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Derivation

1. Initial program 31.1

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

3. Applied flip--31.2

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

4. Applied associate-/l/31.2

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

5. Simplified15.3

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

6. Taylor expanded around inf 15.3

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

7. Simplified15.5

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

9. Applied *-un-lft-identity15.5

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

10. Applied times-frac0.5

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

11. Applied *-un-lft-identity0.5

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

12. Applied times-frac0.2

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

13. Simplified0.2

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

14. Final simplification0.2

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

Runtime

Time bar (total: 31.8s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))