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

Average Error: 35.4 → 27.5

Time: 15.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \le 2.69286171955532038779779213655274361372:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)}\right)}^{3}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r1166100 = x;
        double r1166101 = y;
        double r1166102 = 2.0;
        double r1166103 = r1166101 * r1166102;
        double r1166104 = r1166100 / r1166103;
        double r1166105 = tan(r1166104);
        double r1166106 = sin(r1166104);
        double r1166107 = r1166105 / r1166106;
        return r1166107;
}

↓

double f(double x, double y) {
        double r1166108 = x;
        double r1166109 = 2.0;
        double r1166110 = y;
        double r1166111 = r1166109 * r1166110;
        double r1166112 = r1166108 / r1166111;
        double r1166113 = tan(r1166112);
        double r1166114 = sin(r1166112);
        double r1166115 = r1166113 / r1166114;
        double r1166116 = 2.6928617195553204;
        bool r1166117 = r1166115 <= r1166116;
        double r1166118 = 3.0;
        double r1166119 = pow(r1166115, r1166118);
        double r1166120 = cbrt(r1166119);
        double r1166121 = pow(r1166120, r1166118);
        double r1166122 = pow(r1166121, r1166118);
        double r1166123 = cbrt(r1166122);
        double r1166124 = cbrt(r1166123);
        double r1166125 = 1.0;
        double r1166126 = r1166117 ? r1166124 : r1166125;
        return r1166126;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.4
Target	28.9
Herbie	27.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 2.6928617195553204

   1. Initial program 24.9

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

   2. Simplified 24.9

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

   4. Applied add-cbrt-cube 45.6

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

   5. Applied add-cbrt-cube 45.2

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

   6. Applied cbrt-undiv 45.2

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

   7. Simplified 24.9

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

   9. Applied add-cbrt-cube 24.9

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

   10. Simplified 24.9

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

   12. Applied add-cbrt-cube 45.5

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

   13. Applied add-cbrt-cube 45.2

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

   14. Applied cbrt-undiv 45.2

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

   15. Simplified 24.9

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

   if 2.6928617195553204 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 62.7

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

   2. Simplified 62.7

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

   3. Taylor expanded around 0 34.1

      \[\leadsto \color{blue}{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \le 2.69286171955532038779779213655274361372:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)}\right)}^{3}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))))