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

Percentage Accurate: 43.9% → 57.1%

Time: 19.0s

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: 43.9% 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: 57.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{\frac{x\_m}{2}}{y\_m}\right)}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+232)
   (/ 1.0 (cos (expm1 (pow (sqrt (log1p (/ (/ x_m 2.0) y_m))) 2.0))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+232) {
		tmp = 1.0 / cos(expm1(pow(sqrt(log1p(((x_m / 2.0) / y_m))), 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+232) {
		tmp = 1.0 / Math.cos(Math.expm1(Math.pow(Math.sqrt(Math.log1p(((x_m / 2.0) / y_m))), 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+232:
		tmp = 1.0 / math.cos(math.expm1(math.pow(math.sqrt(math.log1p(((x_m / 2.0) / y_m))), 2.0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+232)
		tmp = Float64(1.0 / cos(expm1((sqrt(log1p(Float64(Float64(x_m / 2.0) / y_m))) ^ 2.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+232], N[(1.0 / N[Cos[N[(Exp[N[Power[N[Sqrt[N[Log[1 + N[(N[(x$95$m / 2.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 4.99999999999999987e232

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 61.7%

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

      1. expm1-log1p-u 58.8%

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

      2. expm1-undefine 58.6%

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

   5. Applied egg-rr 58.6%

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

      1. expm1-define 58.8%

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

      2. associate-*r/ 58.8%

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

      3. *-commutative 58.8%

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

   7. Simplified 58.8%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 58.8%

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

      2. add-sqr-sqrt 30.7%

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

      3. pow2 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 0.5}{y}\right)\right)\right)}\right)}^{2}}\right)\right)} \]

      4. expm1-log1p-u 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 0.5}{y}}\right)}\right)}^{2}\right)\right)} \]

      5. *-commutative 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot x}}{y}\right)}\right)}^{2}\right)\right)} \]

      6. associate-*r/ 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{x}{y}}\right)}\right)}^{2}\right)\right)} \]

      7. metadata-eval 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{1}{2}} \cdot \frac{x}{y}\right)}\right)}^{2}\right)\right)} \]

      8. times-frac 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot x}{2 \cdot y}}\right)}\right)}^{2}\right)\right)} \]

      9. *-un-lft-identity 30.7%

         \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{\color{blue}{x}}{2 \cdot y}\right)}\right)}^{2}\right)\right)} \]

      10. *-commutative 30.7%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{x}{\color{blue}{y \cdot 2}}\right)}\right)}^{2}\right)\right)} \]

      11. add-sqr-sqrt 14.5%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}}}\right)}\right)}^{2}\right)\right)} \]

      12. associate-/l/ 14.5%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x}{\sqrt{y \cdot 2}}}{\sqrt{y \cdot 2}}}\right)}\right)}^{2}\right)\right)} \]

      13. associate-/l/ 14.5%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}}}\right)}\right)}^{2}\right)\right)} \]

      14. add-sqr-sqrt 30.7%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{x}{\color{blue}{y \cdot 2}}\right)}\right)}^{2}\right)\right)} \]

      15. *-commutative 30.7%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{x}{\color{blue}{2 \cdot y}}\right)}\right)}^{2}\right)\right)} \]

      16. associate-/r* 30.7%

          \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x}{2}}{y}}\right)}\right)}^{2}\right)\right)} \]

   9. Applied egg-rr 30.7%

      \[\leadsto \frac{1}{\cos \left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\frac{\frac{x}{2}}{y}\right)}\right)}^{2}}\right)\right)} \]

   if 4.99999999999999987e232 < (/.f64 x (*.f64 y 2))

   1. Initial program 3.3%

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

   3. Taylor expanded in x around 0 15.1%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{\cos \left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\frac{\frac{x}{2}}{y}\right)}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x\_m \cdot 0.5}{\sqrt{y\_m}}}{\sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+234)
   (/ 1.0 (cos (/ (/ (* x_m 0.5) (sqrt y_m)) (sqrt y_m))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / cos((((x_m * 0.5) / sqrt(y_m)) / sqrt(y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+234) then
        tmp = 1.0d0 / cos((((x_m * 0.5d0) / sqrt(y_m)) / sqrt(y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / Math.cos((((x_m * 0.5) / Math.sqrt(y_m)) / Math.sqrt(y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+234:
		tmp = 1.0 / math.cos((((x_m * 0.5) / math.sqrt(y_m)) / math.sqrt(y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+234)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(x_m * 0.5) / sqrt(y_m)) / sqrt(y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+234)
		tmp = 1.0 / cos((((x_m * 0.5) / sqrt(y_m)) / sqrt(y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+234], N[(1.0 / N[Cos[N[(N[(N[(x$95$m * 0.5), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000002e234

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. associate-*r/ 61.5%

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

      2. add-sqr-sqrt 29.9%

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

      3. associate-/r* 30.0%

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

   5. Applied egg-rr 30.0%

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

   if 1.00000000000000002e234 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 15.0%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\sqrt{y}}}{\sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\cos \left(0.5 \cdot \frac{x\_m}{y\_m}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+234)
   (/ 1.0 (pow (cbrt (cos (* 0.5 (/ x_m y_m)))) 3.0))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / pow(cbrt(cos((0.5 * (x_m / y_m)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / Math.pow(Math.cbrt(Math.cos((0.5 * (x_m / y_m)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+234)
		tmp = Float64(1.0 / (cbrt(cos(Float64(0.5 * Float64(x_m / y_m)))) ^ 3.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+234], N[(1.0 / N[Power[N[Power[N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000002e234

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

   5. Applied egg-rr 61.5%

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

   if 1.00000000000000002e234 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 15.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x\_m \cdot 0.5}{y\_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+234)
   (/ 1.0 (expm1 (log1p (cos (/ (* x_m 0.5) y_m)))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / expm1(log1p(cos(((x_m * 0.5) / y_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / Math.expm1(Math.log1p(Math.cos(((x_m * 0.5) / y_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+234:
		tmp = 1.0 / math.expm1(math.log1p(math.cos(((x_m * 0.5) / y_m))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+234)
		tmp = Float64(1.0 / expm1(log1p(cos(Float64(Float64(x_m * 0.5) / y_m)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+234], N[(1.0 / N[(Exp[N[Log[1 + N[Cos[N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000002e234

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. expm1-log1p-u 61.5%

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

      2. expm1-undefine 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. expm1-define 61.5%

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

      2. associate-*r/ 61.5%

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

      3. *-commutative 61.5%

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

   7. Simplified 61.5%

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

   if 1.00000000000000002e234 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 15.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x\_m}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+234) (/ 1.0 (cos (* 0.5 (/ x_m y_m)))) 1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / cos((0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+234) then
        tmp = 1.0d0 / cos((0.5d0 * (x_m / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+234) {
		tmp = 1.0 / Math.cos((0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+234:
		tmp = 1.0 / math.cos((0.5 * (x_m / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+234)
		tmp = Float64(1.0 / cos(Float64(0.5 * Float64(x_m / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+234)
		tmp = 1.0 / cos((0.5 * (x_m / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+234], N[(1.0 / N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000002e234

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf 61.5%

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

   if 1.00000000000000002e234 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 15.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+234}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.7% 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 47.7%

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

3. Taylor expanded in x around 0 56.0%

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

4. Final simplification 56.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 55.7% 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 2024034 
(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)))))