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

Percentage Accurate: 43.6% → 55.7%

Time: 15.4s

Alternatives: 7

Speedup: 211.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t_0}{\sin t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 43.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t_0}{\sin t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 55.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{2}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 1.45:\\ \;\;\;\;\frac{1}{\cos \left(\frac{t_0}{\frac{y \cdot 2}{{\left(\sqrt[3]{\sqrt[3]{t_0}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{3}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 2.0)) (t_1 (/ x (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 1.45)
     (/
      1.0
      (cos
       (/
        t_0
        (/ (* y 2.0) (pow (* (cbrt (cbrt t_0)) (cbrt (cbrt (cbrt x)))) 3.0)))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = pow(cbrt(x), 2.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 1.45) {
		tmp = 1.0 / cos((t_0 / ((y * 2.0) / pow((cbrt(cbrt(t_0)) * cbrt(cbrt(cbrt(x)))), 3.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.cbrt(x), 2.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 1.45) {
		tmp = 1.0 / Math.cos((t_0 / ((y * 2.0) / Math.pow((Math.cbrt(Math.cbrt(t_0)) * Math.cbrt(Math.cbrt(Math.cbrt(x)))), 3.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = cbrt(x) ^ 2.0
	t_1 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 1.45)
		tmp = Float64(1.0 / cos(Float64(t_0 / Float64(Float64(y * 2.0) / (Float64(cbrt(cbrt(t_0)) * cbrt(cbrt(cbrt(x)))) ^ 3.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $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], 1.45], N[(1.0 / N[Cos[N[(t$95$0 / N[(N[(y * 2.0), $MachinePrecision] / N[Power[N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.44999999999999996

   1. Initial program 60.7%

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

   2. Taylor expanded in x around inf 60.7%

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

      1. metadata-eval 60.7%

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

      2. times-frac 60.7%

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

      3. *-un-lft-identity 60.7%

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

      4. *-commutative 60.7%

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

      5. add-cbrt-cube 50.0%

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

      6. cbrt-prod 53.5%

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

      7. associate-/l* 53.6%

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

      8. cbrt-prod 59.5%

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

      9. pow2 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. add-cube-cbrt 60.2%

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

      2. pow3 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. rem-cbrt-cube 60.5%

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

      2. cube-mult 60.5%

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

      3. cbrt-prod 60.8%

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

      4. cbrt-prod 61.0%

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

      5. unpow2 61.0%

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

   8. Applied egg-rr 61.0%

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

      1. *-commutative 61.0%

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

   10. Simplified 61.0%

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

   if 1.44999999999999996 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 2.8%

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

   2. Taylor expanded in x around 0 53.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.45:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\frac{y \cdot 2}{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{3}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 55.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 1.45:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{t_0}^{2}}{\frac{y \cdot 2}{t_0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (cbrt (cbrt x)) 3.0)) (t_1 (/ x (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 1.45)
     (/ 1.0 (cos (/ (pow t_0 2.0) (/ (* y 2.0) t_0))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = pow(cbrt(cbrt(x)), 3.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 1.45) {
		tmp = 1.0 / cos((pow(t_0, 2.0) / ((y * 2.0) / t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.cbrt(Math.cbrt(x)), 3.0);
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 1.45) {
		tmp = 1.0 / Math.cos((Math.pow(t_0, 2.0) / ((y * 2.0) / t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = cbrt(cbrt(x)) ^ 3.0
	t_1 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 1.45)
		tmp = Float64(1.0 / cos(Float64((t_0 ^ 2.0) / Float64(Float64(y * 2.0) / t_0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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], 1.45], N[(1.0 / N[Cos[N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(N[(y * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.44999999999999996

   1. Initial program 60.7%

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

   2. Taylor expanded in x around inf 60.7%

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

      1. metadata-eval 60.7%

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

      2. times-frac 60.7%

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

      3. *-un-lft-identity 60.7%

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

      4. *-commutative 60.7%

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

      5. add-cbrt-cube 50.0%

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

      6. cbrt-prod 53.5%

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

      7. associate-/l* 53.6%

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

      8. cbrt-prod 59.5%

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

      9. pow2 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. add-cube-cbrt 60.2%

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

      2. pow3 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. add-cube-cbrt 60.2%

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

      2. pow3 60.1%

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

   8. Applied egg-rr 60.9%

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

   if 1.44999999999999996 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 2.8%

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

   2. Taylor expanded in x around 0 53.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.45:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}\right)}^{2}}{\frac{y \cdot 2}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \frac{\tan t_0}{\sin t_0}\\ \mathbf{if}\;t_1 \leq 1.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (/ (tan t_0) (sin t_0))))
   (if (<= t_1 1.8) t_1 1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = tan(t_0) / sin(t_0);
	double tmp;
	if (t_1 <= 1.8) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = tan(t_0) / sin(t_0)
    if (t_1 <= 1.8d0) then
        tmp = t_1
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.tan(t_0) / Math.sin(t_0);
	double tmp;
	if (t_1 <= 1.8) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.tan(t_0) / math.sin(t_0)
	tmp = 0
	if t_1 <= 1.8:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = Float64(tan(t_0) / sin(t_0))
	tmp = 0.0
	if (t_1 <= 1.8)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = tan(t_0) / sin(t_0);
	tmp = 0.0;
	if (t_1 <= 1.8)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1.8], t$95$1, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.80000000000000004

