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

Percentage Accurate: 43.8% → 54.8%

Time: 19.8s

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: 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: 54.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

3. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

5. Simplified 57.0%

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

6. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-/l* 57.3%

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

8. Simplified 57.3%

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

   1. div-inv 57.3%

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

   2. clear-num 57.0%

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

   3. *-commutative 57.0%

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

   4. add-cube-cbrt 57.5%

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

   5. associate-*l* 57.5%

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

   6. pow2 57.5%

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

10. Applied egg-rr 57.5%

    \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)}} \]
11. Step-by-step derivation

    1. pow1/3 37.9%

       \[\leadsto \frac{1}{\cos \left({\color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)} \]

    2. div-inv 37.6%

       \[\leadsto \frac{1}{\cos \left({\left({\color{blue}{\left(x \cdot \frac{1}{y}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)} \]

    3. unpow-prod-down 14.5%

       \[\leadsto \frac{1}{\cos \left({\color{blue}{\left({x}^{0.3333333333333333} \cdot {\left(\frac{1}{y}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)} \]

    4. pow1/3 28.9%

       \[\leadsto \frac{1}{\cos \left({\left(\color{blue}{\sqrt[3]{x}} \cdot {\left(\frac{1}{y}\right)}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)} \]

12. Applied egg-rr 28.9%

    \[\leadsto \frac{1}{\cos \left({\color{blue}{\left(\sqrt[3]{x} \cdot {\left(\frac{1}{y}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot 0.5\right)\right)} \]
13. Step-by-step derivation

    1. unpow1/3 57.7%

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

14. Simplified 57.7%

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

15. Final simplification 57.7%

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

Alternative 2: 54.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{x}{y}}\\ \frac{1}{\cos \left(\left(t_0 \cdot 0.5\right) \cdot {t_0}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (/ x y)))) (/ 1.0 (cos (* (* t_0 0.5) (pow t_0 2.0))))))

C

double code(double x, double y) {
	double t_0 = cbrt((x / y));
	return 1.0 / cos(((t_0 * 0.5) * pow(t_0, 2.0)));
}

Java

public static double code(double x, double y) {
	double t_0 = Math.cbrt((x / y));
	return 1.0 / Math.cos(((t_0 * 0.5) * Math.pow(t_0, 2.0)));
}

Julia

function code(x, y)
	t_0 = cbrt(Float64(x / y))
	return Float64(1.0 / cos(Float64(Float64(t_0 * 0.5) * (t_0 ^ 2.0))))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x / y), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[Cos[N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 44.2%

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

3. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

5. Simplified 57.0%

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

6. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-/l* 57.3%

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

8. Simplified 57.3%

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

   1. div-inv 57.3%

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

   2. clear-num 57.0%

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

   3. *-commutative 57.0%

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

   4. add-cube-cbrt 57.5%

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

   5. associate-*l* 57.5%

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

   6. pow2 57.5%

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

10. Applied egg-rr 57.5%

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

11. Final simplification 57.5%

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

Alternative 3: 33.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos((sqrt(((x / y) * 0.5d0)) ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

3. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

5. Simplified 57.0%

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

6. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-/l* 57.3%

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

8. Simplified 57.3%

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

   1. associate-/r/ 57.1%

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

   2. *-commutative 57.1%

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

   3. add-sqr-sqrt 37.7%

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

   4. pow2 37.7%

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

   5. *-commutative 37.7%

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

   6. associate-/r/ 37.9%

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

   7. div-inv 37.9%

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

   8. clear-num 37.9%

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

10. Applied egg-rr 37.9%

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

11. Final simplification 37.9%

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

Alternative 4: 55.1% 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 44.2%

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

3. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

5. Simplified 57.0%

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

6. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-/l* 57.3%

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

8. Simplified 57.3%

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

   1. associate-/r/ 57.1%

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

10. Applied egg-rr 57.1%

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

11. Final simplification 57.1%

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

Alternative 5: 55.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

3. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

5. Simplified 57.0%

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

6. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-/l* 57.3%

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

8. Simplified 57.3%

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

9. Final simplification 57.3%

   \[\leadsto \frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)} \]
10. Add Preprocessing

Alternative 6: 54.8% 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 44.2%

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

3. Taylor expanded in x around 0 56.0%

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

4. Final simplification 56.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 54.5% 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 2024018 
(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)))))