> 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