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

Average Error: 35.4 → 27.2

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 4.5107050099646067:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double code(double x, double y) {
	return ((double) (((double) tan(((double) (x / ((double) (y * 2.0)))))) / ((double) sin(((double) (x / ((double) (y * 2.0))))))));
}

↓

double code(double x, double y) {
	double VAR;
	if ((((double) (((double) tan(((double) (x / ((double) (y * 2.0)))))) / ((double) sin(((double) (x / ((double) (y * 2.0)))))))) <= 4.510705009964607)) {
		VAR = ((double) cbrt(((double) cbrt(((double) pow(((double) pow(((double) cbrt(((double) pow(((double) (((double) tan(((double) (x / ((double) (y * 2.0)))))) / ((double) sin(((double) (x / ((double) (y * 2.0)))))))), 3.0)))), 3.0)), 3.0))))));
	} else {
		VAR = 1.0;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.4
Target	28.8
Herbie	27.2

\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 4.510705009964607

   1. Initial program 25.7

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

   3. Applied add-cbrt-cube 45.6

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

   4. Applied add-cbrt-cube 45.2

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

   5. Applied cbrt-undiv 45.2

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

   6. Simplified 25.7

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

   8. Applied add-cbrt-cube 25.7

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

   9. Simplified 25.7

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

   11. Applied add-cbrt-cube 45.5

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

   12. Applied add-cbrt-cube 45.2

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

   13. Applied cbrt-undiv 45.2

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

   14. Simplified 25.7

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

   if 4.510705009964607 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 63.2

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

   2. Taylor expanded around 0 31.6

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 4.5107050099646067:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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)))))