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Derivation

1. Initial program 30.8

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

3. Applied flip--31.0

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

4. Applied associate-/l/31.0

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

5. Simplified15.6

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

7. Applied associate-*l*15.7

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

9. Applied clear-num15.7

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

11. Applied times-frac0.7

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

12. Applied associate-/r*0.3

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

13. Simplified0.1

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

14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 18.9s)Debug log