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

Percentage Accurate: 44.7% → 56.9%

Time: 15.6s

Alternatives: 8

Speedup: 211.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t_0}{\sin t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 44.7% 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: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 7:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 7.0)
     (/ 1.0 (cos (/ (/ (* x 0.5) (pow (cbrt y) 2.0)) (cbrt y))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 7.0) {
		tmp = 1.0 / cos((((x * 0.5) / pow(cbrt(y), 2.0)) / cbrt(y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 7.0) {
		tmp = 1.0 / Math.cos((((x * 0.5) / Math.pow(Math.cbrt(y), 2.0)) / Math.cbrt(y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 7.0)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(x * 0.5) / (cbrt(y) ^ 2.0)) / cbrt(y))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 7.0], N[(1.0 / N[Cos[N[(N[(N[(x * 0.5), $MachinePrecision] / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 7

   1. Initial program 54.5%

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

   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. add-exp-log 25.6%

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

   4. Applied egg-rr 25.6%

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

      1. add-exp-log 54.5%

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

      2. associate-*r/ 54.5%

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

      3. add-cube-cbrt 56.2%

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

      4. associate-/r* 56.9%

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

      5. pow2 56.9%

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

   6. Applied egg-rr 56.9%

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

   if 7 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 1.0%

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

   2. Taylor expanded in x around 0 56.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 7:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 7:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 7.0)
     (/ 1.0 (cos (* (/ 0.5 (pow (cbrt y) 2.0)) (/ x (cbrt y)))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 7.0) {
		tmp = 1.0 / cos(((0.5 / pow(cbrt(y), 2.0)) * (x / cbrt(y))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 7.0) {
		tmp = 1.0 / Math.cos(((0.5 / Math.pow(Math.cbrt(y), 2.0)) * (x / Math.cbrt(y))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 7.0)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / (cbrt(y) ^ 2.0)) * Float64(x / cbrt(y)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 7.0], N[(1.0 / N[Cos[N[(N[(0.5 / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 7

   1. Initial program 54.5%

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

   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. add-exp-log 25.6%

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

   4. Applied egg-rr 25.6%

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

      1. add-exp-log 54.5%

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

      2. associate-*r/ 54.5%

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

      3. add-cube-cbrt 56.2%

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

      4. times-frac 56.4%

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

      5. pow2 56.4%

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

   6. Applied egg-rr 56.4%

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

   if 7 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 1.0%

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

   2. Taylor expanded in x around 0 56.1%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 7:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot 0.5}{\frac{\sqrt[3]{y}}{{\left(\sqrt[3]{y}\right)}^{-2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 20.0)
     (/ 1.0 (cos (/ (* x 0.5) (/ (cbrt y) (pow (cbrt y) -2.0)))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 20.0) {
		tmp = 1.0 / cos(((x * 0.5) / (cbrt(y) / pow(cbrt(y), -2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 20.0) {
		tmp = 1.0 / Math.cos(((x * 0.5) / (Math.cbrt(y) / Math.pow(Math.cbrt(y), -2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 20.0)
		tmp = Float64(1.0 / cos(Float64(Float64(x * 0.5) / Float64(cbrt(y) / (cbrt(y) ^ -2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 20.0], N[(1.0 / N[Cos[N[(N[(x * 0.5), $MachinePrecision] / N[(N[Power[y, 1/3], $MachinePrecision] / N[Power[N[Power[y, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 20

