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

Average Accuracy: 45.1% → 57.5%

Time: 14.2s

Precision: binary64

Cost: 27272

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 10^{+144}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-2} \cdot 0.25}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (<= t_0 -1e+106)
     1.0
     (if (<= t_0 1e+144)
       (/
        1.0
        (cos (* (cbrt (* 0.5 (/ x y))) (cbrt (* (pow (/ y x) -2.0) 0.25)))))
       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 (t_0 <= -1e+106) {
		tmp = 1.0;
	} else if (t_0 <= 1e+144) {
		tmp = 1.0 / cos((cbrt((0.5 * (x / y))) * cbrt((pow((y / x), -2.0) * 0.25))));
	} 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 (t_0 <= -1e+106) {
		tmp = 1.0;
	} else if (t_0 <= 1e+144) {
		tmp = 1.0 / Math.cos((Math.cbrt((0.5 * (x / y))) * Math.cbrt((Math.pow((y / x), -2.0) * 0.25))));
	} 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 (t_0 <= -1e+106)
		tmp = 1.0;
	elseif (t_0 <= 1e+144)
		tmp = Float64(1.0 / cos(Float64(cbrt(Float64(0.5 * Float64(x / y))) * cbrt(Float64((Float64(y / x) ^ -2.0) * 0.25)))));
	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[t$95$0, -1e+106], 1.0, If[LessEqual[t$95$0, 1e+144], N[(1.0 / N[Cos[N[(N[Power[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Power[N[(y / x), $MachinePrecision], -2.0], $MachinePrecision] * 0.25), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Target

Original	45.1%
Target	55.4%
Herbie	57.5%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < -1.00000000000000009e106 or 1.00000000000000002e144 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.9%

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

   2. Taylor expanded in x around 0 11.9%

      \[\leadsto \color{blue}{1} \]

   if -1.00000000000000009e106 < (/.f64 x (*.f64 y 2)) < 1.00000000000000002e144

   1. Initial program 62.4%

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

   2. Taylor expanded in x around inf 78.1%

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

   3. Applied egg-rr 78.2%

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

      [Start] 78.1	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
      clear-num [=>] 78.2	\[ \frac{1}{\cos \left(0.5 \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)} \]
      un-div-inv [=>] 78.2	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}} \]

   4. Applied egg-rr 78.1%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot {\left(\frac{y}{x}\right)}^{-2}}\right)}} \]
      Proof

      [Start] 78.2	\[ \frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)} \]
      add-cube-cbrt [=>] 78.1	\[ \frac{1}{\cos \color{blue}{\left(\left(\sqrt[3]{\frac{0.5}{\frac{y}{x}}} \cdot \sqrt[3]{\frac{0.5}{\frac{y}{x}}}\right) \cdot \sqrt[3]{\frac{0.5}{\frac{y}{x}}}\right)}} \]
      associate-*l* [=>] 78.1	\[ \frac{1}{\cos \color{blue}{\left(\sqrt[3]{\frac{0.5}{\frac{y}{x}}} \cdot \left(\sqrt[3]{\frac{0.5}{\frac{y}{x}}} \cdot \sqrt[3]{\frac{0.5}{\frac{y}{x}}}\right)\right)}} \]
      div-inv [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{\color{blue}{0.5 \cdot \frac{1}{\frac{y}{x}}}} \cdot \left(\sqrt[3]{\frac{0.5}{\frac{y}{x}}} \cdot \sqrt[3]{\frac{0.5}{\frac{y}{x}}}\right)\right)} \]
      clear-num [<=] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \color{blue}{\frac{x}{y}}} \cdot \left(\sqrt[3]{\frac{0.5}{\frac{y}{x}}} \cdot \sqrt[3]{\frac{0.5}{\frac{y}{x}}}\right)\right)} \]
      cbrt-unprod [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \color{blue}{\sqrt[3]{\frac{0.5}{\frac{y}{x}} \cdot \frac{0.5}{\frac{y}{x}}}}\right)} \]
      div-inv [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\color{blue}{\left(0.5 \cdot \frac{1}{\frac{y}{x}}\right)} \cdot \frac{0.5}{\frac{y}{x}}}\right)} \]
      clear-num [<=] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\left(0.5 \cdot \color{blue}{\frac{x}{y}}\right) \cdot \frac{0.5}{\frac{y}{x}}}\right)} \]
      div-inv [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\left(0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{y}{x}}\right)}}\right)} \]
      clear-num [<=] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\left(0.5 \cdot \frac{x}{y}\right) \cdot \left(0.5 \cdot \color{blue}{\frac{x}{y}}\right)}\right)} \]
      swap-sqr [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)}}\right)} \]
      metadata-eval [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\color{blue}{0.25} \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)}\right)} \]
      clear-num [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right)}\right)} \]
      clear-num [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot \left(\frac{1}{\frac{y}{x}} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)}\right)} \]
      inv-pow [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot \left(\color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \cdot \frac{1}{\frac{y}{x}}\right)}\right)} \]
      inv-pow [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot \left({\left(\frac{y}{x}\right)}^{-1} \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{-1}}\right)}\right)} \]
      pow-prod-up [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{\left(-1 + -1\right)}}}\right)} \]
      metadata-eval [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot {\left(\frac{y}{x}\right)}^{\color{blue}{-2}}}\right)} \]

   5. Simplified 78.1%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-2} \cdot 0.25}\right)}} \]
      Proof

      [Start] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{0.25 \cdot {\left(\frac{y}{x}\right)}^{-2}}\right)} \]
      *-commutative [=>] 78.1	\[ \frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{\color{blue}{{\left(\frac{y}{x}\right)}^{-2} \cdot 0.25}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq -1 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x}{y \cdot 2} \leq 10^{+144}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{0.5 \cdot \frac{x}{y}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-2} \cdot 0.25}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.4%
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 6.6:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	55.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(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)))))