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

Average Error: 35.8 → 27.5

Time: 18.0s

Precision: binary64

Cost: 66052

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{x \cdot 0.5}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 10:\\ \;\;\;\;\frac{1}{\cos \left(\left(\left(t_0 \cdot \sqrt[3]{{t_0}^{2}}\right) \cdot \sqrt[3]{t_0}\right) \cdot \frac{t_0}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (* x 0.5))) (t_1 (/ x (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 10.0)
     (/ 1.0 (cos (* (* (* t_0 (cbrt (pow t_0 2.0))) (cbrt t_0)) (/ t_0 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 = cbrt((x * 0.5));
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 10.0) {
		tmp = 1.0 / cos((((t_0 * cbrt(pow(t_0, 2.0))) * cbrt(t_0)) * (t_0 / 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.cbrt((x * 0.5));
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 10.0) {
		tmp = 1.0 / Math.cos((((t_0 * Math.cbrt(Math.pow(t_0, 2.0))) * Math.cbrt(t_0)) * (t_0 / 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(Float64(x * 0.5))
	t_1 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 10.0)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(t_0 * cbrt((t_0 ^ 2.0))) * cbrt(t_0)) * Float64(t_0 / 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[(x * 0.5), $MachinePrecision], 1/3], $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], 10.0], N[(1.0 / N[Cos[N[(N[(N[(t$95$0 * N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Target

Original	35.8
Target	28.6
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)))) < 10

   1. Initial program 26.3

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

   2. Taylor expanded in x around inf 26.3

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

   3. Simplified 26.4

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}} \]
      Proof
      (/.f64 1 (cos.f64 (*.f64 (/.f64 1/2 y) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (cos.f64 (Rewrite<= associate-/r/_binary64 (/.f64 1/2 (/.f64 y x))))): 17 points increase in error, 21 points decrease in error
      (/.f64 1 (cos.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 x) y)))): 15 points increase in error, 14 points decrease in error
      (/.f64 1 (cos.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 x y))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 45.0

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

   5. Applied egg-rr 26.6

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

   6. Applied egg-rr 26.5

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

   if 10 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 63.7

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

   2. Applied egg-rr 63.7

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

   3. Taylor expanded in x around 0 30.5

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

4. Final simplification 27.5

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

Alternatives

Alternative 1
Error	27.5
Cost	39812

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 6:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot 0.5} \cdot \sqrt[3]{\frac{1}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	27.5
Cost	39684

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 15:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{\frac{y}{x}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	27.5
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 5:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	27.4
Cost	27012

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 15:\\ \;\;\;\;\frac{1}{\cos \left({\left(y \cdot \frac{1}{x \cdot 0.5}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	28.4
Cost	6848

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

Alternative 6
Error	28.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(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)))))