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Derivation

1. Initial program 27.5

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

2. Initial simplification3.0

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

4. Applied associate-/r*2.7

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

6. Applied *-un-lft-identity2.7

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

7. Applied times-frac2.8

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

9. Applied div-inv2.9

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

11. Applied div-inv2.9

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

12. Applied associate-*r*2.9

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

13. Final simplification2.9

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(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))))

Details

Time bar (total: 19.5s)Debug log