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Derivation

1. Initial program 30.6

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

3. Applied flip--30.7

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

4. Applied associate-/l/30.7

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

5. Simplified14.9

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

6. Taylor expanded around inf 14.9

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

7. Simplified15.1

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

9. Applied *-un-lft-identity15.1

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

10. Applied times-frac0.2

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

11. Applied associate-*r*0.2

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

13. Applied un-div-inv0.1

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

14. Applied associate-*l/0.1

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

15. Final simplification0.1

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

Runtime

Time bar (total: 29.1s)Debug logProfile

herbie shell --seed 2018227 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))