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Derivation

1. Initial program 27.4

   \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

2. Initial simplification2.8

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}\]
3. Using strategy rm

4. Applied *-un-lft-identity2.8

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

5. Applied times-frac2.6

   \[\leadsto \color{blue}{\frac{1}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}\]
6. Using strategy rm

7. Applied associate-/r*2.7

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot cos}}{sin}} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}\]
8. Using strategy rm

9. Applied div-inv2.8

   \[\leadsto \color{blue}{\left(\frac{1}{x \cdot cos} \cdot \frac{1}{sin}\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}\]
10. Using strategy rm

11. Applied associate-*l/2.6

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

12. Simplified2.6

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

13. Final simplification2.6

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

Runtime

Time bar (total: 14.8s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))