cos2 (problem 3.4.1)

Average Error: 31.3 → 0.1

Time: 13.0s

Precision: 64

Math

\[\frac{1 - \cos x}{x \cdot x}\]

↓

\[\frac{\frac{\sin x}{\frac{x}{\tan \left(\frac{x}{2}\right)}}}{x}\]

C

double f(double x) {
        double r312433 = 1.0;
        double r312434 = x;
        double r312435 = cos(r312434);
        double r312436 = r312433 - r312435;
        double r312437 = r312434 * r312434;
        double r312438 = r312436 / r312437;
        return r312438;
}

↓

double f(double x) {
        double r312439 = x;
        double r312440 = sin(r312439);
        double r312441 = 2.0;
        double r312442 = r312439 / r312441;
        double r312443 = tan(r312442);
        double r312444 = r312439 / r312443;
        double r312445 = r312440 / r312444;
        double r312446 = r312445 / r312439;
        return r312446;
}

Error

Bits error versus x

Derivation

1. Initial program 31.3

   \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm

3. Applied flip-- 31.4

   \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

4. Simplified 15.7

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x}\]
5. Using strategy rm

6. Applied *-un-lft-identity 15.7

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{1 + \color{blue}{1 \cdot \cos x}}}{x \cdot x}\]

7. Applied *-un-lft-identity 15.7

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{1 \cdot 1} + 1 \cdot \cos x}}{x \cdot x}\]

8. Applied distribute-lft-out 15.7

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x}\]

9. Applied times-frac 15.7

   \[\leadsto \frac{\color{blue}{\frac{\sin x}{1} \cdot \frac{\sin x}{1 + \cos x}}}{x \cdot x}\]

10. Applied times-frac 0.3

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{1}}{x} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}}\]

11. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}\]

12. Simplified 0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}\]
13. Using strategy rm

14. Applied associate-*l/ 0.1

    \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}}\]
15. Using strategy rm

16. Applied *-un-lft-identity 0.1

    \[\leadsto \frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{\color{blue}{1 \cdot x}}\]

17. Applied associate-/r* 0.1

    \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{1}}{x}}\]

18. Simplified 0.1

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{\frac{x}{\tan \left(\frac{x}{2}\right)}}}}{x}\]

19. Final simplification 0.1

    \[\leadsto \frac{\frac{\sin x}{\frac{x}{\tan \left(\frac{x}{2}\right)}}}{x}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))