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

Average Error: 36.0 → 28.4

Time: 15.4s

Precision: binary64

Cost: 72068

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sqrt[3]{\sqrt[3]{y}}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 3.1:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{t_1 \cdot t_1}}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (cbrt y))))
   (if (<= (/ (tan t_0) (sin t_0)) 3.1)
     (/ 1.0 (cos (/ (/ (/ (* x 0.5) (pow (cbrt y) 2.0)) (* t_1 t_1)) 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(cbrt(y));
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 3.1) {
		tmp = 1.0 / cos(((((x * 0.5) / pow(cbrt(y), 2.0)) / (t_1 * t_1)) / 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(Math.cbrt(y));
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 3.1) {
		tmp = 1.0 / Math.cos(((((x * 0.5) / Math.pow(Math.cbrt(y), 2.0)) / (t_1 * t_1)) / 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(cbrt(y))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 3.1)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(Float64(x * 0.5) / (cbrt(y) ^ 2.0)) / Float64(t_1 * t_1)) / 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[Power[y, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 3.1], N[(1.0 / N[Cos[N[(N[(N[(N[(x * 0.5), $MachinePrecision] / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Target

Original	36.0
Target	29.5
Herbie	28.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 3.10000000000000009

   1. Initial program 25.7

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

   2. Applied egg-rr 26.0

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

   3. Taylor expanded in x around inf 25.7

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

   4. Simplified 25.7

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

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

   5. Applied egg-rr 25.9

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

   if 3.10000000000000009 < (/.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 34.7

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

4. Final simplification 28.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 3.1:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	28.3
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 2.85:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{\frac{\frac{x}{y}}{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	28.3
Cost	33220

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.6:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	28.3
Cost	27076

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \frac{\tan t_0}{\sin t_0}\\ \mathbf{if}\;t_1 \leq 1.6:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	29.2
Cost	6848

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

Alternative 5
Error	29.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023018 
(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)))))