Average Error: 28.0 → 2.2
Time: 8.0m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\left(\frac{\frac{\frac{\sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)}}}{\sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{\sqrt[3]{x}}}}{x} \cdot \left(\frac{\frac{\sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)}}}{\sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{\sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\frac{{sin}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)}}}}}{\sqrt[3]{x}}}{{cos}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \frac{\frac{\sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\frac{{sin}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\frac{{sin}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(2 \cdot x\right)}}}}}{\sqrt[3]{x}}}{{cos}^{\left(\frac{2}{2}\right)}}\]
\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}}}}{x} \cdot \left(\frac{\frac{\sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)}} \cdot \sqrt[3]{\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]{\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)}}
double f(double x, double cos, double sin) {
        double r1024435 = 2.0;
        double r1024436 = x;
        double r1024437 = r1024435 * r1024436;
        double r1024438 = cos(r1024437);
        double r1024439 = cos;
        double r1024440 = pow(r1024439, r1024435);
        double r1024441 = sin;
        double r1024442 = pow(r1024441, r1024435);
        double r1024443 = r1024436 * r1024442;
        double r1024444 = r1024443 * r1024436;
        double r1024445 = r1024440 * r1024444;
        double r1024446 = r1024438 / r1024445;
        return r1024446;
}
↓
double f(double x, double cos, double sin) {
        double r1024447 = 2.0;
        double r1024448 = x;
        double r1024449 = r1024447 * r1024448;
        double r1024450 = cos(r1024449);
        double r1024451 = cbrt(r1024450);
        double r1024452 = cbrt(r1024451);
        double r1024453 = sin;
        double r1024454 = 2.0;
        double r1024455 = r1024447 / r1024454;
        double r1024456 = pow(r1024453, r1024455);
        double r1024457 = cbrt(r1024456);
        double r1024458 = r1024452 / r1024457;
        double r1024459 = cbrt(r1024448);
        double r1024460 = cbrt(r1024459);
        double r1024461 = r1024458 / r1024460;
        double r1024462 = r1024461 / r1024448;
        double r1024463 = r1024452 * r1024452;
        double r1024464 = r1024457 * r1024457;
        double r1024465 = r1024463 / r1024464;
        double r1024466 = r1024459 * r1024459;
        double r1024467 = cbrt(r1024466);
        double r1024468 = r1024465 / r1024467;
        double r1024469 = r1024456 / r1024451;
        double r1024470 = r1024451 / r1024469;
        double r1024471 = cbrt(r1024470);
        double r1024472 = r1024471 / r1024459;
        double r1024473 = cos;
        double r1024474 = pow(r1024473, r1024455);
        double r1024475 = r1024472 / r1024474;
        double r1024476 = r1024468 * r1024475;
        double r1024477 = r1024462 * r1024476;
        double r1024478 = r1024471 * r1024471;
        double r1024479 = r1024478 / r1024459;
        double r1024480 = r1024479 / r1024474;
        double r1024481 = r1024477 * r1024480;
        return r1024481;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.0

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

  2. Simplified26.6

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

  4. Applied add-cube-cbrt26.8

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

  5. Applied sqr-pow26.8

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

  6. Applied add-cube-cbrt26.8

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

  7. Applied times-frac26.7

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

  8. Applied times-frac19.9

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

  9. Applied times-frac19.2

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

  10. Simplified17.0

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

  11. Simplified17.0

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

  13. Applied sqr-pow17.0

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

  14. Applied add-cube-cbrt17.0

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

  15. Applied times-frac17.0

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

  16. Applied times-frac10.5

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

  17. Applied associate-*l*5.3

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

  19. Applied *-un-lft-identity5.3

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

  20. Applied add-cube-cbrt5.3

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

  21. Applied cbrt-prod5.4

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

  22. Applied add-cube-cbrt5.4

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

  23. Applied add-cube-cbrt5.4

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

  24. Applied times-frac5.4

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

  25. Applied times-frac5.4

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

  26. Applied times-frac5.4

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

  27. Applied associate-*r*2.2

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

  28. Final simplification2.2

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

Reproduce

herbie shell --seed 2019174 
(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))))