Error

Derivation

1. Split input into 3 regimes

2. if (* sin cos) < -7.130148269305168e+294

   1. Initial program 6.0

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

   3. Applied add-sqr-sqrt6.0

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

   4. Applied simplify6.0

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

   5. Applied simplify0.3

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

   7. Applied cos-20.3

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

   8. Applied div-sub0.3

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

   if -7.130148269305168e+294 < (* sin cos) < -1.3995126005674643e-178

   1. Initial program 29.6

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

   3. Applied add-sqr-sqrt29.6

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

   4. Applied simplify29.6

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

   5. Applied simplify3.8

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

   7. Applied associate-/r*3.5

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

   9. Applied add-sqr-sqrt3.5

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

   10. Applied associate-/r*3.5

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

   11. Taylor expanded around inf 3.5

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

   12. Applied simplify0.4

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

   if -1.3995126005674643e-178 < (* sin cos)

   1. Initial program 29.5

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

   3. Applied add-sqr-sqrt29.5

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

   4. Applied simplify29.5

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

   5. Applied simplify2.7

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

   7. Applied associate-/r*2.5

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

Runtime

Time bar (total: 39.5s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))))