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

Percentage Accurate: 32.2% → 34.9%

Time: 40.1s

Alternatives: 6

Speedup: 211.0×

• could not determine a ground truth

  1. x = -1.4750744640997714e+105
  2. y = 4.312007031122999e+211

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: 32.2% 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: 34.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 1e+253)
   (/ 1.0 (cos (* (cbrt x) (* (/ 0.5 y) (pow (expm1 (log1p (cbrt x))) 2.0)))))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / cos((cbrt(x) * ((0.5 / y) * pow(expm1(log1p(cbrt(x))), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / Math.cos((Math.cbrt(x) * ((0.5 / y) * Math.pow(Math.expm1(Math.log1p(Math.cbrt(x))), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+253)
		tmp = Float64(1.0 / cos(Float64(cbrt(x) * Float64(Float64(0.5 / y) * (expm1(log1p(cbrt(x))) ^ 2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+253], N[(1.0 / N[Cos[N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(0.5 / y), $MachinePrecision] * N[Power[N[(Exp[N[Log[1 + N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.9999999999999994e252

   1. Initial program 40.9%

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

   2. Taylor expanded in x around inf 40.9%

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

      1. clear-num 40.7%

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

      2. un-div-inv 40.7%

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

   4. Applied egg-rr 40.7%

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

      1. associate-/r/ 40.3%

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

      2. add-cube-cbrt 40.4%

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

      3. associate-*r* 40.9%

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

      4. pow2 40.9%

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

   6. Applied egg-rr 40.9%

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

      1. expm1-log1p-u 39.2%

         \[\leadsto \frac{1}{\cos \left(\left(\frac{0.5}{y} \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)\right)}}^{2}\right) \cdot \sqrt[3]{x}\right)} \]

   8. Applied egg-rr 39.2%

      \[\leadsto \frac{1}{\cos \left(\left(\frac{0.5}{y} \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)\right)}}^{2}\right) \cdot \sqrt[3]{x}\right)} \]

   if 9.9999999999999994e252 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 9.4%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 34.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(e^{\log x \cdot 0.3333333333333333}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 1e+253)
   (/
    1.0
    (cos
     (*
      (cbrt x)
      (* (/ 0.5 y) (pow (exp (* (log x) 0.3333333333333333)) 2.0)))))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / cos((cbrt(x) * ((0.5 / y) * pow(exp((log(x) * 0.3333333333333333)), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / Math.cos((Math.cbrt(x) * ((0.5 / y) * Math.pow(Math.exp((Math.log(x) * 0.3333333333333333)), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+253)
		tmp = Float64(1.0 / cos(Float64(cbrt(x) * Float64(Float64(0.5 / y) * (exp(Float64(log(x) * 0.3333333333333333)) ^ 2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+253], N[(1.0 / N[Cos[N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(0.5 / y), $MachinePrecision] * N[Power[N[Exp[N[(N[Log[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.9999999999999994e252

   1. Initial program 40.9%

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

   2. Taylor expanded in x around inf 40.9%

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

      1. clear-num 40.7%

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

      2. un-div-inv 40.7%

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

   4. Applied egg-rr 40.7%

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

      1. associate-/r/ 40.3%

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

      2. add-cube-cbrt 40.4%

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

      3. associate-*r* 40.9%

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

      4. pow2 40.9%

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

   6. Applied egg-rr 40.9%

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

      1. pow1/3 16.7%

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

      2. pow-to-exp 17.7%

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

   8. Applied egg-rr 17.7%

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

   if 9.9999999999999994e252 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 9.4%

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

4. Final simplification 16.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(e^{\log x \cdot 0.3333333333333333}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 34.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 4e+262)
   (/ 1.0 (cos (* (cbrt x) (* (/ 0.5 y) (pow (cbrt x) 2.0)))))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 4e+262) {
		tmp = 1.0 / cos((cbrt(x) * ((0.5 / y) * pow(cbrt(x), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 4e+262) {
		tmp = 1.0 / Math.cos((Math.cbrt(x) * ((0.5 / y) * Math.pow(Math.cbrt(x), 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 4e+262)
		tmp = Float64(1.0 / cos(Float64(cbrt(x) * Float64(Float64(0.5 / y) * (cbrt(x) ^ 2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 4e+262], N[(1.0 / N[Cos[N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(0.5 / y), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 4.0000000000000001e262

