cos2 (problem 3.4.1)

Average Error: 30.9 → 0.1

Time: 24.8s

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 r1107236 = 1.0;
        double r1107237 = x;
        double r1107238 = cos(r1107237);
        double r1107239 = r1107236 - r1107238;
        double r1107240 = r1107237 * r1107237;
        double r1107241 = r1107239 / r1107240;
        return r1107241;
}

↓

double f(double x) {
        double r1107242 = x;
        double r1107243 = 2.0;
        double r1107244 = r1107242 / r1107243;
        double r1107245 = tan(r1107244);
        double r1107246 = sin(r1107242);
        double r1107247 = r1107242 / r1107246;
        double r1107248 = r1107245 / r1107247;
        double r1107249 = r1107248 / r1107242;
        return r1107249;
}

Error

Bits error versus x

Derivation

1. Initial program 30.9

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

3. Applied flip-- 31.1

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

4. Simplified 15.8

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

6. Applied *-un-lft-identity 15.8

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

7. Applied *-un-lft-identity 15.8

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

8. Applied distribute-lft-out 15.8

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

9. Applied times-frac 15.8

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

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

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

19. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))