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

Percentage Accurate: 44.1% → 55.4%

Time: 20.3s

Alternatives: 6

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: 44.1% 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: 55.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (* (/ x (pow (cbrt y) 2.0)) (/ 0.5 (cbrt y))))))

C

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

Java

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

Julia

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(x / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

4. Simplified 57.8%

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

   1. /-rgt-identity 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-*r/ 57.7%

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

6. Applied egg-rr 57.7%

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

   1. associate-*r/ 57.8%

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

   2. *-commutative 57.8%

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

   3. add-cube-cbrt 57.7%

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

   4. pow3 57.9%

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

   5. *-commutative 57.9%

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

   6. associate-*r/ 57.9%

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

8. Applied egg-rr 57.9%

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

   1. rem-cube-cbrt 57.7%

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

   2. associate-*r/ 57.8%

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

   3. add-cube-cbrt 57.9%

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

   4. unpow2 57.9%

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

   5. times-frac 58.1%

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

10. Applied egg-rr 58.1%

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

11. Final simplification 58.1%

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

Alternative 2: 55.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (pow (cbrt (* x (/ 0.5 y))) 3.0))))

C

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

Java

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

Julia

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[Power[N[Power[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

4. Simplified 57.8%

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

   1. /-rgt-identity 57.8%

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

   2. add-log-exp 57.8%

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

   3. /-rgt-identity 57.8%

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

   4. *-commutative 57.8%

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

   5. associate-*r/ 57.7%

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

6. Applied egg-rr 57.7%

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

7. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-*r/ 57.7%

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

9. Simplified 57.7%

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

    1. associate-*r/ 57.8%

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

    2. *-commutative 57.8%

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

    3. add-cube-cbrt 57.7%

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

    4. pow3 57.9%

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

    5. *-commutative 57.9%

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

    6. associate-*r/ 57.9%

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

11. Applied egg-rr 57.9%

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

12. Final simplification 57.9%

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

Alternative 3: 55.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

4. Simplified 57.8%

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

   1. /-rgt-identity 57.8%

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

   2. add-log-exp 57.8%

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

   3. /-rgt-identity 57.8%

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

   4. *-commutative 57.8%

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

   5. associate-*r/ 57.7%

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

6. Applied egg-rr 57.7%

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

   1. associate-*r/ 57.8%

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

8. Applied egg-rr 57.8%

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

9. Final simplification 57.8%

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

Alternative 4: 55.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

4. Simplified 57.8%

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

   1. /-rgt-identity 57.8%

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

   2. add-log-exp 57.8%

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

   3. /-rgt-identity 57.8%

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

   4. *-commutative 57.8%

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

   5. associate-*r/ 57.7%

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

6. Applied egg-rr 57.7%

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

7. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-*r/ 57.7%

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

9. Simplified 57.7%

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

10. Final simplification 57.7%

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

Alternative 5: 55.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

4. Simplified 57.8%

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

5. Final simplification 57.8%

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

Alternative 6: 55.5% 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 45.7%

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

2. Taylor expanded in x around 0 55.1%

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

3. Final simplification 55.1%

   \[\leadsto 1 \]

Developer target: 54.8% 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 2023300 
(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)))))