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

Average Accuracy: 43.4% → 54.6%

Time: 18.2s

Precision: binary64

Cost: 45504

Math

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

↓

\[\sqrt[3]{{\cos \left({\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))

↓

(FPCore (x y)
 :precision binary64
 (cbrt
  (pow
   (cos (* (pow (/ (cbrt 0.5) (cbrt y)) 2.0) (* (cbrt (/ 0.5 y)) x)))
   -3.0)))

C

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

↓

double code(double x, double y) {
	return cbrt(pow(cos((pow((cbrt(0.5) / cbrt(y)), 2.0) * (cbrt((0.5 / y)) * x))), -3.0));
}

Java

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

↓

public static double code(double x, double y) {
	return Math.cbrt(Math.pow(Math.cos((Math.pow((Math.cbrt(0.5) / Math.cbrt(y)), 2.0) * (Math.cbrt((0.5 / y)) * x))), -3.0));
}

Julia

function code(x, y)
	return Float64(tan(Float64(x / Float64(y * 2.0))) / sin(Float64(x / Float64(y * 2.0))))
end

↓

function code(x, y)
	return cbrt((cos(Float64((Float64(cbrt(0.5) / cbrt(y)) ^ 2.0) * Float64(cbrt(Float64(0.5 / y)) * x))) ^ -3.0))
end

Wolfram

code[x_, y_] := N[(N[Tan[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[Power[N[Power[N[Cos[N[(N[Power[N[(N[Power[0.5, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(0.5 / y), $MachinePrecision], 1/3], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]

Target

Original	43.4%
Target	54.1%
Herbie	54.6%

\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

1. Initial program 43.4%

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

2. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-3}}} \]
   Proof

   [Start] 43.4	\[ \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   add-cbrt-cube [=>] 43.4	\[ \color{blue}{\sqrt[3]{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]
   pow3 [=>] 43.4	\[ \sqrt[3]{\color{blue}{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}} \]
   clear-num [=>] 43.4	\[ \sqrt[3]{{\color{blue}{\left(\frac{1}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}}\right)}}^{3}} \]
   inv-pow [=>] 43.4	\[ \sqrt[3]{{\color{blue}{\left({\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}^{-1}\right)}}^{3}} \]
   metadata-eval [<=] 43.4	\[ \sqrt[3]{{\left({\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}^{-1}\right)}^{\color{blue}{\left(1 + 2\right)}}} \]
   pow-pow [=>] 43.4	\[ \sqrt[3]{\color{blue}{{\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}^{\left(-1 \cdot \left(1 + 2\right)\right)}}} \]

3. Applied egg-rr 54.7%

   \[\leadsto \sqrt[3]{{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}^{-3}} \]
   Proof

   [Start] 54.8	\[ \sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-3}} \]
   add-cube-cbrt [=>] 54.6	\[ \sqrt[3]{{\cos \color{blue}{\left(\left(\sqrt[3]{x \cdot \frac{0.5}{y}} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}}^{-3}} \]
   pow3 [=>] 54.7	\[ \sqrt[3]{{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}^{-3}} \]

4. Applied egg-rr 54.6%

   \[\leadsto \sqrt[3]{{\cos \color{blue}{\left({\left(\sqrt[3]{\frac{0.5}{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}}^{-3}} \]
   Proof

   [Start] 54.7	\[ \sqrt[3]{{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}^{-3}} \]
   unpow3 [=>] 54.6	\[ \sqrt[3]{{\cos \color{blue}{\left(\left(\sqrt[3]{x \cdot \frac{0.5}{y}} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}}^{-3}} \]
   add-cube-cbrt [<=] 54.8	\[ \sqrt[3]{{\cos \color{blue}{\left(x \cdot \frac{0.5}{y}\right)}}^{-3}} \]
   *-commutative [=>] 54.8	\[ \sqrt[3]{{\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}}^{-3}} \]
   add-cube-cbrt [=>] 54.6	\[ \sqrt[3]{{\cos \left(\color{blue}{\left(\left(\sqrt[3]{\frac{0.5}{y}} \cdot \sqrt[3]{\frac{0.5}{y}}\right) \cdot \sqrt[3]{\frac{0.5}{y}}\right)} \cdot x\right)}^{-3}} \]
   associate-*l* [=>] 54.6	\[ \sqrt[3]{{\cos \color{blue}{\left(\left(\sqrt[3]{\frac{0.5}{y}} \cdot \sqrt[3]{\frac{0.5}{y}}\right) \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}}^{-3}} \]
   pow2 [=>] 54.6	\[ \sqrt[3]{{\cos \left(\color{blue}{{\left(\sqrt[3]{\frac{0.5}{y}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]

5. Applied egg-rr 54.6%

   \[\leadsto \sqrt[3]{{\cos \left({\color{blue}{\left(\sqrt[3]{0.5} \cdot \frac{1}{\sqrt[3]{y}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   Proof

   [Start] 54.6	\[ \sqrt[3]{{\cos \left({\left(\sqrt[3]{\frac{0.5}{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   cbrt-div [=>] 54.6	\[ \sqrt[3]{{\cos \left({\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{y}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   div-inv [=>] 54.6	\[ \sqrt[3]{{\cos \left({\color{blue}{\left(\sqrt[3]{0.5} \cdot \frac{1}{\sqrt[3]{y}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]

6. Simplified 54.6%

   \[\leadsto \sqrt[3]{{\cos \left({\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{y}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   Proof

   [Start] 54.6	\[ \sqrt[3]{{\cos \left({\left(\sqrt[3]{0.5} \cdot \frac{1}{\sqrt[3]{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   associate-*r/ [=>] 54.6	\[ \sqrt[3]{{\cos \left({\color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot 1}{\sqrt[3]{y}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]
   *-rgt-identity [=>] 54.6	\[ \sqrt[3]{{\cos \left({\left(\frac{\color{blue}{\sqrt[3]{0.5}}}{\sqrt[3]{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]

7. Final simplification 54.6%

   \[\leadsto \sqrt[3]{{\cos \left({\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{0.5}{y}} \cdot x\right)\right)}^{-3}} \]

Alternatives

Alternative 1
Accuracy	54.7%
Cost	32448

\[\sqrt[3]{{\cos \left({\left(\sqrt[3]{\frac{0.5}{y} \cdot x}\right)}^{3}\right)}^{-3}} \]

Alternative 2
Accuracy	54.7%
Cost	32448

\[\sqrt[3]{{\cos \left({\left(\sqrt[3]{y \cdot \frac{2}{x}}\right)}^{-3}\right)}^{-3}} \]

Alternative 3
Accuracy	54.7%
Cost	19712

\[\frac{1}{\cos \left({\left(\sqrt[3]{0.5 \cdot \frac{x}{y}}\right)}^{3}\right)} \]

Alternative 4
Accuracy	54.8%
Cost	19584

\[\sqrt[3]{{\cos \left(\frac{0.5}{y} \cdot x\right)}^{-3}} \]

Alternative 5
Accuracy	54.8%
Cost	6848

\[\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]

Alternative 6
Accuracy	54.9%
Cost	6848

\[\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)} \]

Alternative 7
Accuracy	54.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023153 
(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) 1.0 (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)))))