   1. Initial program 59.6%

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

   if 1.80000000000000004 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 2.3%

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

   2. Taylor expanded in x around 0 55.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.8:\\ \;\;\;\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \tan t_0\\ \mathbf{if}\;\frac{t_1}{\sin t_0} \leq 1.6:\\ \;\;\;\;\frac{t_1}{\sin \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (tan t_0)))
   (if (<= (/ t_1 (sin t_0)) 1.6) (/ t_1 (sin (* x (/ 0.5 y)))) 1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = tan(t_0);
	double tmp;
	if ((t_1 / sin(t_0)) <= 1.6) {
		tmp = t_1 / sin((x * (0.5 / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = tan(t_0)
    if ((t_1 / sin(t_0)) <= 1.6d0) then
        tmp = t_1 / sin((x * (0.5d0 / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.tan(t_0);
	double tmp;
	if ((t_1 / Math.sin(t_0)) <= 1.6) {
		tmp = t_1 / Math.sin((x * (0.5 / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.tan(t_0)
	tmp = 0
	if (t_1 / math.sin(t_0)) <= 1.6:
		tmp = t_1 / math.sin((x * (0.5 / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = tan(t_0)
	tmp = 0.0
	if (Float64(t_1 / sin(t_0)) <= 1.6)
		tmp = Float64(t_1 / sin(Float64(x * Float64(0.5 / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = tan(t_0);
	tmp = 0.0;
	if ((t_1 / sin(t_0)) <= 1.6)
		tmp = t_1 / sin((x * (0.5 / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Tan[t$95$0], $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.6], N[(t$95$1 / N[Sin[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.6000000000000001

   1. Initial program 60.2%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 10.1%

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

      2. *-un-lft-identity 10.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\log \color{blue}{\left(1 \cdot e^{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}} \]

      3. log-prod 10.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\log 1 + \log \left(e^{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}} \]

      4. metadata-eval 10.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{0} + \log \left(e^{\sin \left(\frac{x}{y \cdot 2}\right)}\right)} \]

      5. add-log-exp 60.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{0 + \color{blue}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      6. div-inv 60.3%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{0 + \sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

      7. *-commutative 60.3%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{0 + \sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

      8. associate-/r* 60.3%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{0 + \sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

      9. metadata-eval 60.3%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{0 + \sin \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

   3. Applied egg-rr 60.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{0 + \sin \left(x \cdot \frac{0.5}{y}\right)}} \]

   if 1.6000000000000001 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 2.5%

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

   2. Taylor expanded in x around 0 53.9%

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

4. Final simplification 58.1%

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

Alternative 5: 54.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\right) + -1 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (+ 1.0 (/ 1.0 (cos (* (/ x y) -0.5)))) -1.0))

C

double code(double x, double y) {
	return (1.0 + (1.0 / cos(((x / y) * -0.5)))) + -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (1.0d0 / cos(((x / y) * (-0.5d0))))) + (-1.0d0)
end function

Java

public static double code(double x, double y) {
	return (1.0 + (1.0 / Math.cos(((x / y) * -0.5)))) + -1.0;
}

Python

def code(x, y):
	return (1.0 + (1.0 / math.cos(((x / y) * -0.5)))) + -1.0

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(1.0 / cos(Float64(Float64(x / y) * -0.5)))) + -1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (1.0 / cos(((x / y) * -0.5)))) + -1.0;
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(1.0 / N[Cos[N[(N[(x / y), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

2. Taylor expanded in x around inf 56.4%

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

   1. clear-num 55.7%

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

   2. un-div-inv 55.7%

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

4. Applied egg-rr 55.7%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 53.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}\right)\right)} \]

   2. expm1-udef 53.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}\right)} - 1} \]

6. Applied egg-rr 53.6%

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

   1. log1p-udef 53.6%

      \[\leadsto e^{\color{blue}{\log \left(1 + \frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\right)}} - 1 \]

   2. add-exp-log 56.4%

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

8. Applied egg-rr 56.4%

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

9. Final simplification 56.4%

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

Alternative 6: 54.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (cos (* 0.5 (/ x y)))))

C

double code(double x, double y) {
	return 1.0 / cos((0.5 * (x / y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos((0.5d0 * (x / y)))
end function

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos((0.5 * (x / y)));
}

Python

def code(x, y):
	return 1.0 / math.cos((0.5 * (x / y)))

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(0.5 * Float64(x / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / cos((0.5 * (x / y)));
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

2. Taylor expanded in x around inf 56.4%

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

3. Final simplification 56.4%

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

Alternative 7: 54.5% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 40.4%

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

2. Taylor expanded in x around 0 55.8%

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

3. Final simplification 55.8%

   \[\leadsto 1 \]

Developer target: 53.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \log \left(e^{\cos t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2023173 
(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)))))