Error

Derivation

1. Split input into 2 regimes

2. if sin < -2.9047112100151635e-257

   1. Initial program 26.4

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

   2. Simplified2.3

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

   4. Applied associate-/r*1.9

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

   if -2.9047112100151635e-257 < sin

   1. Initial program 28.0

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

   2. Simplified3.3

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

   3. Taylor expanded around -inf 31.4

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

   4. Simplified2.7

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

   6. Applied add-cube-cbrt2.9

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

   7. Applied times-frac2.7

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

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

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

   10. Applied cbrt-prod2.7

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

   11. Applied times-frac2.7

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

   12. Simplified2.7

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

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin \le -2.9047112100151635 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(cos \cdot x\right)}}{sin \cdot \left(cos \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot sin\right) \cdot cos} \cdot \left(\frac{\frac{1}{x}}{sin} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{cos}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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))))