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

Average Error: 56.09% → 43.64%

Time: 16.5s

Precision: binary64

Cost: 52804

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sqrt[3]{\frac{y}{x}}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 19:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{0.5 \cdot {t_1}^{-2}}\right)}^{3}}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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))) (t_1 (cbrt (/ y x))))
   (if (<= (/ (tan t_0) (sin t_0)) 19.0)
     (/ 1.0 (cos (/ (pow (cbrt (* 0.5 (pow t_1 -2.0))) 3.0) t_1)))
     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 t_1 = cbrt((y / x));
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 19.0) {
		tmp = 1.0 / cos((pow(cbrt((0.5 * pow(t_1, -2.0))), 3.0) / t_1));
	} 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 t_1 = Math.cbrt((y / x));
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 19.0) {
		tmp = 1.0 / Math.cos((Math.pow(Math.cbrt((0.5 * Math.pow(t_1, -2.0))), 3.0) / t_1));
	} 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))
	t_1 = cbrt(Float64(y / x))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 19.0)
		tmp = Float64(1.0 / cos(Float64((cbrt(Float64(0.5 * (t_1 ^ -2.0))) ^ 3.0) / t_1)));
	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]}, Block[{t$95$1 = N[Power[N[(y / x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 19.0], N[(1.0 / N[Cos[N[(N[Power[N[Power[N[(0.5 * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Target

Original	56.09%
Target	45.25%
Herbie	43.64%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 19

   1. Initial program 42.07

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

   2. Taylor expanded in x around inf 42.08

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

   3. Applied egg-rr 42.52

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

   4. Applied egg-rr 42.47

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

   if 19 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 99.72

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

   2. Taylor expanded in x around 0 47.29

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

4. Final simplification 43.64

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 19:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{0.5 \cdot {\left(\sqrt[3]{\frac{y}{x}}\right)}^{-2}}\right)}^{3}}{\sqrt[3]{\frac{y}{x}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	43.57%
Cost	46468

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 8.2:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{0.5 \cdot \frac{x}{y}}\right)}^{2} \cdot \sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	43.49%
Cost	39940

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.9:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}}{\frac{\sqrt[3]{\frac{y}{x}}}{0.5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	43.54%
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 8.2:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y} \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	43.5%
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 8.2:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot {\left(\sqrt[3]{0.5}\right)}^{3}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	43.33%
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.8888:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	44.76%
Cost	6848

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

Alternative 7
Error	44.66%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023115 
(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)))))