cos(2*x)/(cos^2(x)*sin^2(x))

Average Error: 28.8 → 3.4

Time: 32.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;cos \le 1.769257013948251256473583277074330105322 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \cos \left(x \cdot 2\right)}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{elif}\;cos \le 5.853389657751302557139962116291973898598 \cdot 10^{134}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \left(\frac{\cos \left(x \cdot 2\right)}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right) \cdot \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{{cos}^{\left(\frac{2}{2}\right)}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x}}{x}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r91675 = 2.0;
        double r91676 = x;
        double r91677 = r91675 * r91676;
        double r91678 = cos(r91677);
        double r91679 = cos;
        double r91680 = pow(r91679, r91675);
        double r91681 = sin;
        double r91682 = pow(r91681, r91675);
        double r91683 = r91676 * r91682;
        double r91684 = r91683 * r91676;
        double r91685 = r91680 * r91684;
        double r91686 = r91678 / r91685;
        return r91686;
}

↓

double f(double x, double cos, double sin) {
        double r91687 = cos;
        double r91688 = 1.7692570139482513e-278;
        bool r91689 = r91687 <= r91688;
        double r91690 = 1.0;
        double r91691 = sin;
        double r91692 = 2.0;
        double r91693 = 2.0;
        double r91694 = r91692 / r91693;
        double r91695 = pow(r91691, r91694);
        double r91696 = pow(r91687, r91694);
        double r91697 = x;
        double r91698 = r91696 * r91697;
        double r91699 = r91695 * r91698;
        double r91700 = cbrt(r91695);
        double r91701 = r91699 * r91700;
        double r91702 = r91690 / r91701;
        double r91703 = r91697 * r91692;
        double r91704 = cos(r91703);
        double r91705 = r91702 * r91704;
        double r91706 = cbrt(r91697);
        double r91707 = r91705 / r91706;
        double r91708 = r91690 / r91695;
        double r91709 = cbrt(r91708);
        double r91710 = r91709 * r91709;
        double r91711 = r91706 * r91706;
        double r91712 = r91696 * r91711;
        double r91713 = r91710 / r91712;
        double r91714 = r91707 * r91713;
        double r91715 = 5.8533896577513026e+134;
        bool r91716 = r91687 <= r91715;
        double r91717 = cbrt(r91706);
        double r91718 = r91704 / r91717;
        double r91719 = cbrt(r91711);
        double r91720 = r91696 * r91719;
        double r91721 = r91697 * r91695;
        double r91722 = r91720 * r91721;
        double r91723 = r91709 / r91722;
        double r91724 = r91718 * r91723;
        double r91725 = r91713 * r91724;
        double r91726 = r91708 / r91696;
        double r91727 = r91696 * r91695;
        double r91728 = r91727 * r91697;
        double r91729 = r91726 / r91728;
        double r91730 = r91704 * r91729;
        double r91731 = r91730 / r91697;
        double r91732 = r91716 ? r91725 : r91731;
        double r91733 = r91689 ? r91714 : r91732;
        return r91733;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 3 regimes

2. if cos < 1.7692570139482513e-278

   1. Initial program 29.7

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

   2. Simplified 28.6

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

   4. Applied sqr-pow 28.6

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

   5. Applied *-un-lft-identity 28.6

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

   6. Applied times-frac 28.5

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

   7. Applied associate-/l* 23.2

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

   8. Simplified 20.6

      \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\color{blue}{\frac{{cos}^{2}}{\cos \left(2 \cdot x\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}}{x}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 20.6

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

   11. Applied sqr-pow 20.6

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

   12. Applied times-frac 20.6

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

   13. Applied associate-*l* 12.1

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

   14. Simplified 12.1

       \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{1} \cdot \color{blue}{\frac{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}{\cos \left(x \cdot 2\right)}}}}{x}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 12.4

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

   17. Applied add-cube-cbrt 12.5

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

   18. Applied times-frac 9.8

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

   19. Applied times-frac 4.7

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

   20. Simplified 4.6

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

   21. Simplified 4.2

       \[\leadsto \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x} \cdot \cos \left(2 \cdot x\right)}{\sqrt[3]{x}}}\]
   22. Using strategy rm

   23. Applied cbrt-div 4.1

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

   24. Applied associate-/l/ 4.1

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

   25. Simplified 3.2

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

   if 1.7692570139482513e-278 < cos < 5.8533896577513026e+134

   1. Initial program 29.9

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

   2. Simplified 27.8

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

   4. Applied sqr-pow 27.8

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

   5. Applied *-un-lft-identity 27.8

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

   6. Applied times-frac 27.8

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

   7. Applied associate-/l* 22.2

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

   8. Simplified 18.2

      \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\color{blue}{\frac{{cos}^{2}}{\cos \left(2 \cdot x\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}}{x}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 18.2

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

   11. Applied sqr-pow 18.2

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

   12. Applied times-frac 18.2

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

   13. Applied associate-*l* 9.7

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

   14. Simplified 9.7

       \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{1} \cdot \color{blue}{\frac{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}{\cos \left(x \cdot 2\right)}}}}{x}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 10.1

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

   17. Applied add-cube-cbrt 10.3

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

   18. Applied times-frac 9.0

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

   19. Applied times-frac 3.0

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

   20. Simplified 3.0

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

   21. Simplified 3.6

       \[\leadsto \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x} \cdot \cos \left(2 \cdot x\right)}{\sqrt[3]{x}}}\]
   22. Using strategy rm

   23. Applied add-cube-cbrt 3.6

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

   24. Applied cbrt-prod 3.6

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

   25. Applied times-frac 3.6

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

   26. Simplified 3.1

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

   27. Simplified 3.1

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

   if 5.8533896577513026e+134 < cos

   1. Initial program 24.5

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

   2. Simplified 24.2

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

   4. Applied sqr-pow 24.2

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

   5. Applied *-un-lft-identity 24.2

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

   6. Applied times-frac 24.2

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

   7. Applied associate-/l* 19.1

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

   8. Simplified 18.5

      \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\color{blue}{\frac{{cos}^{2}}{\cos \left(2 \cdot x\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}}{x}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 18.5

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

   11. Applied sqr-pow 18.5

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

   12. Applied times-frac 18.5

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

   13. Applied associate-*l* 10.4

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

   14. Simplified 10.4

       \[\leadsto \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{1} \cdot \color{blue}{\frac{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}{\cos \left(x \cdot 2\right)}}}}{x}\]
   15. Using strategy rm

   16. Applied frac-times 10.4

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

   17. Applied associate-/r/ 10.4

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

   18. Simplified 4.5

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 1.769257013948251256473583277074330105322 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \cos \left(x \cdot 2\right)}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{elif}\;cos \le 5.853389657751302557139962116291973898598 \cdot 10^{134}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \left(\frac{\cos \left(x \cdot 2\right)}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right) \cdot \frac{\frac{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}{{cos}^{\left(\frac{2}{2}\right)}}}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))