Average Error: 36.3 → 29.0
Time: 5.0s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\sqrt[3]{{\left(\frac{1}{\cos \left(\frac{1}{y} \cdot \frac{x}{2}\right)}\right)}^{3}}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\sqrt[3]{{\left(\frac{1}{\cos \left(\frac{1}{y} \cdot \frac{x}{2}\right)}\right)}^{3}}
double code(double x, double y) {
	return (tan((x / (y * 2.0))) / sin((x / (y * 2.0))));
}
↓
double code(double x, double y) {
	return cbrt(pow((1.0 / cos(((1.0 / y) * (x / 2.0)))), 3.0));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target29.3
Herbie29.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 36.3

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

  3. Applied tan-quot36.3

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

  4. Applied associate-/l/36.3

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

  6. Applied add-cbrt-cube36.3

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

  7. Applied add-cbrt-cube50.4

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

  8. Applied cbrt-unprod50.4

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

  9. Applied add-cbrt-cube50.2

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

  10. Applied cbrt-undiv50.2

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

  11. Simplified28.9

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

  13. Applied *-un-lft-identity28.9

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

  14. Applied times-frac29.0

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

  15. Final simplification29.0

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

Reproduce

herbie shell --seed 2020100 
(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 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

  (/ (tan (/ x (* y 2))) (sin (/ x (* y 2)))))