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Derivation

1. Initial program 31.3

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

3. Applied flip--31.4

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

4. Applied associate-/l/31.4

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

5. Simplified15.3

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

7. Applied associate-*l*15.4

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

9. Applied times-frac0.3

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

10. Simplified0.3

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

12. Applied *-un-lft-identity0.3

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

13. Applied associate-*r*0.3

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

14. Simplified0.1

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

15. Final simplification0.1

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

Runtime

Time bar (total: 29.4s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))