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

Average Error: 35.8 → 28.3

Time: 4.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \le -195197189105384260:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}\\ \end{array}\]

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) (x / ((double) (y * 2.0)))) <= -1.9519718910538426e+17)) {
		VAR = 1.0;
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (((double) cbrt(((double) cos(((double) (x / ((double) (y * 2.0)))))))) * ((double) cbrt(((double) cos(((double) (x / ((double) (y * 2.0)))))))))))) * ((double) (1.0 / ((double) cbrt(((double) cos(((double) (x / ((double) (y * 2.0))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.8
Target	29.2
Herbie	28.3

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

Derivation

1. Split input into 2 regimes

2. if (/ x (* y 2.0)) < -1.9519718910538426e+17

   1. Initial program 59.1

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

   2. Taylor expanded around 0 56.5

      \[\leadsto \color{blue}{1}\]

   if -1.9519718910538426e+17 < (/ x (* y 2.0))

   1. Initial program 28.5

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

   3. Applied tan-quot 28.5

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

   5. Applied add-cube-cbrt 29.2

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

   6. Applied add-cube-cbrt 29.2

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

   7. Applied add-cube-cbrt 28.5

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

   8. Applied times-frac 28.5

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

   9. Applied times-frac 28.5

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

   10. Simplified 28.5

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

   11. Simplified 19.5

       \[\leadsto \frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \le -195197189105384260:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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)))))