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

Percentage Accurate: 45.0% → 57.5%

Time: 20.0s

Alternatives: 4

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: 45.0% 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: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+114}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 1e+114)
   (/ 1.0 (cos (pow (expm1 (log1p (sqrt (* 0.5 (/ x y))))) 2.0)))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+114) {
		tmp = 1.0 / cos(pow(expm1(log1p(sqrt((0.5 * (x / y))))), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+114) {
		tmp = 1.0 / Math.cos(Math.pow(Math.expm1(Math.log1p(Math.sqrt((0.5 * (x / y))))), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 1e+114:
		tmp = 1.0 / math.cos(math.pow(math.expm1(math.log1p(math.sqrt((0.5 * (x / y))))), 2.0))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+114)
		tmp = Float64(1.0 / cos((expm1(log1p(sqrt(Float64(0.5 * Float64(x / y))))) ^ 2.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+114], N[(1.0 / N[Cos[N[Power[N[(Exp[N[Log[1 + N[Sqrt[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1e114

   1. Initial program 49.5%

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

   2. Taylor expanded in x around inf 64.5%

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

      1. associate-*r/ 64.5%

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

   4. Simplified 64.5%

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

   5. Taylor expanded in x around inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. *-commutative 64.5%

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

      3. associate-*r/ 64.5%

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

   7. Simplified 64.5%

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

      1. add-sqr-sqrt 36.6%

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

      2. pow2 36.6%

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

   9. Applied egg-rr 36.6%

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

       1. add-cube-cbrt 36.8%

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

       2. unpow3 36.6%

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

       3. sqrt-pow1 36.8%

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

       4. expm1-log1p-u 36.8%

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

       5. sqrt-pow1 36.8%

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

       6. rem-cube-cbrt 36.8%

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

       7. associate-*r/ 36.8%

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

       8. *-commutative 36.8%

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

       9. *-un-lft-identity 36.8%

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

       10. times-frac 36.8%

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

       11. metadata-eval 36.8%

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

   11. Applied egg-rr 36.8%

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

   if 1e114 < (/.f64 x (*.f64 y 2))

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0 14.6%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+114}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{\sqrt{x \cdot 0.5}}{\sqrt{y}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 2e+72)
   (/ 1.0 (cos (pow (/ (sqrt (* x 0.5)) (sqrt y)) 2.0)))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+72) {
		tmp = 1.0 / cos(pow((sqrt((x * 0.5)) / sqrt(y)), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x / (y * 2.0d0)) <= 2d+72) then
        tmp = 1.0d0 / cos(((sqrt((x * 0.5d0)) / sqrt(y)) ** 2.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+72) {
		tmp = 1.0 / Math.cos(Math.pow((Math.sqrt((x * 0.5)) / Math.sqrt(y)), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 2e+72:
		tmp = 1.0 / math.cos(math.pow((math.sqrt((x * 0.5)) / math.sqrt(y)), 2.0))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 2e+72)
		tmp = Float64(1.0 / cos((Float64(sqrt(Float64(x * 0.5)) / sqrt(y)) ^ 2.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 2e+72)
		tmp = 1.0 / cos(((sqrt((x * 0.5)) / sqrt(y)) ^ 2.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 2e+72], N[(1.0 / N[Cos[N[Power[N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.99999999999999989e72

   1. Initial program 51.5%

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

   2. Taylor expanded in x around inf 67.2%

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

      1. associate-*r/ 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in x around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*r/ 67.0%

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

   7. Simplified 67.0%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   9. Applied egg-rr 37.7%

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

       1. associate-*r/ 37.7%

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

       2. sqrt-div 14.5%

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

   11. Applied egg-rr 14.5%

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

   if 1.99999999999999989e72 < (/.f64 x (*.f64 y 2))

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 13.9%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{\sqrt{x \cdot 0.5}}{\sqrt{y}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 57.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{0.5 \cdot \frac{x}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 4e+45)
   (/ 1.0 (cos (pow (cbrt (* 0.5 (/ x y))) 3.0)))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 4e+45) {
		tmp = 1.0 / cos(pow(cbrt((0.5 * (x / y))), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 4e+45) {
		tmp = 1.0 / Math.cos(Math.pow(Math.cbrt((0.5 * (x / y))), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 4e+45)
		tmp = Float64(1.0 / cos((cbrt(Float64(0.5 * Float64(x / y))) ^ 3.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 4e+45], N[(1.0 / N[Cos[N[Power[N[Power[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 3.9999999999999997e45

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   4. Simplified 68.0%

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

   5. Taylor expanded in x around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*r/ 67.8%

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

   7. Simplified 67.8%

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

      1. associate-*r/ 68.0%

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

      2. clear-num 67.7%

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

   9. Applied egg-rr 67.7%

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

       1. clear-num 68.0%

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

       2. associate-*r/ 67.8%

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

       3. rem-cube-cbrt 68.2%

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

       4. associate-*r/ 67.8%

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

       5. *-commutative 67.8%

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

       6. *-un-lft-identity 67.8%

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

       7. times-frac 67.8%

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

       8. metadata-eval 67.8%

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

   11. Applied egg-rr 67.8%

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

   if 3.9999999999999997e45 < (/.f64 x (*.f64 y 2))

   1. Initial program 8.2%

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

   2. Taylor expanded in x around 0 14.1%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{0.5 \cdot \frac{x}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 56.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ 1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return 1.0;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := 1.0

Derivation

1. Initial program 43.7%

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

2. Taylor expanded in x around 0 57.2%

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

3. Final simplification 57.2%

   \[\leadsto 1 \]

Developer target: 56.1% 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 2023318 
(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)))))