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Derivation

1. Initial program 27.6

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

2. Initial simplification2.9

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

4. Applied associate-/r*2.6

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

6. Applied add-cube-cbrt2.7

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

7. Applied times-frac2.9

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

8. Taylor expanded around -inf 2.8

   \[\leadsto \frac{\color{blue}{\left(\frac{1}{x \cdot cos} \cdot {\left({\left(\cos \left(2 \cdot x\right)\right)}^{2}\right)}^{\frac{1}{3}}\right)} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{sin}}{\left(x \cdot cos\right) \cdot sin}\]

9. Final simplification2.8

   \[\leadsto \frac{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{sin} \cdot \left(\frac{1}{cos \cdot x} \cdot {\left({\left(\cos \left(2 \cdot x\right)\right)}^{2}\right)}^{\frac{1}{3}}\right)}{\left(cos \cdot x\right) \cdot sin}\]

Runtime

Time bar (total: 26.6s)Debug logProfile

herbie shell --seed 2018295 +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))))