cos2 (problem 3.4.1)

Average Error: 31.2 → 0.1

Time: 14.3s

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 r440760 = 1.0;
        double r440761 = x;
        double r440762 = cos(r440761);
        double r440763 = r440760 - r440762;
        double r440764 = r440761 * r440761;
        double r440765 = r440763 / r440764;
        return r440765;
}

↓

double f(double x) {
        double r440766 = x;
        double r440767 = 2.0;
        double r440768 = r440766 / r440767;
        double r440769 = tan(r440768);
        double r440770 = r440769 / r440766;
        double r440771 = sin(r440766);
        double r440772 = r440771 / r440766;
        double r440773 = r440770 * r440772;
        return r440773;
}

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. Final simplification 0.1

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

Reproduce

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