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

Average Error: 35.8 → 27.4

Time: 16.6s

Precision: binary64

Cost: 65604

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\sqrt[3]{y}\right)}^{2}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 3.5:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{x}{\frac{t_0}{0.5}}}{\sqrt[3]{t_0}}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (cbrt y) 2.0)) (t_1 (/ x (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 3.5)
     (/ 1.0 (cos (/ (/ (/ x (/ t_0 0.5)) (cbrt t_0)) (cbrt (cbrt y)))))
     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 = pow(cbrt(y), 2.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 3.5) {
		tmp = 1.0 / cos((((x / (t_0 / 0.5)) / cbrt(t_0)) / cbrt(cbrt(y))));
	} 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 = Math.pow(Math.cbrt(y), 2.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 3.5) {
		tmp = 1.0 / Math.cos((((x / (t_0 / 0.5)) / Math.cbrt(t_0)) / Math.cbrt(Math.cbrt(y))));
	} 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 = cbrt(y) ^ 2.0
	t_1 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 3.5)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(x / Float64(t_0 / 0.5)) / cbrt(t_0)) / cbrt(cbrt(y)))));
	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[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 3.5], N[(1.0 / N[Cos[N[(N[(N[(x / N[(t$95$0 / 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[y, 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Target

Original	35.8
Target	28.5
Herbie	27.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 3.5

   1. Initial program 25.6

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

   2. Taylor expanded in x around inf 25.6

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

   3. Applied egg-rr 25.7

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

   4. Taylor expanded in x around inf 25.6

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

   5. Simplified 25.7

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

      [Start] 25.6	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
      *-commutative [=>] 25.6	\[ \frac{1}{\cos \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}} \]
      associate-*l/ [=>] 25.6	\[ \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]
      associate-*r/ [<=] 25.7	\[ \frac{1}{\cos \color{blue}{\left(x \cdot \frac{0.5}{y}\right)}} \]

   6. Applied egg-rr 25.8

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

   if 3.5 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 62.9

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

   2. Taylor expanded in x around 0 31.8

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

4. Final simplification 27.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 3.5:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{x}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{0.5}}}{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{2}}}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	27.4
Cost	39812

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.5:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{x}{2}}{\sqrt[3]{y}}}{{\left(\sqrt[3]{y}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	27.5
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.5:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	27.3
Cost	33220

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 2.35:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	27.3
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 2.35:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	60.2
Cost	64

\[-2 \]

Alternative 6
Error	28.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(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)))))