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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. Simplified15.0

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

7. Applied times-frac15.4

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

8. Simplified15.2

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

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

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

11. Applied times-frac0.2

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

12. Final simplification0.2

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

Runtime

Time bar (total: 34.2s)Debug logProfile

herbie shell --seed 2018263 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))