Error

Derivation

1. Initial program 31.2

   \[\frac{1 - \cos x}{x \cdot x}\]

2. Initial simplification31.2

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

4. Applied flip--31.3

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

5. Applied associate-/l/31.3

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

6. Simplified14.9

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

8. Applied flip3-+14.9

   \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

9. Applied associate-*r/14.9

   \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

10. Simplified14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]
11. Using strategy rm

12. Applied *-un-lft-identity14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \color{blue}{1 \cdot \cos x}}}\]

13. Applied *-un-lft-identity14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{1 \cdot (\left(\cos x\right) \cdot \left(\cos x\right) + 1)_*} - 1 \cdot \cos x}}\]

14. Applied distribute-lft-out--14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{1 \cdot \left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)}}}\]

15. Applied times-frac14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\frac{x \cdot x}{1} \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

16. Applied associate-/r*14.9

    \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \sin x}{\frac{x \cdot x}{1}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]
17. Using strategy rm

18. Applied add-sqr-sqrt14.9

    \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{\sqrt{\frac{x \cdot x}{1}} \cdot \sqrt{\frac{x \cdot x}{1}}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}\]

19. Applied times-frac15.2

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{\sqrt{\frac{x \cdot x}{1}}} \cdot \frac{\sin x}{\sqrt{\frac{x \cdot x}{1}}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}\]

20. Simplified15.2

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{\left|x\right|}} \cdot \frac{\sin x}{\sqrt{\frac{x \cdot x}{1}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}\]

21. Simplified0.3

    \[\leadsto \frac{\frac{\sin x}{\left|x\right|} \cdot \color{blue}{\frac{\sin x}{\left|x\right|}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}\]

22. Final simplification0.3

    \[\leadsto \frac{\frac{\sin x}{\left|x\right|} \cdot \frac{\sin x}{\left|x\right|}}{\frac{{\left(\cos x\right)}^{3} + 1}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}\]

Runtime

Time bar (total: 48.0s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))