Average Error: 28.8 → 2.9
Time: 20.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\left(\left(\sqrt[3]{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{1}}}}\right) \cdot \sqrt[3]{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\left(\left(\sqrt[3]{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}}}{\sqrt[3]{\frac{{cos}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{1}}}}\right) \cdot \sqrt[3]{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{1}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}}\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)}
double code(double x, double cos, double sin) {
	return (cos((2.0 * x)) / (pow(cos, 2.0) * ((x * pow(sin, 2.0)) * x)));
}
↓
double code(double x, double cos, double sin) {
	return (((cbrt(((1.0 / pow(cos, (2.0 / 2.0))) * (1.0 / (x * pow(sin, (2.0 / 2.0)))))) * (cbrt(((cbrt(1.0) * cbrt(1.0)) * (1.0 / (x * pow(sin, (2.0 / 2.0)))))) / cbrt((pow(cos, (2.0 / 2.0)) / cbrt(1.0))))) * cbrt(((1.0 / pow(cos, (2.0 / 2.0))) * (1.0 / (x * pow(sin, (2.0 / 2.0))))))) * (cos((2.0 * x)) / (pow(cos, (2.0 / 2.0)) * (x * pow(sin, (2.0 / 2.0))))));
}

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 28.8

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

  3. Applied sqr-pow28.8

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

  4. Applied associate-*r*22.4

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

  5. Applied associate-*l*20.4

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

  6. Applied associate-*r*16.3

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

  8. Applied add-sqr-sqrt40.0

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

  9. Applied add-sqr-sqrt40.0

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

  10. Applied sqr-pow40.0

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

  11. Applied unswap-sqr35.1

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

  12. Applied unswap-sqr33.1

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

  13. Applied *-un-lft-identity33.1

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

  14. Applied times-frac33.0

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

  15. Simplified33.0

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

  16. Simplified2.6

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

  18. Applied add-cube-cbrt3.0

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

  20. Applied add-cube-cbrt3.0

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

  21. Applied associate-/l*3.0

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

  22. Applied associate-*l/3.0

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

  23. Applied cbrt-div2.9

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

  24. Final simplification2.9

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

Reproduce

herbie shell --seed 2020066 +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))))