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Derivation

1. Split input into 2 regimes

2. if x < -61357.354317215366 or -9.238259325850402e-152 < x

   1. Initial program 28.3

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

   3. Applied unpow228.3

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

   4. Applied associate-*l*23.8

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

   6. Applied unpow223.8

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

   7. Applied associate-*r*15.9

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

   9. Applied associate-*r*13.1

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

   11. Applied associate-*r*8.3

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

   if -61357.354317215366 < x < -9.238259325850402e-152

   1. Initial program 27.4

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

   3. Applied unpow227.4

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

   4. Applied associate-*l*23.8

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

   6. Applied unpow223.8

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

   7. Applied associate-*r*21.0

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

   8. Taylor expanded around 0 27.7

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

   9. Simplified9.9

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

4. Final simplification8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -61357.354317215366 \lor \neg \left(x \le -9.238259325850402 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(\left(cos \cdot \left(sin \cdot x\right)\right) \cdot sin\right)\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \frac{2}{3}}{\left(sin \cdot cos\right) \cdot \left(sin \cdot cos\right)} + \left(\frac{\frac{1}{x \cdot x}}{\left(sin \cdot cos\right) \cdot \left(sin \cdot cos\right)} + \frac{-2}{\left(sin \cdot cos\right) \cdot \left(sin \cdot cos\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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))))

Details

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