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

Percentage Accurate: 44.3% → 55.9%

Time: 13.3s

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: 44.3% 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: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := 1 + \frac{x\_m}{y\_m \cdot 2}\\ \frac{1}{\cos t\_0 \cdot \cos 1 + \sin t\_0 \cdot \sin 1} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x_m (* y_m 2.0)))))
   (/ 1.0 (+ (* (cos t_0) (cos 1.0)) (* (sin t_0) (sin 1.0))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = 1.0 + (x_m / (y_m * 2.0));
	return 1.0 / ((cos(t_0) * cos(1.0)) + (sin(t_0) * sin(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
    real(8) :: t_0
    t_0 = 1.0d0 + (x_m / (y_m * 2.0d0))
    code = 1.0d0 / ((cos(t_0) * cos(1.0d0)) + (sin(t_0) * sin(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) {
	double t_0 = 1.0 + (x_m / (y_m * 2.0));
	return 1.0 / ((Math.cos(t_0) * Math.cos(1.0)) + (Math.sin(t_0) * Math.sin(1.0)));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = 1.0 + (x_m / (y_m * 2.0))
	return 1.0 / ((math.cos(t_0) * math.cos(1.0)) + (math.sin(t_0) * math.sin(1.0)))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(1.0 + Float64(x_m / Float64(y_m * 2.0)))
	return Float64(1.0 / Float64(Float64(cos(t_0) * cos(1.0)) + Float64(sin(t_0) * sin(1.0))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	t_0 = 1.0 + (x_m / (y_m * 2.0));
	tmp = 1.0 / ((cos(t_0) * cos(1.0)) + (sin(t_0) * sin(1.0)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in x around inf 61.2%

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

   1. add-cube-cbrt 61.3%

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

   2. pow3 61.3%

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

5. Applied egg-rr 61.3%

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

   1. cube-mult 61.3%

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

   2. add-sqr-sqrt 38.4%

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

   3. associate-*l* 38.4%

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

   4. pow1/3 38.2%

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

   5. sqrt-pow1 38.1%

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

   6. metadata-eval 38.1%

      \[\leadsto \frac{1}{\cos \left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt{\sqrt[3]{\frac{x}{y}}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\right)\right)} \]

   7. pow1/3 38.2%

      \[\leadsto \frac{1}{\cos \left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\frac{x}{y}\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\right)\right)} \]

   8. sqrt-pow1 38.3%

      \[\leadsto \frac{1}{\cos \left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\right)\right)} \]

   9. metadata-eval 38.3%

      \[\leadsto \frac{1}{\cos \left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\right)\right)} \]

   10. pow2 38.3%

       \[\leadsto \frac{1}{\cos \left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}}\right)\right)\right)} \]

7. Applied egg-rr 38.3%

   \[\leadsto \frac{1}{\cos \left(0.5 \cdot \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)\right)}\right)} \]
8. Step-by-step derivation

   1. metadata-eval 38.3%

      \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{1}{2}} \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot \left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)\right)\right)} \]

   2. associate-*r* 38.2%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{2} \cdot \color{blue}{\left(\left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot {\left(\frac{x}{y}\right)}^{0.16666666666666666}\right) \cdot {\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)}\right)} \]

   3. unpow2 38.2%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{2} \cdot \left(\left({\left(\frac{x}{y}\right)}^{0.16666666666666666} \cdot {\left(\frac{x}{y}\right)}^{0.16666666666666666}\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)}\right)\right)} \]

   4. pow-prod-up 38.3%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{2} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{\left(0.16666666666666666 + 0.16666666666666666\right)}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\right)} \]

   5. metadata-eval 38.3%

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

   6. pow1/3 61.3%

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

   7. associate-*r* 61.3%

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

   8. add-cube-cbrt 61.2%

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

   9. times-frac 61.2%

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

   10. *-commutative 61.2%

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

   11. *-un-lft-identity 61.2%

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

   12. div-inv 61.0%

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

   13. metadata-eval 61.0%

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

   14. div-inv 61.0%

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

   15. clear-num 61.0%

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

   16. expm1-log1p-u 58.4%

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

   17. expm1-undefine 58.3%

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

9. Applied egg-rr 61.3%

   \[\leadsto \frac{1}{\color{blue}{\cos \left(1 + \frac{x}{y \cdot 2}\right) \cdot \cos 1 + \sin \left(1 + \frac{x}{y \cdot 2}\right) \cdot \sin 1}} \]
10. Add Preprocessing

Alternative 2: 55.9% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (* 0.5 (* (sqrt x_m) (/ (sqrt x_m) y_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((0.5 * (sqrt(x_m) * (sqrt(x_m) / y_m))));
}

