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

Percentage Accurate: 43.7% → 56.5%

Time: 14.7s

Precision: binary64

Cost: 39876

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 4.1:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + {\left(\frac{x}{y}\right)}^{2} \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 4.1)
     (pow (/ 1.0 (cbrt (cos (/ 1.0 (/ 2.0 (/ x y)))))) 3.0)
     (pow
      (/ 1.0 (expm1 (+ (log 2.0) (* (pow (/ x y) 2.0) -0.020833333333333332))))
      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) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 4.1) {
		tmp = pow((1.0 / cbrt(cos((1.0 / (2.0 / (x / y)))))), 3.0);
	} else {
		tmp = pow((1.0 / expm1((log(2.0) + (pow((x / y), 2.0) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

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 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 4.1) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos((1.0 / (2.0 / (x / y)))))), 3.0);
	} else {
		tmp = Math.pow((1.0 / Math.expm1((Math.log(2.0) + (Math.pow((x / y), 2.0) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

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 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 4.1)
		tmp = Float64(1.0 / cbrt(cos(Float64(1.0 / Float64(2.0 / Float64(x / y)))))) ^ 3.0;
	else
		tmp = Float64(1.0 / expm1(Float64(log(2.0) + Float64((Float64(x / y) ^ 2.0) * -0.020833333333333332)))) ^ 3.0;
	end
	return tmp
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[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 4.1], N[Power[N[(1.0 / N[Power[N[Cos[N[(1.0 / N[(2.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(1.0 / N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]

Target

Original	43.7%
Target	54.7%
Herbie	56.5%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 4.0999999999999996

   1. Initial program 61.5%

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

   2. Applied egg-rr 61.3%

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

      [Start] 61.5%	\[ \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
      add-cube-cbrt [=>] 61.5%	\[ \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]
      pow3 [=>] 61.5%	\[ \color{blue}{{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)}^{3}} \]

   3. Applied egg-rr 61.9%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}}}\right)}^{3} \]
      Step-by-step derivation

      [Start] 61.3%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)}^{3} \]
      clear-num [=>] 61.3%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)}}\right)}^{3} \]
      div-inv [=>] 61.3%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)}}\right)}^{3} \]
      metadata-eval [=>] 61.3%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)}}\right)}^{3} \]
      div-inv [<=] 61.5%	\[ {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}}}\right)}^{3} \]
      associate-/r* [=>] 61.5%	\[ {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}}}\right)}^{3} \]
      clear-num [=>] 61.9%	\[ {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}}}\right)}^{3} \]

   if 4.0999999999999996 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 0.9%

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

   2. Applied egg-rr 53.4%

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

      [Start] 0.9%	\[ \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
      add-cube-cbrt [=>] 0.9%	\[ \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]
      pow3 [=>] 0.9%	\[ \color{blue}{{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)}^{3}} \]

   3. Applied egg-rr 53.4%

      \[\leadsto {\left(\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)}}\right)}^{3} \]
      Step-by-step derivation

      [Start] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)}^{3} \]
      clear-num [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)}}\right)}^{3} \]
      div-inv [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)}}\right)}^{3} \]
      metadata-eval [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)}}\right)}^{3} \]
      div-inv [<=] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}}}\right)}^{3} \]
      *-un-lft-identity [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)}}\right)}^{3} \]
      *-commutative [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)}}\right)}^{3} \]
      times-frac [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}}}\right)}^{3} \]
      metadata-eval [=>] 53.4%	\[ {\left(\frac{1}{\sqrt[3]{\cos \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)}}\right)}^{3} \]
      expm1-log1p-u [=>] 53.4%	\[ {\left(\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)}}\right)}^{3} \]

   4. Taylor expanded in x around 0 58.8%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)}\right)}^{3} \]

   5. Simplified 59.2%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + {\left(\frac{x}{y}\right)}^{2} \cdot -0.020833333333333332}\right)}\right)}^{3} \]
      Step-by-step derivation

      [Start] 58.8%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}\right)}\right)}^{3} \]
      *-commutative [=>] 58.8%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.020833333333333332}\right)}\right)}^{3} \]
      unpow2 [=>] 58.8%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]
      unpow2 [=>] 58.8%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.020833333333333332\right)}\right)}^{3} \]
      times-frac [=>] 59.2%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot -0.020833333333333332\right)}\right)}^{3} \]
      unpow2 [<=] 59.2%	\[ {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 4.1:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + {\left(\frac{x}{y}\right)}^{2} \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.5%
Cost	39876

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 4.1:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + {\left(\frac{x}{y}\right)}^{2} \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternative 2
Accuracy	55.1%
Cost	19840

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

Alternative 3
Accuracy	55.1%
Cost	6976

\[\frac{1}{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)} \]

Alternative 4
Accuracy	55.2%
Cost	6848

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

Alternative 5
Accuracy	55.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023229 
(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)))))