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

Average Error: 35.9 → 27.4

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 3.04266136491719:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double temp;
	if (((tan((x / (y * 2.0))) / sin((x / (y * 2.0)))) <= 3.042661364917194)) {
		temp = log1p(expm1((((cbrt(tan((x / (y * 2.0)))) * cbrt(tan((x / (y * 2.0))))) / (cbrt(sin((x / (y * 2.0)))) * cbrt(sin((x / (y * 2.0)))))) * (cbrt(tan((x / (y * 2.0)))) / cbrt(sin((x / (y * 2.0))))))));
	} else {
		temp = 1.0;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.9
Target	28.7
Herbie	27.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 3.042661364917194

   1. Initial program 25.4

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

   3. Applied log1p-expm1-u 25.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 26.1

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\left(\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}}\right)\right)\]

   6. Applied add-cube-cbrt 25.4

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\left(\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}}{\left(\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right)\]

   7. Applied times-frac 25.4

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}}\right)\right)\]

   if 3.042661364917194 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 63.0

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

   2. Taylor expanded around 0 32.4

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

4. Final simplification 27.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 3.04266136491719:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))