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

Average Error: 28.8 → 2.2

Time: 6.7m

Precision: 64

Math

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

↓

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

C

double f(double x, double cos, double sin) {
        double r881858 = 2.0;
        double r881859 = x;
        double r881860 = r881858 * r881859;
        double r881861 = cos(r881860);
        double r881862 = cos;
        double r881863 = pow(r881862, r881858);
        double r881864 = sin;
        double r881865 = pow(r881864, r881858);
        double r881866 = r881859 * r881865;
        double r881867 = r881866 * r881859;
        double r881868 = r881863 * r881867;
        double r881869 = r881861 / r881868;
        return r881869;
}

↓

double f(double x, double cos, double sin) {
        double r881870 = 2.0;
        double r881871 = x;
        double r881872 = r881870 * r881871;
        double r881873 = cos(r881872);
        double r881874 = cbrt(r881873);
        double r881875 = cbrt(r881874);
        double r881876 = sin;
        double r881877 = 2.0;
        double r881878 = r881870 / r881877;
        double r881879 = pow(r881876, r881878);
        double r881880 = cbrt(r881879);
        double r881881 = r881875 / r881880;
        double r881882 = cbrt(r881871);
        double r881883 = cbrt(r881882);
        double r881884 = r881881 / r881883;
        double r881885 = r881884 / r881882;
        double r881886 = r881874 * r881874;
        double r881887 = cbrt(r881886);
        double r881888 = r881880 * r881880;
        double r881889 = r881887 / r881888;
        double r881890 = r881882 * r881882;
        double r881891 = cbrt(r881890);
        double r881892 = r881889 / r881891;
        double r881893 = r881892 / r881890;
        double r881894 = r881879 / r881874;
        double r881895 = r881874 / r881894;
        double r881896 = cbrt(r881895);
        double r881897 = r881896 / r881882;
        double r881898 = cos;
        double r881899 = pow(r881898, r881878);
        double r881900 = r881897 / r881899;
        double r881901 = r881893 * r881900;
        double r881902 = r881885 * r881901;
        double r881903 = r881896 * r881896;
        double r881904 = r881903 / r881882;
        double r881905 = r881904 / r881899;
        double r881906 = r881902 * r881905;
        return r881906;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Initial program 28.8

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

2. Simplified 27.5

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

4. Applied add-cube-cbrt 27.7

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

5. Applied sqr-pow 27.7

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

6. Applied add-cube-cbrt 27.7

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

7. Applied times-frac 27.6

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

8. Applied times-frac 20.8

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

9. Applied times-frac 20.2

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

10. Simplified 17.7

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

11. Simplified 17.7

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

13. Applied sqr-pow 17.7

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

14. Applied add-cube-cbrt 17.8

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

15. Applied times-frac 17.8

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

16. Applied times-frac 11.0

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

17. Applied associate-*l* 5.6

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

19. Applied add-cube-cbrt 5.8

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

20. Applied add-cube-cbrt 5.8

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

21. Applied cbrt-prod 5.8

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

22. Applied add-cube-cbrt 5.9

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

23. Applied add-cube-cbrt 5.9

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

24. Applied cbrt-prod 5.9

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

25. Applied times-frac 5.9

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

26. Applied times-frac 5.9

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

27. Applied times-frac 5.9

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

28. Applied associate-*r* 2.2

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

29. Final simplification 2.2

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

Reproduce

herbie shell --seed 2019195 
(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))))