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

Average Error: 35.9 → 28.8

Time: 17.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r514599 = x;
        double r514600 = y;
        double r514601 = 2.0;
        double r514602 = r514600 * r514601;
        double r514603 = r514599 / r514602;
        double r514604 = tan(r514603);
        double r514605 = sin(r514603);
        double r514606 = r514604 / r514605;
        return r514606;
}

↓

double f(double x, double y) {
        double r514607 = 1.0;
        double r514608 = x;
        double r514609 = y;
        double r514610 = r514608 / r514609;
        double r514611 = 2.0;
        double r514612 = r514610 / r514611;
        double r514613 = cos(r514612);
        double r514614 = cbrt(r514613);
        double r514615 = r514607 / r514614;
        double r514616 = r514615 / r514614;
        double r514617 = cbrt(r514616);
        double r514618 = r514617 * r514617;
        double r514619 = cbrt(r514615);
        double r514620 = r514619 * r514619;
        double r514621 = r514618 * r514620;
        double r514622 = r514607 / r514613;
        double r514623 = cbrt(r514622);
        double r514624 = r514621 * r514623;
        return r514624;
}

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. Using strategy rm

3. Applied tan-quot 35.9

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

4. Applied associate-/l/ 35.9

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

5. Simplified 35.9

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

7. Applied add-cube-cbrt 35.9

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

8. Simplified 35.9

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

9. Simplified 28.8

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

11. Applied add-cube-cbrt 28.8

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

12. Applied add-cube-cbrt 28.8

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

13. Applied times-frac 28.8

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

14. Applied cbrt-prod 28.8

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

15. Applied add-cube-cbrt 28.8

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

16. Applied add-cube-cbrt 28.8

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

17. Applied times-frac 28.8

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

18. Applied cbrt-prod 28.8

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

19. Applied swap-sqr 28.8

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

20. Simplified 28.8

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

21. Simplified 28.8

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

22. Final simplification 28.8

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

Reproduce

herbie shell --seed 2019196 
(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)))))