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

Percentage Accurate: 43.5% → 56.7%

Time: 19.9s

Alternatives: 7

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: 43.5% 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: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \end{array} \]

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)) 1.98e+113)
   (/ 1.0 (log1p (log (exp (expm1 (cos (* 0.5 (/ x_m y_m))))))))
   (/ -0.5 (cbrt -0.125))))

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)) <= 1.98e+113) {
		tmp = 1.0 / log1p(log(exp(expm1(cos((0.5 * (x_m / y_m)))))));
	} else {
		tmp = -0.5 / cbrt(-0.125);
	}
	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)) <= 1.98e+113) {
		tmp = 1.0 / Math.log1p(Math.log(Math.exp(Math.expm1(Math.cos((0.5 * (x_m / y_m)))))));
	} else {
		tmp = -0.5 / Math.cbrt(-0.125);
	}
	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)) <= 1.98e+113)
		tmp = Float64(1.0 / log1p(log(exp(expm1(cos(Float64(0.5 * Float64(x_m / y_m))))))));
	else
		tmp = Float64(-0.5 / cbrt(-0.125));
	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], 1.98e+113], N[(1.0 / N[Log[1 + N[Log[N[Exp[N[(Exp[N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.97999999999999985e113

   1. Initial program 50.8%

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

   2. Taylor expanded in x around inf 64.4%

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

      1. associate-*r/ 64.4%

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

   4. Simplified 64.4%

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

      1. /-rgt-identity 64.4%

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

      2. log1p-expm1-u 64.4%

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

      3. /-rgt-identity 64.4%

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

      4. div-inv 64.1%

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

      5. associate-*l* 64.1%

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

      6. div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

      1. add-log-exp 64.4%

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

   8. Applied egg-rr 64.4%

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

   if 1.97999999999999985e113 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 1.6%

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

      1. associate-*r/ 1.6%

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

   4. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 1.6%

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

      2. log1p-udef 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      3. div-inv 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}\right)\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      5. associate-/r* 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)\right)\right)} \]

      6. metadata-eval 1.6%

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

   6. Applied egg-rr 1.6%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{0.5}{y}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 1.6%

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

      2. *-commutative 1.6%

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

      3. add-cbrt-cube 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt[3]{\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}}\right)}\right)\right)} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

      5. pow3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\color{blue}{\left({\left(\frac{0.5 \cdot x}{y}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\left({\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

   9. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \]

Alternative 2: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \end{array} \]

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)) 1.98e+113)
   (log1p (expm1 (/ 1.0 (cos (* 0.5 (/ x_m y_m))))))
   (/ -0.5 (cbrt -0.125))))

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)) <= 1.98e+113) {
		tmp = log1p(expm1((1.0 / cos((0.5 * (x_m / y_m))))));
	} else {
		tmp = -0.5 / cbrt(-0.125);
	}
	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)) <= 1.98e+113) {
		tmp = Math.log1p(Math.expm1((1.0 / Math.cos((0.5 * (x_m / y_m))))));
	} else {
		tmp = -0.5 / Math.cbrt(-0.125);
	}
	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)) <= 1.98e+113)
		tmp = log1p(expm1(Float64(1.0 / cos(Float64(0.5 * Float64(x_m / y_m))))));
	else
		tmp = Float64(-0.5 / cbrt(-0.125));
	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], 1.98e+113], N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.5 / N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.97999999999999985e113

   1. Initial program 50.8%

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

   2. Taylor expanded in x around inf 64.4%

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

      1. associate-*r/ 64.4%

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

   4. Simplified 64.4%

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

      1. log1p-expm1-u 64.4%

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

      2. div-inv 64.1%

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

      3. associate-*l* 64.1%

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

      4. div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

   if 1.97999999999999985e113 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 1.6%

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

      1. associate-*r/ 1.6%

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

   4. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 1.6%

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

      2. log1p-udef 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      3. div-inv 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}\right)\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      5. associate-/r* 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)\right)\right)} \]

      6. metadata-eval 1.6%

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

   6. Applied egg-rr 1.6%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{0.5}{y}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 1.6%

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

      2. *-commutative 1.6%

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

      3. add-cbrt-cube 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt[3]{\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}}\right)}\right)\right)} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

      5. pow3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\color{blue}{\left({\left(\frac{0.5 \cdot x}{y}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\left({\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

   9. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \]

Alternative 3: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \end{array} \]

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)) 1.98e+113)
   (/ 1.0 (log (exp (cos (* 0.5 (/ x_m y_m))))))
   (/ -0.5 (cbrt -0.125))))

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)) <= 1.98e+113) {
		tmp = 1.0 / log(exp(cos((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / cbrt(-0.125);
	}
	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)) <= 1.98e+113) {
		tmp = 1.0 / Math.log(Math.exp(Math.cos((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / Math.cbrt(-0.125);
	}
	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)) <= 1.98e+113)
		tmp = Float64(1.0 / log(exp(cos(Float64(0.5 * Float64(x_m / y_m))))));
	else
		tmp = Float64(-0.5 / cbrt(-0.125));
	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], 1.98e+113], N[(1.0 / N[Log[N[Exp[N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.97999999999999985e113

   1. Initial program 50.8%

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

   2. Taylor expanded in x around inf 64.4%

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

      1. associate-*r/ 64.4%

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

   4. Simplified 64.4%

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

      1. /-rgt-identity 64.4%

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

      2. add-log-exp 64.4%

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

      3. /-rgt-identity 64.4%

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

      4. div-inv 64.1%

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

      5. associate-*l* 64.1%

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

      6. div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

   if 1.97999999999999985e113 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 1.6%

