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

Percentage Accurate: 43.5% → 56.0%

Time: 19.0s

Alternatives: 5

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: 56.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := x \cdot \frac{0.5}{y}\\ t_1 := \sqrt[3]{t_0}\\ \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{t_1}\right)}^{2} \cdot \left({t_1}^{2} \cdot {t_0}^{0.1111111111111111}\right)\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ 0.5 y))) (t_1 (cbrt t_0)))
   (if (<= (/ x (* y 2.0)) 5e+246)
     (pow
      (/
       1.0
       (cbrt
        (cos
         (*
          (pow (cbrt t_1) 2.0)
          (* (pow t_1 2.0) (pow t_0 0.1111111111111111))))))
      3.0)
     1.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = x * (0.5 / y);
	double t_1 = cbrt(t_0);
	double tmp;
	if ((x / (y * 2.0)) <= 5e+246) {
		tmp = pow((1.0 / cbrt(cos((pow(cbrt(t_1), 2.0) * (pow(t_1, 2.0) * pow(t_0, 0.1111111111111111)))))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = x * (0.5 / y);
	double t_1 = Math.cbrt(t_0);
	double tmp;
	if ((x / (y * 2.0)) <= 5e+246) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos((Math.pow(Math.cbrt(t_1), 2.0) * (Math.pow(t_1, 2.0) * Math.pow(t_0, 0.1111111111111111)))))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(x * Float64(0.5 / y))
	t_1 = cbrt(t_0)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 5e+246)
		tmp = Float64(1.0 / cbrt(cos(Float64((cbrt(t_1) ^ 2.0) * Float64((t_1 ^ 2.0) * (t_0 ^ 0.1111111111111111)))))) ^ 3.0;
	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_] := Block[{t$95$0 = N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 5e+246], N[Power[N[(1.0 / N[Power[N[Cos[N[(N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[Power[t$95$0, 0.1111111111111111], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 4.99999999999999976e246

   1. Initial program 46.9%

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

      1. add-cube-cbrt 46.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

      2. pow3 46.9%

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

   3. Applied egg-rr 57.4%

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

      1. add-cube-cbrt 58.4%

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

      2. pow3 58.4%

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

   5. Applied egg-rr 58.4%

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

      1. cube-mult 58.4%

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

      2. add-cube-cbrt 58.4%

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

      3. associate-*l* 58.6%

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

      4. pow2 58.6%

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

      5. pow2 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. pow1/3 37.9%

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

      2. pow1/3 37.6%

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

      3. pow-pow 38.1%

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

      4. metadata-eval 38.1%

         \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}}\right)}^{2} \cdot \left({\left(x \cdot \frac{0.5}{y}\right)}^{\color{blue}{0.1111111111111111}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

   9. Applied egg-rr 38.1%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}}\right)}^{2} \cdot \left(\color{blue}{{\left(x \cdot \frac{0.5}{y}\right)}^{0.1111111111111111}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

   if 4.99999999999999976e246 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.4%

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

   2. Taylor expanded in x around 0 9.5%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}}\right)}^{2} \cdot \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2} \cdot {\left(x \cdot \frac{0.5}{y}\right)}^{0.1111111111111111}\right)\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 54.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := x \cdot \frac{0.5}{y}\\ t_1 := \sqrt[3]{t_0}\\ t_2 := \sqrt[3]{{t_0}^{0.16666666666666666}}\\ {\left(\frac{1}{\sqrt[3]{\cos \left({\left(t_2 \cdot t_2\right)}^{2} \cdot \left(\sqrt[3]{t_1} \cdot {t_1}^{2}\right)\right)}}\right)}^{3} \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ 0.5 y)))
        (t_1 (cbrt t_0))
        (t_2 (cbrt (pow t_0 0.16666666666666666))))
   (pow
    (/ 1.0 (cbrt (cos (* (pow (* t_2 t_2) 2.0) (* (cbrt t_1) (pow t_1 2.0))))))
    3.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = x * (0.5 / y);
	double t_1 = cbrt(t_0);
	double t_2 = cbrt(pow(t_0, 0.16666666666666666));
	return pow((1.0 / cbrt(cos((pow((t_2 * t_2), 2.0) * (cbrt(t_1) * pow(t_1, 2.0)))))), 3.0);
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = x * (0.5 / y);
	double t_1 = Math.cbrt(t_0);
	double t_2 = Math.cbrt(Math.pow(t_0, 0.16666666666666666));
	return Math.pow((1.0 / Math.cbrt(Math.cos((Math.pow((t_2 * t_2), 2.0) * (Math.cbrt(t_1) * Math.pow(t_1, 2.0)))))), 3.0);
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(x * Float64(0.5 / y))
	t_1 = cbrt(t_0)
	t_2 = cbrt((t_0 ^ 0.16666666666666666))
	return Float64(1.0 / cbrt(cos(Float64((Float64(t_2 * t_2) ^ 2.0) * Float64(cbrt(t_1) * (t_1 ^ 2.0)))))) ^ 3.0
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[t$95$0, 0.16666666666666666], $MachinePrecision], 1/3], $MachinePrecision]}, N[Power[N[(1.0 / N[Power[N[Cos[N[(N[Power[N[(t$95$2 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Initial program 43.8%