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 / cos((0.5d0 * (sqrt(x_m) * (sqrt(x_m) / y_m))))
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 / Math.cos((0.5 * (Math.sqrt(x_m) * (Math.sqrt(x_m) / y_m))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos((0.5 * (math.sqrt(x_m) * (math.sqrt(x_m) / y_m))))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(0.5 * N[(N[Sqrt[x$95$m], $MachinePrecision] * N[(N[Sqrt[x$95$m], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in x around inf 61.2%

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

   1. add-sqr-sqrt 26.5%

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

   2. associate-/l* 27.0%

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

5. Applied egg-rr 27.0%

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

Alternative 3: 55.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{1}{\frac{2}{\frac{x\_m}{y\_m}}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ 1.0 (/ 2.0 (/ x_m y_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((1.0 / (2.0 / (x_m / y_m))));
}

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 / cos((1.0d0 / (2.0d0 / (x_m / y_m))))
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 / Math.cos((1.0 / (2.0 / (x_m / y_m))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos((1.0 / (2.0 / (x_m / y_m))))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(1.0 / Float64(2.0 / Float64(x_m / y_m)))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((1.0 / (2.0 / (x_m / y_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(1.0 / N[(2.0 / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in x around inf 61.2%

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

   1. add-sqr-sqrt 26.5%

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

   2. associate-/l* 27.0%

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

5. Applied egg-rr 27.0%

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

   1. associate-*r/ 26.5%

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

   2. add-sqr-sqrt 61.2%

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

   3. *-commutative 61.2%

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

   4. metadata-eval 61.2%

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

   5. div-inv 61.2%

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

   6. clear-num 61.3%

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

7. Applied egg-rr 61.3%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]
8. Add Preprocessing

Alternative 4: 55.9% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* 0.5 (/ x_m y_m)))))

C

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

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 / cos((0.5d0 * (x_m / y_m)))
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 / Math.cos((0.5 * (x_m / y_m)));
}

Python

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

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(0.5 * Float64(x_m / y_m))))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in x around inf 61.2%

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

Alternative 5: 56.0% 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 48.3%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. remove-double-neg 48.3%

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

   2. distribute-frac-neg 48.3%

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

   3. tan-neg 48.3%

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

   4. distribute-frac-neg2 48.3%

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

   5. distribute-lft-neg-out 48.3%

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

   6. distribute-frac-neg2 48.3%

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

   7. distribute-lft-neg-out 48.3%

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

   8. distribute-frac-neg2 48.3%

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

   9. distribute-frac-neg 48.3%

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

   10. neg-mul-1 48.3%

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

   11. *-commutative 48.3%

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

   12. associate-/l* 47.9%

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

   13. *-commutative 47.9%

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

   14. associate-/r* 47.9%

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

   15. metadata-eval 47.9%

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

   16. sin-neg 47.9%

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

   17. distribute-frac-neg 47.9%

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

3. Simplified 48.1%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 59.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 6: 6.6% 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 48.3%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. remove-double-neg 48.3%

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

   2. distribute-frac-neg 48.3%

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

   3. tan-neg 48.3%

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

   4. distribute-frac-neg2 48.3%

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

   5. distribute-lft-neg-out 48.3%

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

   6. distribute-frac-neg2 48.3%

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

   7. distribute-lft-neg-out 48.3%

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

   8. distribute-frac-neg2 48.3%

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

   9. distribute-frac-neg 48.3%

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

   10. neg-mul-1 48.3%

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

   11. *-commutative 48.3%

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

   12. associate-/l* 47.9%

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

   13. *-commutative 47.9%

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

   14. associate-/r* 47.9%

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

   15. metadata-eval 47.9%

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

   16. sin-neg 47.9%

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

   17. distribute-frac-neg 47.9%

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

3. Simplified 48.1%

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

   1. add-sqr-sqrt 26.7%

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

   2. sqrt-unprod 16.7%

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

   3. frac-times 16.1%

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

   4. metadata-eval 16.1%

      \[\leadsto \frac{\tan \left(x \cdot \sqrt{\frac{\color{blue}{0.25}}{y \cdot y}}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)} \]

   5. pow2 16.1%

      \[\leadsto \frac{\tan \left(x \cdot \sqrt{\frac{0.25}{\color{blue}{{y}^{2}}}}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)} \]

6. Applied egg-rr 16.1%

   \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\sqrt{\frac{0.25}{{y}^{2}}}}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)} \]

7. Taylor expanded in x around 0 6.0%

   \[\leadsto \color{blue}{-1} \]
8. Add Preprocessing

Developer Target 1: 56.0% 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 2024180 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))