Error

Derivation

1. Split input into 2 regimes

2. if (/ (/ (cos (* 2 x)) (* cos (* x sin))) (* cos (* x sin))) < 1.3649867799669e-317 or 1.2738224146149126e+242 < (/ (/ (cos (* 2 x)) (* cos (* x sin))) (* cos (* x sin)))

   1. Initial program 18.7

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

   2. Taylor expanded around 0 3.1

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

   4. Applied cos-23.2

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

   5. Applied div-sub3.2

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

   7. Applied *-un-lft-identity3.2

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

   8. Applied times-frac3.2

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

   9. Applied add-sqr-sqrt3.2

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

   10. Applied times-frac3.2

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

   11. Applied prod-diff3.2

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

   12. Applied simplify2.6

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

   13. Applied simplify1.5

       \[\leadsto (\left(\frac{\cos x}{\left|\left(cos \cdot sin\right) \cdot x\right|}\right) \cdot \left(\frac{\cos x}{\left|\left(cos \cdot sin\right) \cdot x\right|}\right) + \left(\frac{\left(-\sin x\right) \cdot \sin x}{\left(\left(cos \cdot sin\right) \cdot x\right) \cdot \left(\left(cos \cdot sin\right) \cdot x\right)}\right))_* + \color{blue}{0}\]

   if 1.3649867799669e-317 < (/ (/ (cos (* 2 x)) (* cos (* x sin))) (* cos (* x sin))) < 1.2738224146149126e+242

   1. Initial program 44.0

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

   2. Taylor expanded around 0 1.4

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

   4. Applied unpow21.4

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

   5. Applied associate-/r*0.9

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

4. Applied simplify1.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(x \cdot 2\right)}{cos \cdot \left(sin \cdot x\right)}}{cos \cdot \left(sin \cdot x\right)} \le 1.3649867799669 \cdot 10^{-317}:\\ \;\;\;\;(\left(\frac{\cos x}{\left|\left(sin \cdot cos\right) \cdot x\right|}\right) \cdot \left(\frac{\cos x}{\left|\left(sin \cdot cos\right) \cdot x\right|}\right) + \left(\frac{\sin x \cdot \left(-\sin x\right)}{\left(\left(sin \cdot cos\right) \cdot x\right) \cdot \left(\left(sin \cdot cos\right) \cdot x\right)}\right))_*\\ \mathbf{if}\;\frac{\frac{\cos \left(x \cdot 2\right)}{cos \cdot \left(sin \cdot x\right)}}{cos \cdot \left(sin \cdot x\right)} \le 1.2738224146149126 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{cos \cdot \left(sin \cdot x\right)}}{cos \cdot \left(sin \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\cos x}{\left|\left(sin \cdot cos\right) \cdot x\right|}\right) \cdot \left(\frac{\cos x}{\left|\left(sin \cdot cos\right) \cdot x\right|}\right) + \left(\frac{\sin x \cdot \left(-\sin x\right)}{\left(\left(sin \cdot cos\right) \cdot x\right) \cdot \left(\left(sin \cdot cos\right) \cdot x\right)}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018195 +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))))