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Derivation

1. Initial program 30.6

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

2. Initial simplification30.6

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

4. Applied flip--30.7

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

5. Applied associate-/l/30.8

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

6. Simplified15.2

   \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
7. Using strategy rm

8. Applied times-frac15.6

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

9. Simplified15.5

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

11. Applied *-un-lft-identity15.5

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

12. Applied times-frac0.2

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

14. Applied associate-*r/0.2

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

15. Applied associate-*l/0.2

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

16. Simplified0.1

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

17. Final simplification0.1

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

Runtime

Time bar (total: 24.6s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))