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

      1. associate-*r/ 1.6%

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

   4. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 1.6%

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

      2. log1p-udef 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      3. div-inv 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}\right)\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      5. associate-/r* 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)\right)\right)} \]

      6. metadata-eval 1.6%

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

   6. Applied egg-rr 1.6%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{0.5}{y}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 1.6%

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

      2. *-commutative 1.6%

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

      3. add-cbrt-cube 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt[3]{\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}}\right)}\right)\right)} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

      5. pow3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\color{blue}{\left({\left(\frac{0.5 \cdot x}{y}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\left({\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

   9. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \]

Alternative 4: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \end{array} \]

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)) 1.98e+113)
   (/ 1.0 (log1p (expm1 (cos (* 0.5 (/ x_m y_m))))))
   (/ -0.5 (cbrt -0.125))))

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)) <= 1.98e+113) {
		tmp = 1.0 / log1p(expm1(cos((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / cbrt(-0.125);
	}
	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)) <= 1.98e+113) {
		tmp = 1.0 / Math.log1p(Math.expm1(Math.cos((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / Math.cbrt(-0.125);
	}
	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)) <= 1.98e+113)
		tmp = Float64(1.0 / log1p(expm1(cos(Float64(0.5 * Float64(x_m / y_m))))));
	else
		tmp = Float64(-0.5 / cbrt(-0.125));
	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], 1.98e+113], N[(1.0 / N[Log[1 + N[(Exp[N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.97999999999999985e113

   1. Initial program 50.8%

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

   2. Taylor expanded in x around inf 64.4%

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

      1. associate-*r/ 64.4%

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

   4. Simplified 64.4%

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

      1. /-rgt-identity 64.4%

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

      2. log1p-expm1-u 64.4%

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

      3. /-rgt-identity 64.4%

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

      4. div-inv 64.1%

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

      5. associate-*l* 64.1%

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

      6. div-inv 64.4%

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

   6. Applied egg-rr 64.4%

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

   if 1.97999999999999985e113 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 1.6%

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

      1. associate-*r/ 1.6%

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

   4. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 1.6%

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

      2. log1p-udef 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      3. div-inv 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}\right)\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      5. associate-/r* 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)\right)\right)} \]

      6. metadata-eval 1.6%

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

   6. Applied egg-rr 1.6%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{0.5}{y}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 1.6%

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

      2. *-commutative 1.6%

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

      3. add-cbrt-cube 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt[3]{\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}}\right)}\right)\right)} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

      5. pow3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\color{blue}{\left({\left(\frac{0.5 \cdot x}{y}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\left({\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

   9. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \]

Alternative 5: 56.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x_m \cdot 0.5}{y_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \end{array} \]

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)) 1.98e+113)
   (/ 1.0 (cos (/ (* x_m 0.5) y_m)))
   (/ -0.5 (cbrt -0.125))))

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)) <= 1.98e+113) {
		tmp = 1.0 / cos(((x_m * 0.5) / y_m));
	} else {
		tmp = -0.5 / cbrt(-0.125);
	}
	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)) <= 1.98e+113) {
		tmp = 1.0 / Math.cos(((x_m * 0.5) / y_m));
	} else {
		tmp = -0.5 / Math.cbrt(-0.125);
	}
	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)) <= 1.98e+113)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * 0.5) / y_m)));
	else
		tmp = Float64(-0.5 / cbrt(-0.125));
	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], 1.98e+113], N[(1.0 / N[Cos[N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.97999999999999985e113

   1. Initial program 50.8%

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

   2. Taylor expanded in x around inf 64.4%

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

      1. associate-*r/ 64.4%

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

   4. Simplified 64.4%

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

   if 1.97999999999999985e113 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 1.6%

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

      1. associate-*r/ 1.6%

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

   4. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot x}{y}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 1.6%

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

      2. log1p-udef 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      3. div-inv 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}\right)\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      5. associate-/r* 1.6%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)\right)\right)} \]

      6. metadata-eval 1.6%

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

   6. Applied egg-rr 1.6%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \left(x \cdot \frac{0.5}{y}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 1.6%

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

      2. *-commutative 1.6%

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

      3. add-cbrt-cube 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt[3]{\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}}\right)}\right)\right)} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\left(\frac{0.5 \cdot x}{y} \cdot \frac{0.5 \cdot x}{y}\right) \cdot \frac{0.5 \cdot x}{y}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

      5. pow3 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\color{blue}{\left({\left(\frac{0.5 \cdot x}{y}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \left({\left({\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{0.5 \cdot x}{y}}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

   9. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.98 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot 0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt[3]{-0.125}}\\ \end{array} \]

Alternative 6: 55.0% 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 43.1%

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

2. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

4. Simplified 54.5%

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

   1. /-rgt-identity 54.5%

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

   2. log1p-expm1-u 54.5%

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

   3. /-rgt-identity 54.5%

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

   4. div-inv 54.2%

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

   5. associate-*l* 54.2%

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

   6. div-inv 54.5%

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

6. Applied egg-rr 54.5%

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

   1. add-log-exp 54.5%

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

8. Applied egg-rr 54.5%

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

   1. rem-log-exp 54.5%

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

   2. log1p-expm1-u 54.5%

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

   3. associate-*r/ 54.5%

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

   4. associate-/l* 54.4%

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

10. Applied egg-rr 54.4%

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

11. Final simplification 54.4%

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

Alternative 7: 55.1% 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 43.1%

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

2. Taylor expanded in x around 0 54.3%

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

3. Final simplification 54.3%

   \[\leadsto 1 \]

Developer target: 55.1% 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 2023334 
(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)))))