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Derivation

1. Initial program 28.0

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

2. Initial simplification3.0

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

4. Applied cos-23.0

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

6. Applied difference-of-squares3.0

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

7. Applied times-frac2.8

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

9. Applied associate-/r*2.8

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

10. Final simplification2.8

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

Runtime

Time bar (total: 45.0s)Debug logProfile

herbie shell --seed 2018216 +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))))