Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5

Average Error: 35.9 → 28.8

Time: 18.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r622670 = x;
        double r622671 = y;
        double r622672 = 2.0;
        double r622673 = r622671 * r622672;
        double r622674 = r622670 / r622673;
        double r622675 = tan(r622674);
        double r622676 = sin(r622674);
        double r622677 = r622675 / r622676;
        return r622677;
}

↓

double f(double x, double y) {
        double r622678 = 1.0;
        double r622679 = x;
        double r622680 = y;
        double r622681 = 2.0;
        double r622682 = r622680 * r622681;
        double r622683 = r622679 / r622682;
        double r622684 = cos(r622683);
        double r622685 = r622678 / r622684;
        double r622686 = cbrt(r622685);
        double r622687 = cbrt(r622684);
        double r622688 = r622678 / r622687;
        double r622689 = r622688 / r622687;
        double r622690 = cbrt(r622689);
        double r622691 = r622690 * r622686;
        double r622692 = cbrt(r622688);
        double r622693 = r622691 * r622692;
        double r622694 = r622686 * r622693;
        return r622694;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.9
Target	29.1
Herbie	28.8

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

Derivation

1. Initial program 35.9

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

2. Simplified 35.9

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

4. Applied tan-quot 35.9

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

5. Applied associate-/l/ 35.9

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

6. Simplified 35.9

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

8. Applied add-cube-cbrt 35.9

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

9. Simplified 35.9

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

10. Simplified 28.8

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

12. Applied add-cube-cbrt 28.8

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

13. Applied *-un-lft-identity 28.8

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

14. Applied times-frac 28.8

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

15. Applied cbrt-prod 28.8

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

16. Applied associate-*r* 28.8

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

17. Simplified 28.8

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

18. Final simplification 28.8

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"

  :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)))))