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

Percentage Accurate: 45.1% → 57.4%

Time: 21.9s

Precision: binary64

Cost: 59268

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.5:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\frac{\sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (<= (/ (tan t_0) (sin t_0)) 1.5)
     (log1p
      (expm1
       (/
        1.0
        (cos
         (* (pow (/ (cbrt (* x 0.5)) (cbrt y)) 2.0) (cbrt (* x (/ 0.5 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 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.5) {
		tmp = log1p(expm1((1.0 / cos((pow((cbrt((x * 0.5)) / cbrt(y)), 2.0) * cbrt((x * (0.5 / 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 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.5) {
		tmp = Math.log1p(Math.expm1((1.0 / Math.cos((Math.pow((Math.cbrt((x * 0.5)) / Math.cbrt(y)), 2.0) * Math.cbrt((x * (0.5 / 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 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.5)
		tmp = log1p(expm1(Float64(1.0 / cos(Float64((Float64(cbrt(Float64(x * 0.5)) / cbrt(y)) ^ 2.0) * cbrt(Float64(x * Float64(0.5 / 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[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.5], N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(N[Power[N[(N[Power[N[(x * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 1.0]]

Target

Original	45.1%
Target	55.5%
Herbie	57.4%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.5

   1. Initial program 66.6%

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

   2. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
      Step-by-step derivation

      [Start] 66.6	\[ \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
      log1p-expm1-u [=>] 66.6	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]
      div-inv [=>] 63.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]
      tan-quot [=>] 63.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      associate-*l/ [=>] 63.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]
      pow1 [=>] 63.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      inv-pow [=>] 63.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      pow-prod-up [=>] 66.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      metadata-eval [=>] 66.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      metadata-eval [=>] 66.6	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]
      div-inv [=>] 66.7	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}}\right)\right) \]
      *-commutative [=>] 66.7	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}\right)\right) \]
      associate-/r* [=>] 66.7	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}\right)\right) \]
      metadata-eval [=>] 66.7	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)}\right)\right) \]

   3. Applied egg-rr 66.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}\right)\right) \]
      Step-by-step derivation

      [Start] 66.7	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right) \]
      add-cube-cbrt [=>] 67.0	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(\left(\sqrt[3]{x \cdot \frac{0.5}{y}} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}}\right)\right) \]
      pow3 [=>] 66.9	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}\right)\right) \]

   4. Applied egg-rr 67.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}}\right)\right) \]
      Step-by-step derivation

      [Start] 66.9	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}\right)\right) \]
      metadata-eval [<=] 66.9	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{\color{blue}{\left(2 + 1\right)}}\right)}\right)\right) \]
      pow-prod-up [<=] 67.0	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{1}\right)}}\right)\right) \]
      pow1 [<=] 67.0	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2} \cdot \color{blue}{\sqrt[3]{x \cdot \frac{0.5}{y}}}\right)}\right)\right) \]

   5. Applied egg-rr 68.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\color{blue}{\left(\frac{\sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right) \]
      Step-by-step derivation

      [Start] 67.0	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right) \]
      associate-*r/ [=>] 67.5	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{\color{blue}{\frac{x \cdot 0.5}{y}}}\right)}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right) \]
      cbrt-div [=>] 68.1	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\color{blue}{\left(\frac{\sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right) \]

   if 1.5 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 2.2%

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

   2. Taylor expanded in x around 0 39.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.5:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\frac{\sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}^{2} \cdot \sqrt[3]{x \cdot \frac{0.5}{y}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.3%
Cost	52484

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.12:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\frac{\sqrt[3]{x \cdot 0.5}}{\sqrt[3]{y}}\right)}^{3}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	56.6%
Cost	33284

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.208:\\ \;\;\;\;\sqrt{{\cos \left(\frac{1}{\frac{y}{x \cdot 0.5}}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	56.5%
Cost	33156

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.208:\\ \;\;\;\;\sqrt{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	56.5%
Cost	33156

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.12:\\ \;\;\;\;\sqrt{{\cos \left(\frac{x \cdot 0.5}{y}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	56.1%
Cost	32512

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}\right)\right) \]

Alternative 6
Accuracy	56.5%
Cost	26820

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.208:\\ \;\;\;\;\left|\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	56.1%
Cost	19712

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

Alternative 8
Accuracy	56.3%
Cost	6976

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

Alternative 9
Accuracy	56.3%
Cost	6848

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

Alternative 10
Accuracy	56.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023160 
(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)))))