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Derivation

1. Initial program 30.1

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

3. Applied flip--30.3

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

4. Applied associate-/l/30.3

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

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 associate-/r*15.0

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

9. Applied *-un-lft-identity15.0

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

10. Applied times-frac0.3

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

11. Applied times-frac0.3

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

12. Simplified0.3

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

13. Simplified0.1

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

14. Final simplification0.1

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

Runtime

Time bar (total: 19.9s)Debug logProfile

herbie shell --seed 2018295 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))