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

   1. add-cube-cbrt 43.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

   2. pow3 43.8%

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

3. Applied egg-rr 53.5%

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

   1. add-cube-cbrt 54.5%

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

   2. pow3 54.4%

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

5. Applied egg-rr 54.4%

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

   1. cube-mult 54.5%

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

   2. add-cube-cbrt 54.3%

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

   3. associate-*l* 54.6%

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

   4. pow2 54.6%

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

   5. pow2 54.6%

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

7. Applied egg-rr 54.6%

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

   1. pow1/3 34.8%

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

   2. add-sqr-sqrt 34.8%

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

   3. unpow-prod-down 35.2%

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

   4. pow1/3 35.2%

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

   5. sqrt-pow1 35.1%

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

   6. metadata-eval 35.1%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{\color{blue}{0.16666666666666666}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\sqrt[3]{x \cdot \frac{0.5}{y}}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

   7. pow1/3 35.0%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\color{blue}{{\left(x \cdot \frac{0.5}{y}\right)}^{0.3333333333333333}}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

   8. sqrt-pow1 35.0%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left({\left(x \cdot \frac{0.5}{y}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

   9. metadata-eval 35.0%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333} \cdot {\left({\left(x \cdot \frac{0.5}{y}\right)}^{\color{blue}{0.16666666666666666}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

9. Applied egg-rr 35.0%

   \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\color{blue}{\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333} \cdot {\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]
10. Step-by-step derivation

    1. unpow1/3 35.0%

       \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left(\color{blue}{\sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}}} \cdot {\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}\right)}^{0.3333333333333333}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

    2. unpow1/3 35.5%

       \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}} \cdot \color{blue}{\sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}}}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

11. Simplified 35.5%

    \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\color{blue}{\left(\sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}}\right)}}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

12. Final simplification 35.5%

    \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{{\left(x \cdot \frac{0.5}{y}\right)}^{0.16666666666666666}}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{x \cdot \frac{0.5}{y}}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)\right)}}\right)}^{3} \]

Alternative 3: 56.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{x \cdot \frac{0.5}{y}}}\\ t_1 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_1}{\sin t_1} \leq 3:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(t_0 \cdot t_0\right)}^{3}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt (* x (/ 0.5 y))))) (t_1 (/ x (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 3.0)
     (pow (/ 1.0 (cbrt (cos (pow (* t_0 t_0) 3.0)))) 3.0)
     1.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = cbrt(sqrt((x * (0.5 / y))));
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 3.0) {
		tmp = pow((1.0 / cbrt(cos(pow((t_0 * t_0), 3.0)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = Math.cbrt(Math.sqrt((x * (0.5 / y))));
	double t_1 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 3.0) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(Math.pow((t_0 * t_0), 3.0)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = cbrt(sqrt(Float64(x * Float64(0.5 / y))))
	t_1 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 3.0)
		tmp = Float64(1.0 / cbrt(cos((Float64(t_0 * t_0) ^ 3.0)))) ^ 3.0;
	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_] := Block[{t$95$0 = N[Power[N[Sqrt[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 3.0], N[Power[N[(1.0 / N[Power[N[Cos[N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $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)))) < 3

   1. Initial program 62.0%

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

      1. add-cube-cbrt 62.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

      2. pow3 62.0%

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

   3. Applied egg-rr 61.9%

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

      1. add-cube-cbrt 63.0%

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

      2. pow3 63.0%

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

   5. Applied egg-rr 63.0%

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

      1. pow1/3 35.7%

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

      2. add-sqr-sqrt 35.9%

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

      3. unpow-prod-down 35.8%

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

   7. Applied egg-rr 35.8%

      \[\leadsto {\left(\frac{1}{\sqrt[3]{\cos \left({\color{blue}{\left({\left(\sqrt{x \cdot \frac{0.5}{y}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{x \cdot \frac{0.5}{y}}\right)}^{0.3333333333333333}\right)}}^{3}\right)}}\right)}^{3} \]
   8. Step-by-step derivation

      1. unpow1/3 35.3%

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

      2. unpow1/3 36.2%

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

   9. Simplified 36.2%

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

   if 3 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.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 39.6%

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

4. Final simplification 37.2%

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

Alternative 4: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 12.6:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 12.6)
     (pow (/ 1.0 (cbrt (cos (pow (cbrt (* x (/ 0.5 y))) 3.0)))) 3.0)
     1.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 12.6) {
		tmp = pow((1.0 / cbrt(cos(pow(cbrt((x * (0.5 / y))), 3.0)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 12.6) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(Math.pow(Math.cbrt((x * (0.5 / y))), 3.0)))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 12.6)
		tmp = Float64(1.0 / cbrt(cos((cbrt(Float64(x * Float64(0.5 / y))) ^ 3.0)))) ^ 3.0;
	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_] := 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], 12.6], N[Power[N[(1.0 / N[Power[N[Cos[N[Power[N[Power[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $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)))) < 12.5999999999999996

   1. Initial program 58.9%

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

      1. add-cube-cbrt 58.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

      2. pow3 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. add-cube-cbrt 60.2%

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

      2. pow3 60.1%

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

   5. Applied egg-rr 60.1%

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

   if 12.5999999999999996 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 0.3%

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

   2. Taylor expanded in x around 0 44.2%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 12.6:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 54.6% 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 43.8%

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

2. Taylor expanded in x around 0 54.4%

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

3. Final simplification 54.4%

   \[\leadsto 1 \]

Developer target: 54.6% 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 2023214 
(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)))))