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

Average Error: 56.18% → 44.49%

Time: 15.0s

Precision: binary64

Cost: 39168

Math

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

↓

\[\frac{\sqrt{\sqrt[3]{{\cos \left(0.5 \cdot \frac{x}{y}\right)}^{-4}}}}{\sqrt[3]{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/
  (sqrt (cbrt (pow (cos (* 0.5 (/ x y))) -4.0)))
  (cbrt (cos (/ 0.5 (/ y x))))))

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 sqrt(cbrt(pow(cos((0.5 * (x / y))), -4.0))) / cbrt(cos((0.5 / (y / x))));
}

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.sqrt(Math.cbrt(Math.pow(Math.cos((0.5 * (x / y))), -4.0))) / Math.cbrt(Math.cos((0.5 / (y / x))));
}

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(sqrt(cbrt((cos(Float64(0.5 * Float64(x / y))) ^ -4.0))) / cbrt(cos(Float64(0.5 / Float64(y / x)))))
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[(N[Sqrt[N[Power[N[Power[N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] / N[Power[N[Cos[N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Target

Original	56.18%
Target	45.13%
Herbie	44.49%

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

Derivation

1. Initial program 56.18

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

2. Taylor expanded in x around inf 44.58

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

3. Applied egg-rr 44.6

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

4. Applied egg-rr 44.56

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

5. Simplified 44.52

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

   [Start] 44.56	\[ \frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}} \cdot \sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-2}} \]
   associate-*l/ [=>] 44.56	\[ \color{blue}{\frac{1 \cdot \sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-2}}}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}}} \]
   *-lft-identity [=>] 44.56	\[ \frac{\color{blue}{\sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-2}}}}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   *-commutative [=>] 44.56	\[ \frac{\sqrt[3]{{\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}}^{-2}}}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   associate-/r/ [<=] 44.56	\[ \frac{\sqrt[3]{{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}^{-2}}}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   *-commutative [=>] 44.56	\[ \frac{\sqrt[3]{{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}^{-2}}}{\sqrt[3]{\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}}} \]
   associate-/r/ [<=] 44.52	\[ \frac{\sqrt[3]{{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}^{-2}}}{\sqrt[3]{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}} \]

6. Applied egg-rr 44.49

   \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{\cos \left(0.5 \cdot \frac{x}{y}\right)}^{-4}}}}}{\sqrt[3]{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}} \]

7. Final simplification 44.49

   \[\leadsto \frac{\sqrt{\sqrt[3]{{\cos \left(0.5 \cdot \frac{x}{y}\right)}^{-4}}}}{\sqrt[3]{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}} \]

Alternatives

Alternative 1
Error	44.49%
Cost	32768

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

Alternative 2
Error	44.6%
Cost	19648

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right) \]

Alternative 3
Error	44.58%
Cost	6848

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

Alternative 4
Error	44.77%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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)))))