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

Average Error: 35.4 → 28.4

Time: 26.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r424415 = x;
        double r424416 = y;
        double r424417 = 2.0;
        double r424418 = r424416 * r424417;
        double r424419 = r424415 / r424418;
        double r424420 = tan(r424419);
        double r424421 = sin(r424419);
        double r424422 = r424420 / r424421;
        return r424422;
}

↓

double f(double x, double y) {
        double r424423 = 1.0;
        double r424424 = x;
        double r424425 = y;
        double r424426 = 2.0;
        double r424427 = r424425 * r424426;
        double r424428 = r424424 / r424427;
        double r424429 = cos(r424428);
        double r424430 = r424423 / r424429;
        double r424431 = cbrt(r424430);
        double r424432 = cbrt(r424431);
        double r424433 = 6.0;
        double r424434 = pow(r424432, r424433);
        double r424435 = r424431 * r424434;
        return r424435;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.4
Target	28.9
Herbie	28.4

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

Derivation

1. Initial program 35.4

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

3. Applied tan-quot 35.4

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

4. Applied associate-/l/ 35.4

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

6. Applied add-cube-cbrt 35.4

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

7. Simplified 35.4

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

8. Simplified 28.4

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

10. Applied *-un-lft-identity 28.4

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

11. Applied add-cube-cbrt 28.4

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

12. Applied times-frac 28.4

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

13. Applied cbrt-prod 28.4

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

14. Applied associate-*l* 28.4

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

15. Simplified 28.4

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

16. Final simplification 28.4

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

Reproduce

herbie shell --seed 2019326 
(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)))))