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

Percentage Accurate: 43.8% → 56.7%

Time: 19.4s

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: 43.8% 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: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+77) (/ 1.0 (log (exp (cos (* x (/ 0.5 y)))))) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+77) {
		tmp = 1.0 / log(exp(cos((x * (0.5 / y)))));
	} 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+77) then
        tmp = 1.0d0 / log(exp(cos((x * (0.5d0 / y)))))
    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+77) {
		tmp = 1.0 / Math.log(Math.exp(Math.cos((x * (0.5 / y)))));
	} 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+77:
		tmp = 1.0 / math.log(math.exp(math.cos((x * (0.5 / y)))))
	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+77)
		tmp = Float64(1.0 / log(exp(cos(Float64(x * Float64(0.5 / y))))));
	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+77)
		tmp = 1.0 / log(exp(cos((x * (0.5 / y)))));
	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+77], N[(1.0 / N[Log[N[Exp[N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.99999999999999997e77

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 69.1%

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

      1. associate-*r/ 69.1%

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

   4. Simplified 69.1%

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

      1. /-rgt-identity 69.1%

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

      2. add-log-exp 69.1%

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

      3. /-rgt-identity 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-*r/ 69.0%

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

   6. Applied egg-rr 69.0%

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

   if 1.99999999999999997e77 < (/.f64 x (*.f64 y 2))

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 14.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \frac{1}{\cos \left({\left(\sqrt{{\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}}\right)}^{2}\right)} \end{array} \]

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 (cos (pow (sqrt (pow (cbrt (* x (/ 0.5 y))) 3.0)) 2.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos(pow(sqrt(pow(cbrt((x * (0.5 / y))), 3.0)), 2.0));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return 1.0 / Math.cos(Math.pow(Math.sqrt(Math.pow(Math.cbrt((x * (0.5 / y))), 3.0)), 2.0));
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return Float64(1.0 / cos((sqrt((cbrt(Float64(x * Float64(0.5 / y))) ^ 3.0)) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. associate-*r/ 59.9%

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

7. Simplified 59.9%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. add-sqr-sqrt 40.7%

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

   4. pow2 40.7%

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

   5. *-commutative 40.7%

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

   6. associate-*r/ 40.7%

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

9. Applied egg-rr 40.7%

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

    1. associate-*r/ 40.7%

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

    2. *-commutative 40.7%

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

    3. add-cube-cbrt 40.6%

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

    4. pow3 40.6%

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

    5. *-commutative 40.6%

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

    6. associate-*r/ 40.5%

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

11. Applied egg-rr 40.5%

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

12. Final simplification 40.5%

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

Alternative 3: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+77)
   (log1p (expm1 (/ 1.0 (cos (* x (/ 0.5 y))))))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+77) {
		tmp = log1p(expm1((1.0 / cos((x * (0.5 / y))))));
	} 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)) <= 2e+77) {
		tmp = Math.log1p(Math.expm1((1.0 / Math.cos((x * (0.5 / y))))));
	} 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+77:
		tmp = math.log1p(math.expm1((1.0 / math.cos((x * (0.5 / y))))))
	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+77)
		tmp = log1p(expm1(Float64(1.0 / cos(Float64(x * Float64(0.5 / y))))));
	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], 2e+77], N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.99999999999999997e77

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 69.1%

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

      1. associate-*r/ 69.1%

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

   4. Simplified 69.1%

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

      1. log1p-expm1-u 69.1%

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

      2. *-commutative 69.1%

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

      3. associate-*r/ 68.9%

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

   6. Applied egg-rr 68.9%

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

   if 1.99999999999999997e77 < (/.f64 x (*.f64 y 2))

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 14.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \frac{1}{\cos \left({\left(\sqrt{x \cdot \frac{0.5}{y}}\right)}^{2}\right)} \end{array} \]

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 (cos (pow (sqrt (* x (/ 0.5 y))) 2.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos(pow(sqrt((x * (0.5 / y))), 2.0));
}

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 / cos((sqrt((x * (0.5d0 / y))) ** 2.0d0))
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return 1.0 / Math.cos(Math.pow(Math.sqrt((x * (0.5 / y))), 2.0));
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0 / math.cos(math.pow(math.sqrt((x * (0.5 / y))), 2.0))

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return Float64(1.0 / cos((sqrt(Float64(x * Float64(0.5 / y))) ^ 2.0)))
end

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((sqrt((x * (0.5 / y))) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. associate-*r/ 59.9%

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

7. Simplified 59.9%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. add-sqr-sqrt 40.7%

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

   4. pow2 40.7%

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

   5. *-commutative 40.7%

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

   6. associate-*r/ 40.7%

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

9. Applied egg-rr 40.7%

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

10. Final simplification 40.7%

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

Alternative 5: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \frac{1}{\cos \left({\left(\sqrt{\frac{0.5}{\frac{y}{x}}}\right)}^{2}\right)} \end{array} \]

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 (cos (pow (sqrt (/ 0.5 (/ y x))) 2.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos(pow(sqrt((0.5 / (y / x))), 2.0));
}

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 / cos((sqrt((0.5d0 / (y / x))) ** 2.0d0))
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return 1.0 / Math.cos(Math.pow(Math.sqrt((0.5 / (y / x))), 2.0));
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0 / math.cos(math.pow(math.sqrt((0.5 / (y / x))), 2.0))

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return Float64(1.0 / cos((sqrt(Float64(0.5 / Float64(y / x))) ^ 2.0)))
end

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((sqrt((0.5 / (y / x))) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 45.9%

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

2. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. associate-*r/ 59.9%

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

7. Simplified 59.9%

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

   1. associate-*r/ 60.0%

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

   2. *-commutative 60.0%

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

   3. add-sqr-sqrt 40.7%

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

   4. pow2 40.7%

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

   5. *-commutative 40.7%

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

   6. associate-*r/ 40.7%

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

9. Applied egg-rr 40.7%

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

    1. *-commutative 40.7%

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

    2. associate-/r/ 41.2%

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

11. Applied egg-rr 41.2%

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

12. Final simplification 41.2%

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

Alternative 6: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+77) (/ 1.0 (cos (* x (/ 0.5 y)))) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+77) {
		tmp = 1.0 / cos((x * (0.5 / y)));
	} 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+77) then
        tmp = 1.0d0 / cos((x * (0.5d0 / y)))
    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+77) {
		tmp = 1.0 / Math.cos((x * (0.5 / y)));
	} 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+77:
		tmp = 1.0 / math.cos((x * (0.5 / y)))
	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+77)
		tmp = Float64(1.0 / cos(Float64(x * Float64(0.5 / y))));
	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+77)
		tmp = 1.0 / cos((x * (0.5 / y)));
	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+77], N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.99999999999999997e77

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 69.1%

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

      1. associate-*r/ 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in x around inf 69.1%

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

      1. associate-*r/ 69.1%

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

      2. *-commutative 69.1%

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

      3. associate-*r/ 69.0%

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

   7. Simplified 69.0%

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

   if 1.99999999999999997e77 < (/.f64 x (*.f64 y 2))

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 14.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 55.0% 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 45.9%

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

2. Taylor expanded in x around 0 59.8%

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

3. Final simplification 59.8%

   \[\leadsto 1 \]

Developer target: 55.0% 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 2023310 
(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)))))