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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. Taylor expanded around -inf 15.0

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

8. Simplified15.2

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

10. Applied *-un-lft-identity15.2

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

11. Applied times-frac0.4

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

12. Simplified0.4

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

14. Applied tan-quot0.5

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

15. Applied associate-/l/0.5

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

16. Final simplification0.5

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

Runtime

Time bar (total: 23.3s)Debug logProfile

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