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf 53.3%

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

      1. add-exp-log 25.2%

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

   4. Applied egg-rr 25.2%

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

      1. add-exp-log 53.3%

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

      2. associate-*r/ 53.3%

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

      3. associate-*l/ 53.3%

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

   6. Applied egg-rr 53.3%

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

      1. associate-*l/ 53.3%

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

      2. add-cube-cbrt 55.1%

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

      3. unpow2 55.1%

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

      4. associate-/r* 55.5%

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

      5. div-inv 55.3%

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

      6. associate-/l* 55.2%

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

      7. pow-flip 55.3%

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

      8. metadata-eval 55.3%

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

   8. Applied egg-rr 55.3%

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

   if 20 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 0.4%

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

   2. Taylor expanded in x around 0 59.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot 0.5}{\frac{\sqrt[3]{y}}{{\left(\sqrt[3]{y}\right)}^{-2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 56.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 7.4:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt{0.5}\right)}^{2} \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 7.4)
     (/ 1.0 (cos (* (pow (sqrt 0.5) 2.0) (/ x y))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 7.4) {
		tmp = 1.0 / cos((pow(sqrt(0.5), 2.0) * (x / y)));
	} 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) :: tmp
    t_0 = x / (y * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 7.4d0) then
        tmp = 1.0d0 / cos(((sqrt(0.5d0) ** 2.0d0) * (x / y)))
    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 tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 7.4) {
		tmp = 1.0 / Math.cos((Math.pow(Math.sqrt(0.5), 2.0) * (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 7.4:
		tmp = 1.0 / math.cos((math.pow(math.sqrt(0.5), 2.0) * (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 7.4)
		tmp = Float64(1.0 / cos(Float64((sqrt(0.5) ^ 2.0) * Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 7.4)
		tmp = 1.0 / cos(((sqrt(0.5) ^ 2.0) * (x / y)));
	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]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 7.4], N[(1.0 / N[Cos[N[(N[Power[N[Sqrt[0.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 7.4000000000000004

   1. Initial program 54.2%

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

   2. Taylor expanded in x around inf 54.2%

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

      1. add-exp-log 25.5%

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

   4. Applied egg-rr 25.5%

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

      1. add-exp-log 54.2%

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

      2. associate-*r/ 54.2%

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

      3. add-cube-cbrt 55.9%

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

      4. associate-/r* 56.6%

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

      5. pow2 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-/r* 55.9%

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

      2. unpow2 55.9%

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

      3. add-cube-cbrt 54.2%

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

      4. associate-*r/ 54.2%

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

      5. add-sqr-sqrt 25.2%

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

      6. unpow2 25.2%

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

      7. sqrt-prod 24.9%

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

      8. unpow-prod-down 24.5%

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

      9. pow2 24.5%

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

      10. add-sqr-sqrt 55.9%

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

   8. Applied egg-rr 55.9%

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

   if 7.4000000000000004 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 1.0%

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

   2. Taylor expanded in x around 0 56.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 7.4:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt{0.5}\right)}^{2} \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 34.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 5e+55) (/ 1.0 (cos (exp (log (* 0.5 (/ x y)))))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 5e+55) {
		tmp = 1.0 / cos(exp(log((0.5 * (x / y)))));
	} 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) :: tmp
    if ((x / (y * 2.0d0)) <= 5d+55) then
        tmp = 1.0d0 / cos(exp(log((0.5d0 * (x / y)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 5e+55) {
		tmp = 1.0 / Math.cos(Math.exp(Math.log((0.5 * (x / y)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 5e+55:
		tmp = 1.0 / math.cos(math.exp(math.log((0.5 * (x / y)))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 5e+55)
		tmp = Float64(1.0 / cos(exp(log(Float64(0.5 * Float64(x / y))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 5e+55)
		tmp = 1.0 / cos(exp(log((0.5 * (x / y)))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 5e+55], N[(1.0 / N[Cos[N[Exp[N[Log[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5.00000000000000046e55

   1. Initial program 44.8%

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

   2. Taylor expanded in x around inf 62.7%

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

      1. add-exp-log 38.0%

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

   4. Applied egg-rr 38.0%

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

   if 5.00000000000000046e55 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.5%

