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

Average Accuracy: 44.3% → 57.2%

Time: 18.4s

Precision: binary64

Cost: 39812

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt[3]{x}}{\frac{y \cdot 2}{{\left(\sqrt[3]{x}\right)}^{2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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)) 20.0)
     (/ 1.0 (cos (/ (cbrt x) (/ (* y 2.0) (pow (cbrt x) 2.0)))))
     1.0)))

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)) <= 20.0) {
		tmp = 1.0 / cos((cbrt(x) / ((y * 2.0) / pow(cbrt(x), 2.0))));
	} else {
		tmp = 1.0;
	}
	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)) <= 20.0) {
		tmp = 1.0 / Math.cos((Math.cbrt(x) / ((y * 2.0) / Math.pow(Math.cbrt(x), 2.0))));
	} else {
		tmp = 1.0;
	}
	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)) <= 20.0)
		tmp = Float64(1.0 / cos(Float64(cbrt(x) / Float64(Float64(y * 2.0) / (cbrt(x) ^ 2.0)))));
	else
		tmp = 1.0;
	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], 20.0], N[(1.0 / N[Cos[N[(N[Power[x, 1/3], $MachinePrecision] / N[(N[(y * 2.0), $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Target

Original	44.3%
Target	55.2%
Herbie	57.2%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 20

   1. Initial program 58.5%

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

   2. Taylor expanded in x around inf 58.5%

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

   3. Applied egg-rr 58.4%

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

      [Start] 58.5	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
      log1p-expm1-u [=>] 58.4	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

   4. Taylor expanded in x around inf 58.5%

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

   5. Simplified 58.5%

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

      [Start] 58.5	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
      *-commutative [=>] 58.5	\[ \frac{1}{\cos \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}} \]
      associate-*l/ [=>] 58.5	\[ \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]
      associate-*r/ [<=] 58.5	\[ \frac{1}{\cos \color{blue}{\left(x \cdot \frac{0.5}{y}\right)}} \]

   6. Applied egg-rr 58.3%

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

      [Start] 58.5	\[ \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)} \]
      associate-*r/ [=>] 58.5	\[ \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]
      associate-/l* [=>] 58.5	\[ \frac{1}{\cos \color{blue}{\left(\frac{x}{\frac{y}{0.5}}\right)}} \]
      add-cube-cbrt [=>] 58.3	\[ \frac{1}{\cos \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{y}{0.5}}\right)} \]
      *-commutative [=>] 58.3	\[ \frac{1}{\cos \left(\frac{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}{\frac{y}{0.5}}\right)} \]
      associate-/l* [=>] 58.3	\[ \frac{1}{\cos \color{blue}{\left(\frac{\sqrt[3]{x}}{\frac{\frac{y}{0.5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)}} \]
      div-inv [=>] 58.3	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x}}{\frac{\color{blue}{y \cdot \frac{1}{0.5}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)} \]
      metadata-eval [=>] 58.3	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x}}{\frac{y \cdot \color{blue}{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)} \]
      pow2 [=>] 58.3	\[ \frac{1}{\cos \left(\frac{\sqrt[3]{x}}{\frac{y \cdot 2}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}}\right)} \]

   if 20 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 0.5%

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

   2. Taylor expanded in x around 0 53.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt[3]{x}}{\frac{y \cdot 2}{{\left(\sqrt[3]{x}\right)}^{2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.3%
Cost	33220

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 10:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	57.3%
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 10:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	55.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023137 
(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)))))