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Derivation

1. Initial program 26.7

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

2. Initial simplification2.7

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

4. Applied add-cube-cbrt2.9

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}\]

5. Applied times-frac2.7

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

7. Applied cbrt-unprod2.6

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)}}}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot cos\right) \cdot sin}\]

8. Final simplification2.6

   \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)}}{\left(cos \cdot x\right) \cdot sin}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018251 +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))))