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

Average Error: 27.8 → 2.8

Time: 59.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{sin}^{2} \le 4.94066 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \left(\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\sqrt[3]{\sqrt[3]{x}}}\right)\\ \mathbf{elif}\;{sin}^{2} \le 1.81775767047105466 \cdot 10^{306}:\\ \;\;\;\;\frac{\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {1}^{\left(\frac{2}{2}\right)}\right)}}{\sqrt[3]{x} \cdot {sin}^{\left(\frac{2}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{\sqrt[3]{x}}\\ \end{array}\]

C

double code(double x, double cos, double sin) {
	return ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(cos, 2.0)) * ((double) (((double) (x * ((double) pow(sin, 2.0)))) * x))))));
}

↓

double code(double x, double cos, double sin) {
	double VAR;
	if ((((double) pow(sin, 2.0)) <= 4.9406564584125e-324)) {
		VAR = ((double) (((double) (((double) (1.0 / ((double) pow(cos, ((double) (2.0 / 2.0)))))) / ((double) (((double) pow(sin, ((double) (2.0 / 2.0)))) * ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) * ((double) (((double) (((double) (1.0 / ((double) (((double) pow(cos, ((double) (2.0 / 2.0)))) * ((double) pow(sin, ((double) (2.0 / 2.0)))))))) / ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) * ((double) (((double) (((double) cos(((double) (2.0 * x)))) / x)) / ((double) cbrt(((double) cbrt(x))))))))));
	} else {
		double VAR_1;
		if ((((double) pow(sin, 2.0)) <= 1.8177576704710547e+306)) {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / ((double) pow(cos, ((double) (2.0 / 2.0)))))) / ((double) (((double) pow(sin, ((double) (2.0 / 2.0)))) * ((double) (((double) cbrt(x)) * ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))))))) / ((double) cbrt(((double) cbrt(x)))))) * ((double) (((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(cos, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(1.0, ((double) (2.0 / 2.0)))))))))) / ((double) (((double) cbrt(x)) * ((double) pow(sin, ((double) (2.0 / 2.0))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / ((double) pow(cos, ((double) (2.0 / 2.0)))))) / ((double) (((double) pow(sin, ((double) (2.0 / 2.0)))) * ((double) (((double) cbrt(x)) * ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))))))) / ((double) cbrt(((double) cbrt(x)))))) * ((double) (((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(cos, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(sin, ((double) (2.0 / 2.0)))))))))) / ((double) cbrt(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 3 regimes

2. if (pow sin 2.0) < 4.9406564584125e-324

   1. Initial program 63.9

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

   3. Applied sqr-pow 63.9

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

   4. Applied associate-*r* 35.7

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

   5. Applied associate-*l* 27.8

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

   6. Applied associate-*r* 22.6

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

   7. Applied associate-/r* 22.5

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

   9. Applied add-cube-cbrt 22.8

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

   10. Applied associate-*r* 22.8

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

   11. Applied sqr-pow 22.8

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

   12. Applied associate-*l* 11.7

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

   13. Applied *-un-lft-identity 11.7

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

   14. Applied times-frac 11.0

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

   15. Applied times-frac 6.4

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

   17. Applied add-cube-cbrt 6.4

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

   18. Applied cbrt-prod 6.5

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

   19. Applied *-commutative 6.5

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

   20. Applied associate-*r* 8.2

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

   21. Applied *-un-lft-identity 8.2

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

   22. Applied times-frac 9.0

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

   23. Applied times-frac 10.5

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

   if 4.9406564584125e-324 < (pow sin 2.0) < 1.8177576704710547e+306

   1. Initial program 21.1

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

   3. Applied sqr-pow 21.1

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

   4. Applied associate-*r* 20.9

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

   5. Applied associate-*l* 20.9

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

   6. Applied associate-*r* 16.1

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

   7. Applied associate-/r* 16.0

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

   9. Applied add-cube-cbrt 16.2

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

   10. Applied associate-*r* 16.2

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

   11. Applied sqr-pow 16.2

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

   12. Applied associate-*l* 7.5

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

   13. Applied *-un-lft-identity 7.5

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

   14. Applied times-frac 7.2

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

   15. Applied times-frac 2.4

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

   17. Applied add-cube-cbrt 2.4

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

   18. Applied cbrt-prod 2.5

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

   19. Applied associate-*r* 2.5

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

   20. Applied associate-*r* 2.5

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

   21. Applied associate-/r* 2.5

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

   23. Applied *-un-lft-identity 2.5

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

   24. Applied unpow-prod-down 2.5

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

   25. Applied associate-*r* 2.5

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

   26. Applied associate-*r* 1.0

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

   27. Applied associate-/r* 1.0

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

   28. Applied associate-/l/ 1.0

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

   if 1.8177576704710547e+306 < (pow sin 2.0)

   1. Initial program 24.5

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

   3. Applied sqr-pow 24.5

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

   4. Applied associate-*r* 17.9

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

   5. Applied associate-*l* 15.5

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

   6. Applied associate-*r* 13.4

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

   7. Applied associate-/r* 13.3

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

   9. Applied add-cube-cbrt 13.5

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

   10. Applied associate-*r* 13.5

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

   11. Applied sqr-pow 13.5

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

   12. Applied associate-*l* 5.3

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

   13. Applied *-un-lft-identity 5.3

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

   14. Applied times-frac 5.3

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

   15. Applied times-frac 2.7

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

   17. Applied add-cube-cbrt 2.7

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

   18. Applied cbrt-prod 2.8

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

   19. Applied associate-*r* 2.8

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

   20. Applied associate-*r* 2.7

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

   21. Applied associate-/r* 2.7

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;{sin}^{2} \le 4.94066 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \left(\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\sqrt[3]{\sqrt[3]{x}}}\right)\\ \mathbf{elif}\;{sin}^{2} \le 1.81775767047105466 \cdot 10^{306}:\\ \;\;\;\;\frac{\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {1}^{\left(\frac{2}{2}\right)}\right)}}{\sqrt[3]{x} \cdot {sin}^{\left(\frac{2}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{\sqrt[3]{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))