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

Average Error: 28.5 → 3.2

Time: 36.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{cos}^{2} \le 0.0:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{sin}\right)}^{\left(\frac{2}{2}\right)}}}{\left(\left(x \cdot {\left({cos}^{1} \cdot {sin}^{1}\right)}^{1}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(2 \cdot x\right)}}\right) \cdot \sqrt[3]{x}} \cdot \frac{\frac{1}{{\left(\sqrt[3]{sin} \cdot \sqrt[3]{sin}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{elif}\;{cos}^{2} \le 2.153259979754124549423327232650568646761 \cdot 10^{301}:\\ \;\;\;\;\frac{1}{\left(\frac{{cos}^{2}}{\cos \left(2 \cdot x\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \frac{x}{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{x} \cdot \left(\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r98632 = 2.0;
        double r98633 = x;
        double r98634 = r98632 * r98633;
        double r98635 = cos(r98634);
        double r98636 = cos;
        double r98637 = pow(r98636, r98632);
        double r98638 = sin;
        double r98639 = pow(r98638, r98632);
        double r98640 = r98633 * r98639;
        double r98641 = r98640 * r98633;
        double r98642 = r98637 * r98641;
        double r98643 = r98635 / r98642;
        return r98643;
}

↓

double f(double x, double cos, double sin) {
        double r98644 = cos;
        double r98645 = 2.0;
        double r98646 = pow(r98644, r98645);
        double r98647 = 0.0;
        bool r98648 = r98646 <= r98647;
        double r98649 = 1.0;
        double r98650 = sin;
        double r98651 = cbrt(r98650);
        double r98652 = 2.0;
        double r98653 = r98645 / r98652;
        double r98654 = pow(r98651, r98653);
        double r98655 = r98649 / r98654;
        double r98656 = x;
        double r98657 = 1.0;
        double r98658 = pow(r98644, r98657);
        double r98659 = pow(r98650, r98657);
        double r98660 = r98658 * r98659;
        double r98661 = pow(r98660, r98657);
        double r98662 = r98656 * r98661;
        double r98663 = r98645 * r98656;
        double r98664 = cos(r98663);
        double r98665 = r98649 / r98664;
        double r98666 = cbrt(r98665);
        double r98667 = r98662 * r98666;
        double r98668 = cbrt(r98656);
        double r98669 = r98667 * r98668;
        double r98670 = r98655 / r98669;
        double r98671 = r98651 * r98651;
        double r98672 = pow(r98671, r98653);
        double r98673 = r98649 / r98672;
        double r98674 = pow(r98644, r98653);
        double r98675 = cbrt(r98664);
        double r98676 = r98675 * r98675;
        double r98677 = r98674 / r98676;
        double r98678 = r98668 * r98668;
        double r98679 = r98677 * r98678;
        double r98680 = r98673 / r98679;
        double r98681 = r98670 * r98680;
        double r98682 = 2.1532599797541245e+301;
        bool r98683 = r98646 <= r98682;
        double r98684 = r98646 / r98664;
        double r98685 = pow(r98650, r98653);
        double r98686 = r98685 * r98656;
        double r98687 = r98684 * r98686;
        double r98688 = r98649 / r98685;
        double r98689 = r98656 / r98688;
        double r98690 = r98687 * r98689;
        double r98691 = r98649 / r98690;
        double r98692 = cbrt(r98688);
        double r98693 = r98692 * r98692;
        double r98694 = r98693 / r98677;
        double r98695 = r98694 / r98678;
        double r98696 = r98674 / r98675;
        double r98697 = r98696 * r98686;
        double r98698 = r98668 * r98697;
        double r98699 = r98692 / r98698;
        double r98700 = r98695 * r98699;
        double r98701 = r98683 ? r98691 : r98700;
        double r98702 = r98648 ? r98681 : r98701;
        return r98702;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 3 regimes

2. if (pow cos 2.0) < 0.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied sqr-pow 64.0

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

   5. Applied *-un-lft-identity 64.0

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

   6. Applied times-frac 64.0

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

   7. Applied associate-/l* 64.0

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

   8. Simplified 64.0

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

   10. Applied add-cube-cbrt 64.0

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

   11. Applied sqr-pow 64.0

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

   12. Applied times-frac 64.0

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

   13. Applied associate-*l* 28.0

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

   14. Simplified 21.5

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

   16. Applied add-cube-cbrt 21.6

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

   17. Applied add-cube-cbrt 21.8

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

   18. Applied unpow-prod-down 21.8

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

   19. Applied add-sqr-sqrt 21.8

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

   20. Applied times-frac 21.8

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

   21. Applied times-frac 16.5

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

   22. Applied times-frac 9.0

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

   23. Simplified 5.9

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

   24. Simplified 11.8

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

   25. Taylor expanded around inf 31.5

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

   26. Simplified 8.9

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

   if 0.0 < (pow cos 2.0) < 2.1532599797541245e+301

   1. Initial program 20.6

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

   2. Simplified 18.1

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

   4. Applied sqr-pow 18.1

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

   5. Applied *-un-lft-identity 18.1

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

   6. Applied times-frac 17.8

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

   7. Applied associate-/l* 10.8

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

   8. Simplified 5.9

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

   10. Applied clear-num 6.1

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

   11. Simplified 1.0

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

   if 2.1532599797541245e+301 < (pow cos 2.0)

   1. Initial program 26.6

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

   2. Simplified 26.4

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

   4. Applied sqr-pow 26.4

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

   5. Applied *-un-lft-identity 26.4

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

   6. Applied times-frac 26.4

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

   7. Applied associate-/l* 20.8

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

   8. Simplified 20.7

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

   10. Applied add-cube-cbrt 20.7

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

   11. Applied sqr-pow 20.7

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

   12. Applied times-frac 20.7

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

   13. Applied associate-*l* 10.5

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

   14. Simplified 8.8

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

   16. Applied add-cube-cbrt 9.0

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

   17. Applied add-cube-cbrt 9.0

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

   18. Applied times-frac 7.1

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

   19. Applied times-frac 4.3

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

   20. Simplified 4.3

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

   21. Simplified 4.1

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;{cos}^{2} \le 0.0:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{sin}\right)}^{\left(\frac{2}{2}\right)}}}{\left(\left(x \cdot {\left({cos}^{1} \cdot {sin}^{1}\right)}^{1}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(2 \cdot x\right)}}\right) \cdot \sqrt[3]{x}} \cdot \frac{\frac{1}{{\left(\sqrt[3]{sin} \cdot \sqrt[3]{sin}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{elif}\;{cos}^{2} \le 2.153259979754124549423327232650568646761 \cdot 10^{301}:\\ \;\;\;\;\frac{1}{\left(\frac{{cos}^{2}}{\cos \left(2 \cdot x\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \frac{x}{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{\frac{1}{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{x} \cdot \left(\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))