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Derivation

1. Split input into 2 regimes

2. if cos < -8.862615262515132e-182 or 1.661889731014095e-188 < cos

   1. Initial program 23.2

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

   2. Initial simplification2.6

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

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

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

   5. Applied associate-/l*2.6

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

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

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

   8. Applied times-frac2.6

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

   9. Applied associate-/r*2.4

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

   11. Applied *-un-lft-identity2.4

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

   12. Applied *-un-lft-identity2.4

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

   13. Applied times-frac2.4

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

   14. Simplified2.4

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

   15. Simplified1.9

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

   if -8.862615262515132e-182 < cos < 1.661889731014095e-188

   1. Initial program 61.9

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

   2. Initial simplification5.4

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

   4. Applied *-un-lft-identity5.4

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

   5. Applied associate-/l*5.4

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

   7. Applied *-un-lft-identity5.4

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

   8. Applied times-frac5.4

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

   9. Applied associate-/r*5.1

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

   11. Applied *-un-lft-identity5.1

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

   12. Applied associate-/l*5.4

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\left(x \cdot cos\right) \cdot sin}{\cos \left(2 \cdot x\right)}}{\frac{1}{\frac{\left(x \cdot cos\right) \cdot sin}{1}}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -8.862615262515132 \cdot 10^{-182} \lor \neg \left(cos \le 1.661889731014095 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(sin \cdot x\right) \cdot cos\right) \cdot \left(\left(sin \cdot x\right) \cdot cos\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{sin \cdot \left(x \cdot cos\right)}{\cos \left(2 \cdot x\right)}}{\frac{1}{sin \cdot \left(x \cdot cos\right)}}}\\ \end{array}\]

Runtime

Time bar (total: 29.7s)Debug logProfile

herbie shell --seed 2018339 
(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))))