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

   2. Taylor expanded in x around 0 14.0%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 55.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+90}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{t_0}^{3} + -1}{t_0 \cdot t_0 + \left(1 + t_0\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (/ x y)))))
   (if (<= (/ x (* y 2.0)) 1e+90)
     (/ 1.0 (cos (/ (+ (pow t_0 3.0) -1.0) (+ (* t_0 t_0) (+ 1.0 t_0)))))
     1.0)))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x / y));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+90) {
		tmp = 1.0 / cos(((pow(t_0, 3.0) + -1.0) / ((t_0 * t_0) + (1.0 + 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) :: tmp
    t_0 = 1.0d0 + (0.5d0 * (x / y))
    if ((x / (y * 2.0d0)) <= 1d+90) then
        tmp = 1.0d0 / cos((((t_0 ** 3.0d0) + (-1.0d0)) / ((t_0 * t_0) + (1.0d0 + t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x / y));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+90) {
		tmp = 1.0 / Math.cos(((Math.pow(t_0, 3.0) + -1.0) / ((t_0 * t_0) + (1.0 + t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.5 * (x / y))
	tmp = 0
	if (x / (y * 2.0)) <= 1e+90:
		tmp = 1.0 / math.cos(((math.pow(t_0, 3.0) + -1.0) / ((t_0 * t_0) + (1.0 + t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x / y)))
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+90)
		tmp = Float64(1.0 / cos(Float64(Float64((t_0 ^ 3.0) + -1.0) / Float64(Float64(t_0 * t_0) + Float64(1.0 + t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.5 * (x / y));
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 1e+90)
		tmp = 1.0 / cos((((t_0 ^ 3.0) + -1.0) / ((t_0 * t_0) + (1.0 + t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+90], N[(1.0 / N[Cos[N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.99999999999999966e89

   1. Initial program 44.2%

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

   2. Taylor expanded in x around inf 61.8%

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

      1. add-exp-log 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. add-exp-log 61.8%

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

      2. expm1-log1p-u 58.8%

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

      3. expm1-udef 58.8%

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

      4. flip3-- 58.8%

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{{\left(e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)}} \]

      5. log1p-udef 58.8%

         \[\leadsto \frac{1}{\cos \left(\frac{{\left(e^{\color{blue}{\log \left(1 + 0.5 \cdot \frac{x}{y}\right)}}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      6. add-exp-log 58.7%

         \[\leadsto \frac{1}{\cos \left(\frac{{\color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)}}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      7. +-commutative 58.7%

         \[\leadsto \frac{1}{\cos \left(\frac{{\color{blue}{\left(0.5 \cdot \frac{x}{y} + 1\right)}}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      8. metadata-eval 58.7%

         \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      9. log1p-udef 58.7%

         \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot \frac{x}{y}\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      10. add-exp-log 58.6%

          \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{\color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      11. +-commutative 58.6%

          \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{\color{blue}{\left(0.5 \cdot \frac{x}{y} + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      12. log1p-udef 58.6%

          \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{\left(0.5 \cdot \frac{x}{y} + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot \frac{x}{y}\right)}} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      13. add-exp-log 58.6%

          \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{\left(0.5 \cdot \frac{x}{y} + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

      14. +-commutative 58.6%

          \[\leadsto \frac{1}{\cos \left(\frac{{\left(0.5 \cdot \frac{x}{y} + 1\right)}^{3} - 1}{\left(0.5 \cdot \frac{x}{y} + 1\right) \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} + 1\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(0.5 \cdot \frac{x}{y}\right)} \cdot 1\right)}\right)} \]

   6. Applied egg-rr 61.4%

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

   if 9.99999999999999966e89 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.0%

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

   2. Taylor expanded in x around 0 14.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+90}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(1 + 0.5 \cdot \frac{x}{y}\right)}^{3} + -1}{\left(1 + 0.5 \cdot \frac{x}{y}\right) \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right) + \left(1 + \left(1 + 0.5 \cdot \frac{x}{y}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 2e+16) (/ 1.0 (cos (* 0.5 (/ x y)))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+16) {
		tmp = 1.0 / cos((0.5 * (x / y)));
	} 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) :: tmp
    if ((x / (y * 2.0d0)) <= 2d+16) then
        tmp = 1.0d0 / cos((0.5d0 * (x / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+16) {
		tmp = 1.0 / Math.cos((0.5 * (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 2e+16:
		tmp = 1.0 / math.cos((0.5 * (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 2e+16)
		tmp = Float64(1.0 / cos(Float64(0.5 * Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 2e+16)
		tmp = 1.0 / cos((0.5 * (x / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 2e+16], 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)) < 2e16

   1. Initial program 46.7%

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

   2. Taylor expanded in x around inf 65.5%

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

   if 2e16 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 14.0%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 55.5% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 38.2%

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

2. Taylor expanded in x around 0 53.2%

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

3. Final simplification 53.2%

   \[\leadsto 1 \]

Developer target: 54.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \log \left(e^{\cos t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2023263 
(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)))))