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

Average Error: 35.8 → 27.5

Time: 16.8s

Precision: binary64

Cost: 52612

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.49:\\ \;\;\;\;\frac{1}{2 \cdot \log \left(\sqrt{\sqrt[3]{{\left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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)) 3.49)
     (/ 1.0 (* 2.0 (log (sqrt (cbrt (pow (exp (cos (* 0.5 (/ x y)))) 3.0))))))
     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)) <= 3.49) {
		tmp = 1.0 / (2.0 * log(sqrt(cbrt(pow(exp(cos((0.5 * (x / y)))), 3.0)))));
	} 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)) <= 3.49) {
		tmp = 1.0 / (2.0 * Math.log(Math.sqrt(Math.cbrt(Math.pow(Math.exp(Math.cos((0.5 * (x / y)))), 3.0)))));
	} 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)) <= 3.49)
		tmp = Float64(1.0 / Float64(2.0 * log(sqrt(cbrt((exp(cos(Float64(0.5 * Float64(x / y)))) ^ 3.0))))));
	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], 3.49], N[(1.0 / N[(2.0 * N[Log[N[Sqrt[N[Power[N[Power[N[Exp[N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Target

Original	35.8
Target	28.7
Herbie	27.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 3.4900000000000002

   1. Initial program 25.7

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

   2. Taylor expanded in x around inf 25.7

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

   3. Applied egg-rr 25.7

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

   4. Applied egg-rr 25.7

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

   5. Simplified 25.7

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

      [Start] 25.7	\[ \frac{1}{\log \left(\sqrt{e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}}\right)} \]
      count-2 [=>] 25.7	\[ \frac{1}{\color{blue}{2 \cdot \log \left(\sqrt{e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}}\right)}} \]
      associate-*r/ [=>] 25.7	\[ \frac{1}{2 \cdot \log \left(\sqrt{e^{\cos \color{blue}{\left(\frac{0.5 \cdot x}{y}\right)}}}\right)} \]

   6. Applied egg-rr 25.7

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

   if 3.4900000000000002 < (/.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 32.4

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

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 3.49:\\ \;\;\;\;\frac{1}{2 \cdot \log \left(\sqrt{\sqrt[3]{{\left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	27.5
Cost	33220

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.49:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	27.5
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.49:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	28.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(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)))))