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Derivation

1. Split input into 2 regimes

2. if (pow cos 2) < 1.5034798266949682e-306 or 1.5266182343325565e+96 < (pow cos 2)

   1. Initial program 31.1

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

   3. Applied unpow231.1

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

   4. Applied associate-*r*27.1

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

   6. Applied unpow227.1

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

   7. Applied associate-*l*19.1

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

   9. Applied associate-*r*15.2

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

   11. Applied associate-*r*8.6

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

   if 1.5034798266949682e-306 < (pow cos 2) < 1.5266182343325565e+96

   1. Initial program 21.3

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

   3. Applied unpow221.3

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

   4. Applied associate-*r*10.8

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

   6. Applied associate-*l*6.6

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

4. Final simplification8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;{cos}^{2} \le 1.5034798266949682 \cdot 10^{-306} \lor \neg \left({cos}^{2} \le 1.5266182343325565 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(sin \cdot \left(\left(sin \cdot x\right) \cdot cos\right)\right)\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{2} \cdot \left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +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))))

Details

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