Average Error: 35.9 → 28.8
Time: 12.1s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}
double f(double x, double y) {
        double r402699 = x;
        double r402700 = y;
        double r402701 = 2.0;
        double r402702 = r402700 * r402701;
        double r402703 = r402699 / r402702;
        double r402704 = tan(r402703);
        double r402705 = sin(r402703);
        double r402706 = r402704 / r402705;
        return r402706;
}
↓
double f(double x, double y) {
        double r402707 = 1.0;
        double r402708 = x;
        double r402709 = y;
        double r402710 = 2.0;
        double r402711 = r402709 * r402710;
        double r402712 = r402708 / r402711;
        double r402713 = cos(r402712);
        double r402714 = r402707 / r402713;
        return r402714;
}

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
Target29.3
Herbie28.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.230369091130699363447511617672816900781 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.102852406811913849731222630299032206502 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

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 add-cbrt-cube50.7

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

  4. Applied add-cbrt-cube50.5

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

  5. Applied cbrt-undiv50.5

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

  6. Simplified35.9

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

  8. Applied tan-quot35.9

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

  10. Applied *-un-lft-identity35.9

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

  11. Applied *-un-lft-identity35.9

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

  12. Applied times-frac35.9

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

  13. Applied associate-/l*35.9

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

  14. Simplified28.8

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

  15. Final simplification28.8

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.23036909113069936e114) 1 (if (< y -9.1028524068119138e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

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