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

Average Error: 35.4 → 28.8

Time: 17.3s

Precision: binary64

Cost: 58496

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/
  1.0
  (cos
   (*
    (pow (cbrt (cbrt x)) 3.0)
    (/ (/ 0.5 (pow (cbrt y) 2.0)) (/ (cbrt y) (pow (cbrt x) 2.0)))))))

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 1.0 / cos((pow(cbrt(cbrt(x)), 3.0) * ((0.5 / pow(cbrt(y), 2.0)) / (cbrt(y) / pow(cbrt(x), 2.0)))));
}

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 1.0 / Math.cos((Math.pow(Math.cbrt(Math.cbrt(x)), 3.0) * ((0.5 / Math.pow(Math.cbrt(y), 2.0)) / (Math.cbrt(y) / Math.pow(Math.cbrt(x), 2.0)))));
}

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 Float64(1.0 / cos(Float64((cbrt(cbrt(x)) ^ 3.0) * Float64(Float64(0.5 / (cbrt(y) ^ 2.0)) / Float64(cbrt(y) / (cbrt(x) ^ 2.0))))))
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[(1.0 / N[Cos[N[(N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(0.5 / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[y, 1/3], $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	35.4
Target	29.2
Herbie	28.8

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

Derivation

1. Initial program 35.4

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

2. Taylor expanded in x around inf 28.7

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

3. Applied egg-rr 28.7

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

4. Simplified 28.7

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

   [Start] 28.7	\[ \frac{1}{\cos \left(\frac{\frac{0.5}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{x}}}\right)} \]
   associate-/r/ [=>] 28.7	\[ \frac{1}{\cos \color{blue}{\left(\frac{\frac{0.5}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right)}} \]
   *-commutative [=>] 28.7	\[ \frac{1}{\cos \color{blue}{\left(\sqrt[3]{x} \cdot \frac{\frac{0.5}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{{\left(\sqrt[3]{x}\right)}^{2}}}}{\sqrt[3]{y}}\right)}} \]
   associate-/r/ [=>] 28.7	\[ \frac{1}{\cos \left(\sqrt[3]{x} \cdot \frac{\color{blue}{\frac{0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{y}}\right)} \]
   associate-/l* [=>] 28.7	\[ \frac{1}{\cos \left(\sqrt[3]{x} \cdot \color{blue}{\frac{\frac{0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\frac{\sqrt[3]{y}}{{\left(\sqrt[3]{x}\right)}^{2}}}}\right)} \]

5. Applied egg-rr 28.8

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

6. Final simplification 28.8

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

Alternatives

Alternative 1
Error	28.7
Cost	32576

\[\frac{1}{\log \log \left(1 + \mathsf{expm1}\left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

Alternative 2
Error	28.7
Cost	19648

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

Alternative 3
Error	28.7
Cost	6848

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

Alternative 4
Error	28.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023023 
(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)))))