cos2 (problem 3.4.1)

Average Error: 31.2 → 0.1

Time: 18.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r560783 = 1.0;
        double r560784 = x;
        double r560785 = cos(r560784);
        double r560786 = r560783 - r560785;
        double r560787 = r560784 * r560784;
        double r560788 = r560786 / r560787;
        return r560788;
}

↓

double f(double x) {
        double r560789 = x;
        double r560790 = 2.0;
        double r560791 = r560789 / r560790;
        double r560792 = tan(r560791);
        double r560793 = sin(r560789);
        double r560794 = r560789 / r560793;
        double r560795 = r560792 / r560794;
        double r560796 = r560795 / r560789;
        return r560796;
}

Error

Bits error versus x

Derivation

1. Initial program 31.2

   \[\frac{1 - \cos x}{x \cdot x}\]
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 15.9

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

6. Applied *-un-lft-identity 15.9

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

7. Applied *-un-lft-identity 15.9

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

8. Applied distribute-lft-out 15.9

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

9. Applied times-frac 15.9

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

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 clear-num 0.1

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sin x}}} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]
15. Using strategy rm

16. Applied associate-*r/ 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))