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

Average Error: 27.8 → 2.9

Time: 48.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{sin}^{2} \le 5.7216187708641599 \cdot 10^{-292} \lor \neg \left({sin}^{2} \le 3.870929869616987 \cdot 10^{212}\right):\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{\sqrt[3]{1}}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{x}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{x} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{sin}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

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)) <= 5.72161877086416e-292) || !(((double) pow(sin, 2.0)) <= 3.870929869616987e+212))) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))) / 1.0)) / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) (((double) (((double) cbrt(1.0)) / ((double) pow(cos, ((double) (2.0 / 2.0)))))) / ((double) pow(sin, ((double) (2.0 / 2.0)))))))) * ((double) (((double) (((double) (1.0 / ((double) (((double) pow(cos, ((double) (2.0 / 2.0)))) * ((double) pow(sin, ((double) (2.0 / 2.0)))))))) * ((double) (((double) cos(((double) (2.0 * x)))) / x)))) / ((double) cbrt(x))))));
	} else {
		VAR = ((double) (((double) (((double) (1.0 / ((double) pow(cos, ((double) (2.0 / 2.0)))))) / 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) pow(sin, ((double) (2.0 / 2.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if (pow sin 2.0) < 5.72161877086416e-292 or 3.870929869616987e+212 < (pow sin 2.0)

   1. Initial program 32.1

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

   3. Applied sqr-pow 32.1

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

   4. Applied associate-*r* 21.8

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

   5. Applied associate-*l* 18.7

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

   6. Applied associate-*r* 15.8

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

   7. Applied associate-/r* 15.6

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

   9. Applied add-cube-cbrt 15.9

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

   10. Applied associate-*r* 15.9

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

   11. Applied sqr-pow 15.9

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

   12. Applied associate-*l* 7.2

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

   13. Applied *-un-lft-identity 7.2

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

   14. Applied times-frac 6.9

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

   15. Applied times-frac 3.5

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

   17. Applied *-commutative 3.5

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

   18. Applied associate-*r* 4.7

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

   19. Applied *-un-lft-identity 4.7

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

   20. Applied times-frac 4.9

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

   22. Applied *-commutative 4.9

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

   23. Applied *-un-lft-identity 4.9

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

   24. Applied unpow-prod-down 4.9

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

   25. Applied add-cube-cbrt 4.9

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

   26. Applied times-frac 4.9

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

   27. Applied times-frac 3.6

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

   28. Simplified 3.6

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

   if 5.72161877086416e-292 < (pow sin 2.0) < 3.870929869616987e+212

   1. Initial program 21.8

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

   3. Applied sqr-pow 21.8

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

   4. Applied associate-*r* 21.8

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

   5. Applied associate-*l* 21.8

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

   6. Applied associate-*r* 16.5

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

   7. Applied associate-/r* 16.3

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

   9. Applied *-commutative 16.3

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

   10. Applied sqr-pow 16.3

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

   11. Applied associate-*l* 7.1

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

   12. Applied *-un-lft-identity 7.1

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

   13. Applied times-frac 6.8

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

   14. Applied times-frac 1.9

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;{sin}^{2} \le 5.7216187708641599 \cdot 10^{-292} \lor \neg \left({sin}^{2} \le 3.870929869616987 \cdot 10^{212}\right):\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{\sqrt[3]{1}}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{x}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{x} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{sin}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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))))