cos2 (problem 3.4.1)

Average Error: 31.3 → 0.1

Time: 10.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r701889 = 1.0;
        double r701890 = x;
        double r701891 = cos(r701890);
        double r701892 = r701889 - r701891;
        double r701893 = r701890 * r701890;
        double r701894 = r701892 / r701893;
        return r701894;
}

↓

double f(double x) {
        double r701895 = x;
        double r701896 = 2.0;
        double r701897 = r701895 / r701896;
        double r701898 = tan(r701897);
        double r701899 = r701898 / r701895;
        double r701900 = sin(r701895);
        double r701901 = r701900 / r701895;
        double r701902 = r701899 * r701901;
        return r701902;
}

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. Final simplification 0.1

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

Reproduce

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