Average Error: 35.2 → 28.1
Time: 6.8s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}
double code(double x, double y) {
	return (((double) tan((x / ((double) (y * 2.0))))) / ((double) sin((x / ((double) (y * 2.0))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (((double) cbrt((1.0 / ((double) (((double) cbrt(((double) cos((x / ((double) (y * 2.0))))))) * ((double) cbrt(((double) cos((x / ((double) (y * 2.0)))))))))))) * ((double) cbrt((1.0 / ((double) (((double) cbrt(((double) cos((x / ((double) (y * 2.0))))))) * ((double) cbrt(((double) cos((x / ((double) (y * 2.0)))))))))))))) * ((double) (((double) cbrt((1.0 / ((double) cbrt(((double) cos((x / ((double) (y * 2.0)))))))))) * ((double) cbrt((1.0 / ((double) cbrt(((double) cos((x / ((double) (y * 2.0)))))))))))))) * ((double) cbrt((1.0 / ((double) cos((x / ((double) (y * 2.0))))))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.2
Target28.6
Herbie28.1
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 35.2

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

  3. Applied tan-quot35.2

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

  4. Applied associate-/l/35.2

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

  6. Applied add-cube-cbrt35.2

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

  7. Simplified35.2

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

  8. Simplified28.1

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

  10. Applied add-cube-cbrt28.1

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

  11. Applied add-cube-cbrt28.1

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

  12. Applied times-frac28.1

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

  13. Applied cbrt-prod28.1

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

  14. Applied add-cube-cbrt28.1

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

  15. Applied add-cube-cbrt28.1

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

  16. Applied times-frac28.1

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

  17. Applied cbrt-prod28.1

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

  18. Applied swap-sqr28.1

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

  19. Simplified28.1

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

  20. Simplified28.1

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

  21. Final simplification28.1

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

Reproduce

herbie shell --seed 2020196 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))