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

Average Error: 36.4 → 28.2

Time: 18.1s

Precision: binary64

Cost: 58884

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{y}}\\ \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+56}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{0.5}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{x}}}{t_0 \cdot t_0}}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt y))))
   (if (<= (/ x (* y 2.0)) 1e+56)
     (/ 1.0 (cos (/ (/ (/ 0.5 (/ (pow (cbrt y) 2.0) x)) (* t_0 t_0)) t_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 = cbrt(cbrt(y));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+56) {
		tmp = 1.0 / cos((((0.5 / (pow(cbrt(y), 2.0) / x)) / (t_0 * t_0)) / t_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 = Math.cbrt(Math.cbrt(y));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+56) {
		tmp = 1.0 / Math.cos((((0.5 / (Math.pow(Math.cbrt(y), 2.0) / x)) / (t_0 * t_0)) / t_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 = cbrt(cbrt(y))
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+56)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(0.5 / Float64((cbrt(y) ^ 2.0) / x)) / Float64(t_0 * t_0)) / t_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[Power[N[Power[y, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+56], N[(1.0 / N[Cos[N[(N[(N[(0.5 / N[(N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Target

Original	36.4
Target	29.2
Herbie	28.2

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000009e56

   1. Initial program 30.4

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

   2. Taylor expanded in x around inf 20.7

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

   3. Applied egg-rr 20.7

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

   4. Taylor expanded in x around inf 20.7

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

   5. Simplified 20.7

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

      [Start] 20.7	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
      associate-*r/ [=>] 20.7	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5 \cdot x}{y}\right)}} \]
      associate-/l* [=>] 20.7	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}} \]
      associate-/r/ [=>] 20.7	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}} \]

   6. Applied egg-rr 20.6

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

   if 1.00000000000000009e56 < (/.f64 x (*.f64 y 2))

   1. Initial program 59.0

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

   2. Taylor expanded in x around 0 56.5

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

4. Final simplification 28.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+56}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{0.5}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{x}}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	28.0
Cost	40132

\[\begin{array}{l} t_0 := \sqrt[3]{x \cdot 0.5}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 1.5:\\ \;\;\;\;\frac{1}{\cos \left(\frac{t_0}{\frac{\frac{y}{t_0}}{t_0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	60.1
Cost	64

\[-2 \]

Alternative 3
Error	28.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023054 
(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)))))