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

Percentage Accurate: 44.0% → 56.6%

Time: 19.6s

Alternatives: 3

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: 44.0% 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: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\cos \left(\frac{{\left({x_m}^{0.3333333333333333}\right)}^{2}}{\frac{y_m \cdot 2}{\sqrt[3]{x_m}}}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{1}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+277)
   (/
    1.0
    (pow
     (cbrt
      (cos
       (/ (pow (pow x_m 0.3333333333333333) 2.0) (/ (* y_m 2.0) (cbrt x_m)))))
     3.0))
   (* x_m (/ 1.0 x_m))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+277) {
		tmp = 1.0 / pow(cbrt(cos((pow(pow(x_m, 0.3333333333333333), 2.0) / ((y_m * 2.0) / cbrt(x_m))))), 3.0);
	} else {
		tmp = x_m * (1.0 / x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+277) {
		tmp = 1.0 / Math.pow(Math.cbrt(Math.cos((Math.pow(Math.pow(x_m, 0.3333333333333333), 2.0) / ((y_m * 2.0) / Math.cbrt(x_m))))), 3.0);
	} else {
		tmp = x_m * (1.0 / x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+277)
		tmp = Float64(1.0 / (cbrt(cos(Float64(((x_m ^ 0.3333333333333333) ^ 2.0) / Float64(Float64(y_m * 2.0) / cbrt(x_m))))) ^ 3.0));
	else
		tmp = Float64(x_m * Float64(1.0 / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+277], N[(1.0 / N[Power[N[Power[N[Cos[N[(N[Power[N[Power[x$95$m, 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(y$95$m * 2.0), $MachinePrecision] / N[Power[x$95$m, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 4.99999999999999982e277

   1. Initial program 39.0%

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

   2. Taylor expanded in x around inf 50.8%

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

      1. associate-*r/ 50.8%

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

   4. Simplified 50.8%

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

      1. add-cube-cbrt 50.8%

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

      2. pow3 50.8%

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

      3. *-commutative 50.8%

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

      4. associate-/l* 50.8%

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

      5. div-inv 50.8%

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

      6. metadata-eval 50.8%

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

      7. *-un-lft-identity 50.8%

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

      8. *-commutative 50.8%

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

      9. times-frac 50.8%

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

      10. metadata-eval 50.8%

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

   6. Applied egg-rr 50.8%

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

      1. *-commutative 50.8%

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

      2. associate-/r/ 50.8%

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

      3. add-cube-cbrt 50.6%

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

      4. associate-/l* 50.3%

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

      5. pow2 50.3%

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

      6. div-inv 50.3%

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

      7. metadata-eval 50.3%

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

   8. Applied egg-rr 50.3%

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

      1. pow1/3 23.1%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{\cos \left(\frac{{\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{2}}{\frac{y \cdot 2}{\sqrt[3]{x}}}\right)}\right)}^{3}} \]

   10. Applied egg-rr 23.1%

       \[\leadsto \frac{1}{{\left(\sqrt[3]{\cos \left(\frac{{\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{2}}{\frac{y \cdot 2}{\sqrt[3]{x}}}\right)}\right)}^{3}} \]

   if 4.99999999999999982e277 < (/.f64 x (*.f64 y 2))

   1. Initial program 0.7%

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

   2. Taylor expanded in x around 0 0.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   3. Step-by-step derivation

      1. associate-*r/ 0.1%

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

   4. Simplified 0.1%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. div-inv 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-*r/ 0.1%

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

      4. associate-*l* 0.1%

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

      5. *-un-lft-identity 0.1%

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

      6. *-commutative 0.1%

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

      7. times-frac 0.1%

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

      8. metadata-eval 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-rgt-identity 0.1%

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

      3. associate-*r/ 0.1%

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

      4. *-commutative 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in y around inf 13.8%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\cos \left(\frac{{\left({x}^{0.3333333333333333}\right)}^{2}}{\frac{y \cdot 2}{\sqrt[3]{x}}}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \end{array} \]

Alternative 2: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 5 \cdot 10^{+83}:\\ \;\;\;\;{\cos \left(\frac{0.5}{\frac{y_m}{x_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{1}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+83)
   (pow (cos (/ 0.5 (/ y_m x_m))) -1.0)
   (* x_m (/ 1.0 x_m))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+83) {
		tmp = pow(cos((0.5 / (y_m / x_m))), -1.0);
	} else {
		tmp = x_m * (1.0 / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+83) then
        tmp = cos((0.5d0 / (y_m / x_m))) ** (-1.0d0)
    else
        tmp = x_m * (1.0d0 / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+83) {
		tmp = Math.pow(Math.cos((0.5 / (y_m / x_m))), -1.0);
	} else {
		tmp = x_m * (1.0 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+83:
		tmp = math.pow(math.cos((0.5 / (y_m / x_m))), -1.0)
	else:
		tmp = x_m * (1.0 / x_m)
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+83)
		tmp = cos(Float64(0.5 / Float64(y_m / x_m))) ^ -1.0;
	else
		tmp = Float64(x_m * Float64(1.0 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+83)
		tmp = cos((0.5 / (y_m / x_m))) ^ -1.0;
	else
		tmp = x_m * (1.0 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+83], N[Power[N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], N[(x$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5.00000000000000029e83

   1. Initial program 44.6%

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

   2. Taylor expanded in x around inf 58.5%

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

      1. associate-*r/ 58.5%

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

   4. Simplified 58.5%

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

      1. inv-pow 58.5%

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

      2. div-inv 58.4%

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

      3. associate-*l* 58.4%

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

      4. div-inv 58.5%

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

   6. Applied egg-rr 58.5%

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

      1. associate-*r/ 58.5%

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

      2. associate-/l* 58.6%

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

   8. Applied egg-rr 58.6%

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

   if 5.00000000000000029e83 < (/.f64 x (*.f64 y 2))

   1. Initial program 4.9%

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

   2. Taylor expanded in x around 0 1.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   3. Step-by-step derivation

      1. associate-*r/ 1.9%

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

   4. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. div-inv 1.9%

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

      2. *-commutative 1.9%

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

      3. associate-*r/ 1.9%

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

      4. associate-*l* 1.9%

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

      5. *-un-lft-identity 1.9%

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

      6. *-commutative 1.9%

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

      7. times-frac 1.9%

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

      8. metadata-eval 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. associate-*r/ 1.9%

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

      2. *-rgt-identity 1.9%

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

      3. associate-*r/ 1.9%

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

      4. *-commutative 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in y around inf 11.8%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+83}:\\ \;\;\;\;{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \end{array} \]

Alternative 3: 55.5% accurate, 211.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 1.0)

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := 1.0

Derivation

1. Initial program 35.2%

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

2. Taylor expanded in x around 0 46.4%

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

3. Final simplification 46.4%

   \[\leadsto 1 \]

Developer target: 55.5% 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 2023322 
(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)))))