Average Error: 35.9 → 28.9
Time: 5.6s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot {\left({\left(\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}^{6}\right)}^{0.3333333333333333}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot {\left({\left(\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}^{6}\right)}^{0.3333333333333333}
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (*
  (cbrt (/ 1.0 (cos (/ x (* y 2.0)))))
  (pow (pow (cbrt (/ 1.0 (cos (/ x (* y 2.0))))) 6.0) 0.3333333333333333)))
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(1.0 / cos(x / (y * 2.0))) * pow(pow(cbrt(1.0 / cos(x / (y * 2.0))), 6.0), 0.3333333333333333);
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.9
Target29.5
Herbie28.9
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 35.9

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

  3. Applied tan-quot_binary6435.9

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

  5. Applied add-cube-cbrt_binary6435.9

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

  6. Simplified35.9

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

  7. Simplified28.9

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

  9. Applied pow1/3_binary6430.7

    \[\leadsto \left(\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \color{blue}{{\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

  10. Applied pow1/3_binary6430.7

    \[\leadsto \left(\color{blue}{{\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}^{0.3333333333333333}} \cdot {\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

  11. Applied pow-prod-down_binary6428.9

    \[\leadsto \color{blue}{{\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

  12. Simplified28.9

    \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}^{6}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

  13. Final simplification28.9

    \[\leadsto \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot {\left({\left(\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}^{6}\right)}^{0.3333333333333333}\]

Reproduce

herbie shell --seed 2020220 
(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)))))