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

Average Error: 35.4 → 28.4

Time: 25.4s

Precision: 64

Math

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

↓

\[\left({\left(\sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

C

double f(double x, double y) {
        double r383187 = x;
        double r383188 = y;
        double r383189 = 2.0;
        double r383190 = r383188 * r383189;
        double r383191 = r383187 / r383190;
        double r383192 = tan(r383191);
        double r383193 = sin(r383191);
        double r383194 = r383192 / r383193;
        return r383194;
}

↓

double f(double x, double y) {
        double r383195 = 1.0;
        double r383196 = x;
        double r383197 = y;
        double r383198 = 2.0;
        double r383199 = r383197 * r383198;
        double r383200 = r383196 / r383199;
        double r383201 = cos(r383200);
        double r383202 = r383195 / r383201;
        double r383203 = cbrt(r383202);
        double r383204 = cbrt(r383203);
        double r383205 = 5.0;
        double r383206 = pow(r383204, r383205);
        double r383207 = r383206 * r383204;
        double r383208 = r383207 * r383203;
        return r383208;
}

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 add-cube-cbrt 28.4

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

11. Applied associate-*r* 28.4

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

12. Simplified 28.4

    \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

13. Final simplification 28.4

    \[\leadsto \left({\left(\sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))))