   1. Initial program 40.4%

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

   2. Taylor expanded in x around inf 40.4%

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

      1. clear-num 40.4%

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

      2. un-div-inv 40.4%

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

   4. Applied egg-rr 40.4%

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

      1. associate-/r/ 39.9%

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

      2. add-cube-cbrt 40.1%

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

      3. associate-*r* 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

   if 4.0000000000000001e262 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.2%

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

   2. Taylor expanded in x around 0 8.8%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x} \cdot \left(\frac{0.5}{y} \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{{\left(0.5 \cdot \frac{x}{y}\right)}^{1.5}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 2e+200)
   (/ 1.0 (cos (pow (cbrt (pow (* 0.5 (/ x y)) 1.5)) 2.0)))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+200) {
		tmp = 1.0 / cos(pow(cbrt(pow((0.5 * (x / y)), 1.5)), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+200) {
		tmp = 1.0 / Math.cos(Math.pow(Math.cbrt(Math.pow((0.5 * (x / y)), 1.5)), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 2e+200)
		tmp = Float64(1.0 / cos((cbrt((Float64(0.5 * Float64(x / y)) ^ 1.5)) ^ 2.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 2e+200], N[(1.0 / N[Cos[N[Power[N[Power[N[Power[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.9999999999999999e200

   1. Initial program 42.7%

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

   2. Taylor expanded in x around inf 42.7%

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

      1. add-sqr-sqrt 16.9%

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

      2. pow2 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. add-cbrt-cube 17.1%

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

      2. pow1/3 16.5%

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

      3. add-sqr-sqrt 16.6%

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

      4. pow1 16.6%

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

      5. pow1/2 16.6%

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

      6. pow-prod-up 16.6%

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

      7. metadata-eval 16.6%

         \[\leadsto \frac{1}{\cos \left({\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right)} \]

   6. Applied egg-rr 16.6%

      \[\leadsto \frac{1}{\cos \left({\color{blue}{\left({\left({\left(0.5 \cdot \frac{x}{y}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow1/3 17.2%

         \[\leadsto \frac{1}{\cos \left({\color{blue}{\left(\sqrt[3]{{\left(0.5 \cdot \frac{x}{y}\right)}^{1.5}}\right)}}^{2}\right)} \]

   8. Simplified 17.2%

      \[\leadsto \frac{1}{\cos \left({\color{blue}{\left(\sqrt[3]{{\left(0.5 \cdot \frac{x}{y}\right)}^{1.5}}\right)}}^{2}\right)} \]

   if 1.9999999999999999e200 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 8.9%

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

4. Final simplification 15.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{{\left(0.5 \cdot \frac{x}{y}\right)}^{1.5}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 35.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 1e+253) (/ 1.0 (cos (* 0.5 (/ x y)))) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / cos((0.5 * (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x / (y * 2.0d0)) <= 1d+253) then
        tmp = 1.0d0 / cos((0.5d0 * (x / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+253) {
		tmp = 1.0 / Math.cos((0.5 * (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 1e+253:
		tmp = 1.0 / math.cos((0.5 * (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+253)
		tmp = Float64(1.0 / cos(Float64(0.5 * Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 1e+253)
		tmp = 1.0 / cos((0.5 * (x / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+253], N[(1.0 / N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.9999999999999994e252

   1. Initial program 40.9%

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

   2. Taylor expanded in x around inf 40.9%

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

   if 9.9999999999999994e252 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 9.4%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 33.7% accurate, 211.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ 1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 1.0)

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return 1.0;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return 1.0
end

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := 1.0

Derivation

1. Initial program 35.8%

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

2. Taylor expanded in x around 0 35.4%

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

3. Final simplification 35.4%

   \[\leadsto 1 \]

Developer target: 33.7% 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 2023278 
(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)))))