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

Average Error: 35.4 → 27.2

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 4.5107050099646067:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(\frac{x}{y \cdot 2}\right)\right)\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

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) log1p(((double) log1p(((double) expm1(((double) expm1(((double) (((double) log1p(((double) expm1(((double) tan(((double) (x / ((double) (y * 2.0)))))))))) / ((double) sin(((double) (x / ((double) (y * 2.0))))))))))))))));
	} 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 log1p-expm1-u 25.7

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

   5. Applied log1p-expm1-u 25.7

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

   7. Applied log1p-expm1-u 25.7

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 4.5107050099646067:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(\frac{x}{y \cdot 2}\right)\right)\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +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)))))