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

Percentage Accurate: 43.6% → 55.5%

Time: 17.9s

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: 43.6% 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: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left({\left({\left({\left(0.5 \cdot \frac{x\_m}{y\_m}\right)}^{0.25}\right)}^{2}\right)}^{2}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (pow (pow (pow (* 0.5 (/ x_m y_m)) 0.25) 2.0) 2.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(pow(pow(pow((0.5 * (x_m / y_m)), 0.25), 2.0), 2.0));
}

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 / cos(((((0.5d0 * (x_m / y_m)) ** 0.25d0) ** 2.0d0) ** 2.0d0))
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 / Math.cos(Math.pow(Math.pow(Math.pow((0.5 * (x_m / y_m)), 0.25), 2.0), 2.0));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos(math.pow(math.pow(math.pow((0.5 * (x_m / y_m)), 0.25), 2.0), 2.0))

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)) ^ 0.25) ^ 2.0) ^ 2.0)))
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)) ^ 0.25) ^ 2.0) ^ 2.0));
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[Power[N[Power[N[Power[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in x around inf 49.1%

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

   1. add-sqr-sqrt 28.6%

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

   2. metadata-eval 28.6%

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

   3. fabs-sqr 28.6%

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

   4. add-sqr-sqrt 49.1%

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

   5. fabs-mul 49.1%

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

   6. add-sqr-sqrt 48.8%

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

   7. pow2 48.8%

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

   8. fabs-mul 48.8%

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

   9. metadata-eval 48.8%

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

   10. sqrt-prod 48.5%

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

   11. add-sqr-sqrt 29.1%

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

   12. fabs-sqr 29.1%

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

   13. add-sqr-sqrt 29.1%

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

   14. sqrt-prod 29.5%

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

   15. associate-*r/ 29.5%

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

   16. *-commutative 29.5%

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

   17. associate-/l* 29.6%

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

5. Applied egg-rr 29.6%

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

   1. add-sqr-sqrt 29.5%

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

   2. pow2 29.5%

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

   3. pow1/2 29.5%

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

   4. add-sqr-sqrt 29.5%

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

   5. unpow2 29.5%

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

   6. sqrt-pow1 29.5%

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

   7. unpow2 29.5%

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

   8. add-sqr-sqrt 29.5%

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

   9. *-commutative 29.5%

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

   10. associate-*l/ 29.6%

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

   11. associate-*r/ 29.6%

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

   12. metadata-eval 29.6%

       \[\leadsto \frac{1}{\cos \left({\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{\color{blue}{0.25}}\right)}^{2}\right)}^{2}\right)} \]

7. Applied egg-rr 29.6%

   \[\leadsto \frac{1}{\cos \left({\color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{0.25}\right)}^{2}\right)}}^{2}\right)} \]

8. Final simplification 29.6%

   \[\leadsto \frac{1}{\cos \left({\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{0.25}\right)}^{2}\right)}^{2}\right)} \]
9. Add Preprocessing

Alternative 2: 55.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((sqrt(x_m) * ((0.5 / y_m) * sqrt(x_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((sqrt(x_m) * ((0.5d0 / y_m) * sqrt(x_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((Math.sqrt(x_m) * ((0.5 / y_m) * Math.sqrt(x_m))));
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((sqrt(x_m) * ((0.5 / y_m) * sqrt(x_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[(N[Sqrt[x$95$m], $MachinePrecision] * N[(N[(0.5 / y$95$m), $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in x around inf 49.1%

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

   1. metadata-eval 49.1%

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

   2. times-frac 49.1%

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

   3. *-un-lft-identity 49.1%

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

   4. *-commutative 49.1%

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

   5. clear-num 49.1%

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

   6. associate-/l* 49.2%

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

5. Applied egg-rr 49.2%

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

   1. associate-*r/ 49.1%

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

   2. associate-/r/ 49.0%

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

   3. *-commutative 49.0%

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

   4. associate-/r* 49.0%

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

   5. metadata-eval 49.0%

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

   6. add-sqr-sqrt 27.8%

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

   7. associate-*r* 28.0%

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

7. Applied egg-rr 28.0%

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

8. Final simplification 28.0%

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

Alternative 3: 55.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((1.0 / (y_m * (2.0 / x_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 / (y_m * (2.0d0 / x_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 / (y_m * (2.0 / x_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 / (y_m * (2.0 / x_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(y_m * Float64(2.0 / x_m)))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((1.0 / (y_m * (2.0 / x_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[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in x around inf 49.1%

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

   1. metadata-eval 49.1%

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

   2. times-frac 49.1%

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

   3. *-un-lft-identity 49.1%

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

   4. *-commutative 49.1%

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

   5. clear-num 49.1%

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

   6. associate-/l* 49.2%

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

5. Applied egg-rr 49.2%

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

6. Final simplification 49.2%

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

Alternative 4: 55.6% 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 39.7%

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

3. Taylor expanded in x around inf 49.1%

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

4. Final simplification 49.1%

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

Alternative 5: 55.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((0.5 / (y_m / x_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[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in x around inf 49.1%

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

   1. clear-num 49.1%

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

   2. un-div-inv 49.1%

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

5. Applied egg-rr 49.1%

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

6. Final simplification 49.1%

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

Alternative 6: 55.8% 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 39.7%

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

   1. remove-double-neg 39.7%

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

   2. distribute-frac-neg 39.7%

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

   3. tan-neg 39.7%

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

   4. distribute-frac-neg2 39.7%

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

   5. distribute-lft-neg-out 39.7%

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

   6. distribute-frac-neg2 39.7%

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

   7. distribute-lft-neg-out 39.7%

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

   8. distribute-frac-neg2 39.7%

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

   9. distribute-frac-neg 39.7%

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

   10. neg-mul-1 39.7%

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

   11. *-commutative 39.7%

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

   12. associate-/l* 39.6%

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

   13. *-commutative 39.6%

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

   14. associate-/r* 39.6%

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

   15. metadata-eval 39.6%

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

   16. sin-neg 39.6%

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

   17. distribute-frac-neg 39.6%

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

3. Simplified 39.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Final simplification 48.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 55.8% 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 2024057 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (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)))))