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

Average Error: 35.5 → 28.5

Time: 13.9s

Precision: binary64

Cost: 26240

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (/ (/ (* x 0.5) (pow (cbrt y) 2.0)) (cbrt 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) {
	return 1.0 / cos((((x * 0.5) / pow(cbrt(y), 2.0)) / cbrt(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) {
	return 1.0 / Math.cos((((x * 0.5) / Math.pow(Math.cbrt(y), 2.0)) / Math.cbrt(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)
	return Float64(1.0 / cos(Float64(Float64(Float64(x * 0.5) / (cbrt(y) ^ 2.0)) / cbrt(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_] := N[(1.0 / N[Cos[N[(N[(N[(x * 0.5), $MachinePrecision] / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	35.5
Target	28.9
Herbie	28.5

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

Derivation

1. Initial program 35.5

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

2. Applied egg-rr 35.7

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

3. Taylor expanded in x around inf 28.4

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

4. Simplified 28.4

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

   [Start] 28.4	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
   *-commutative [=>] 28.4	\[ \frac{1}{\cos \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}} \]
   associate-*l/ [=>] 28.4	\[ \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]
   associate-*r/ [<=] 28.4	\[ \frac{1}{\cos \color{blue}{\left(x \cdot \frac{0.5}{y}\right)}} \]

5. Applied egg-rr 28.5

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

6. Final simplification 28.5

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

Alternatives

Alternative 1
Error	28.5
Cost	26240

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

Alternative 2
Error	28.4
Cost	26240

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

Alternative 3
Error	28.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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)))))