call that computer algebra. As soon as you start doing that, you aren't doing automatic differentiation, you are doing (at least in part) symbolic differentiation.
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
double f(const double x) {
return pow(x, 1000) / pow(x, 998);
}
int main(int argc, char* argv[]) {
if(argc != 2) {
fprintf(stderr, "Usage: %s x\n", argv[0]);
exit(1);
}
const double x = atof(argv[1]); // demo code without error checking
printf("x = %g, f(x) = %g\n", x, f(x));
exit(0);
}
gcc -Wall -g -o test test.c -lm
petschge@localhost:~$ ./test 0
x = 0, f(x) = -nan
petschge@localhost:~$ ./test 1
x = 1, f(x) = 1
petschge@localhost:~$ ./test 2
x = 2, f(x) = 4
gcc -Wall -O3 -ffast-math -g -o test2 test.c -lm
petschge@localhost:~$ ./test2 0
x = 0, f(x) = 0