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

Average Error: 35.8 → 28.8

Time: 15.1s

Precision: binary64

Cost: 26368

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{x \cdot 0.5}\\ \frac{1}{\cos \left({t_0}^{2} \cdot \frac{t_0}{y}\right)} \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (* x 0.5)))) (/ 1.0 (cos (* (pow t_0 2.0) (/ t_0 y))))))

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 t_0 = cbrt((x * 0.5));
	return 1.0 / cos((pow(t_0, 2.0) * (t_0 / y)));
}

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) {
	double t_0 = Math.cbrt((x * 0.5));
	return 1.0 / Math.cos((Math.pow(t_0, 2.0) * (t_0 / y)));
}

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)
	t_0 = cbrt(Float64(x * 0.5))
	return Float64(1.0 / cos(Float64((t_0 ^ 2.0) * Float64(t_0 / y))))
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_] := Block[{t$95$0 = N[Power[N[(x * 0.5), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[Cos[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	35.8
Target	28.9
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 35.8

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

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 28.8

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

4. Simplified 28.8

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

   [Start] 28.8	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{0.5 \cdot x} \cdot \sqrt[3]{0.5 \cdot x}}{\frac{y}{\sqrt[3]{0.5 \cdot x}}}\right)} \]
   associate-/l* [=>] 28.8	\[ \frac{1}{\cos \color{blue}{\left(\frac{\sqrt[3]{0.5 \cdot x}}{\frac{\frac{y}{\sqrt[3]{0.5 \cdot x}}}{\sqrt[3]{0.5 \cdot x}}}\right)}} \]
   *-commutative [=>] 28.8	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{\color{blue}{x \cdot 0.5}}}{\frac{\frac{y}{\sqrt[3]{0.5 \cdot x}}}{\sqrt[3]{0.5 \cdot x}}}\right)} \]
   associate-/l/ [=>] 28.8	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x \cdot 0.5}}{\color{blue}{\frac{y}{\sqrt[3]{0.5 \cdot x} \cdot \sqrt[3]{0.5 \cdot x}}}}\right)} \]
   *-commutative [=>] 28.8	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x \cdot 0.5}}{\frac{y}{\sqrt[3]{\color{blue}{x \cdot 0.5}} \cdot \sqrt[3]{0.5 \cdot x}}}\right)} \]
   *-commutative [=>] 28.8	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x \cdot 0.5}}{\frac{y}{\sqrt[3]{x \cdot 0.5} \cdot \sqrt[3]{\color{blue}{x \cdot 0.5}}}}\right)} \]

5. Applied egg-rr 28.8

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

6. Final simplification 28.8

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

Alternatives

Alternative 1
Error	28.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022354 
(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)))))