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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/30.7

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

6. Simplified15.0

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

8. Applied times-frac15.4

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

9. Simplified15.2

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

11. Applied associate-/r*0.2

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

13. Applied tan-quot0.2

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

14. Applied frac-times0.1

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

15. Final simplification0.1

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

Runtime

Time bar (total: 22.5s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))