Average Error: 28.0 → 3.3
Time: 35.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{1}{\left(\left(x \cdot \frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(x \cdot 2\right)}}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{x}} \cdot \frac{\frac{1}{\sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \frac{1}{\sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(x \cdot 2\right)} \cdot \sqrt[3]{\cos \left(x \cdot 2\right)}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{1}{\left(\left(x \cdot \frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\cos \left(x \cdot 2\right)}}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{x}} \cdot \frac{\frac{1}{\sqrt[3]{{sin}^{\left(\frac{2}{2}\right)}}} \cdot \frac{1}{\sqrt[3]{{sin}^{\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)}}}
double f(double x, double cos, double sin) {
        double r101377 = 2.0;
        double r101378 = x;
        double r101379 = r101377 * r101378;
        double r101380 = cos(r101379);
        double r101381 = cos;
        double r101382 = pow(r101381, r101377);
        double r101383 = sin;
        double r101384 = pow(r101383, r101377);
        double r101385 = r101378 * r101384;
        double r101386 = r101385 * r101378;
        double r101387 = r101382 * r101386;
        double r101388 = r101380 / r101387;
        return r101388;
}
↓
double f(double x, double cos, double sin) {
        double r101389 = 1.0;
        double r101390 = x;
        double r101391 = cos;
        double r101392 = 2.0;
        double r101393 = 2.0;
        double r101394 = r101392 / r101393;
        double r101395 = pow(r101391, r101394);
        double r101396 = r101390 * r101392;
        double r101397 = cos(r101396);
        double r101398 = cbrt(r101397);
        double r101399 = r101395 / r101398;
        double r101400 = r101390 * r101399;
        double r101401 = sin;
        double r101402 = pow(r101401, r101394);
        double r101403 = r101400 * r101402;
        double r101404 = cbrt(r101402);
        double r101405 = r101403 * r101404;
        double r101406 = r101389 / r101405;
        double r101407 = cbrt(r101390);
        double r101408 = r101406 / r101407;
        double r101409 = r101389 / r101404;
        double r101410 = r101409 * r101409;
        double r101411 = r101407 * r101407;
        double r101412 = r101398 * r101398;
        double r101413 = r101395 / r101412;
        double r101414 = r101411 * r101413;
        double r101415 = r101410 / r101414;
        double r101416 = r101408 * r101415;
        return r101416;
}

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. Simplified26.5

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

  4. Applied sqr-pow26.5

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

  5. Applied *-un-lft-identity26.5

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

  6. Applied times-frac26.5

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

  7. Applied associate-/l*21.3

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

  8. Simplified18.4

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

  10. Applied add-cube-cbrt18.5

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

  11. Applied sqr-pow18.5

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

  12. Applied times-frac18.5

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

  13. Applied associate-*l*10.4

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

  14. Simplified9.7

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

  16. Applied add-cube-cbrt9.9

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

  17. Applied add-cube-cbrt10.0

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

  18. Applied add-cube-cbrt10.0

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

  19. Applied times-frac10.0

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

  20. Applied times-frac8.7

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

  21. Applied times-frac4.0

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

  22. Simplified3.3

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

  23. Simplified3.3

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

  24. Final simplification3.3

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

Reproduce

herbie shell --seed 2019179 +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))))