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

Average Error: 28.5 → 3.2

Time: 37.5s

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 r104371 = 2.0;
        double r104372 = x;
        double r104373 = r104371 * r104372;
        double r104374 = cos(r104373);
        double r104375 = cos;
        double r104376 = pow(r104375, r104371);
        double r104377 = sin;
        double r104378 = pow(r104377, r104371);
        double r104379 = r104372 * r104378;
        double r104380 = r104379 * r104372;
        double r104381 = r104376 * r104380;
        double r104382 = r104374 / r104381;
        return r104382;
}

↓

double f(double x, double cos, double sin) {
        double r104383 = cos;
        double r104384 = 2.0;
        double r104385 = pow(r104383, r104384);
        double r104386 = 0.0;
        bool r104387 = r104385 <= r104386;
        double r104388 = 1.0;
        double r104389 = sin;
        double r104390 = cbrt(r104389);
        double r104391 = 2.0;
        double r104392 = r104384 / r104391;
        double r104393 = pow(r104390, r104392);
        double r104394 = r104388 / r104393;
        double r104395 = x;
        double r104396 = 1.0;
        double r104397 = pow(r104383, r104396);
        double r104398 = pow(r104389, r104396);
        double r104399 = r104397 * r104398;
        double r104400 = pow(r104399, r104396);
        double r104401 = r104395 * r104400;
        double r104402 = r104384 * r104395;
        double r104403 = cos(r104402);
        double r104404 = r104388 / r104403;
        double r104405 = cbrt(r104404);
        double r104406 = r104401 * r104405;
        double r104407 = cbrt(r104395);
        double r104408 = r104406 * r104407;
        double r104409 = r104394 / r104408;
        double r104410 = r104390 * r104390;
        double r104411 = pow(r104410, r104392);
        double r104412 = r104388 / r104411;
        double r104413 = pow(r104383, r104392);
        double r104414 = cbrt(r104403);
        double r104415 = r104414 * r104414;
        double r104416 = r104413 / r104415;
        double r104417 = r104407 * r104407;
        double r104418 = r104416 * r104417;
        double r104419 = r104412 / r104418;
        double r104420 = r104409 * r104419;
        double r104421 = 2.1532599797541245e+301;
        bool r104422 = r104385 <= r104421;
        double r104423 = r104385 / r104403;
        double r104424 = pow(r104389, r104392);
        double r104425 = r104424 * r104395;
        double r104426 = r104423 * r104425;
        double r104427 = r104388 / r104424;
        double r104428 = r104395 / r104427;
        double r104429 = r104426 * r104428;
        double r104430 = r104388 / r104429;
        double r104431 = cbrt(r104427);
        double r104432 = r104431 * r104431;
        double r104433 = r104432 / r104416;
        double r104434 = r104433 / r104417;
        double r104435 = r104413 / r104414;
        double r104436 = r104435 * r104425;
        double r104437 = r104407 * r104436;
        double r104438 = r104431 / r104437;
        double r104439 = r104434 * r104438;
        double r104440 = r104422 ? r104430 : r104439;
        double r104441 = r104387 ? r104420 : r104440;
        return r104441;
}

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))))