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Derivation

1. Initial program 27.8

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

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 *-un-lft-identity3.0

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

5. Applied times-frac2.8

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

7. Applied associate-/r*2.9

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

9. Applied cos-22.9

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

10. Applied div-sub2.9

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

11. Applied div-sub2.9

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

12. Final simplification2.9

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

Runtime

Time bar (total: 17.2s)Debug logProfile

herbie shell --seed 2018354 +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))))