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

Percentage Accurate: 44.9% → 57.6%

Time: 16.3s

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: 44.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := {\left(\sqrt{\frac{x \cdot 0.5}{y}}\right)}^{3}\\ \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+85}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\sqrt[3]{t_0 \cdot t_0}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 (pow (sqrt (/ (* x 0.5) y)) 3.0)))
   (if (<= (/ x (* y 2.0)) 1e+85)
     (pow (/ 1.0 (cbrt (cos (cbrt (* t_0 t_0))))) 3.0)
     1.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = pow(sqrt(((x * 0.5) / y)), 3.0);
	double tmp;
	if ((x / (y * 2.0)) <= 1e+85) {
		tmp = pow((1.0 / cbrt(cos(cbrt((t_0 * t_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.pow(Math.sqrt(((x * 0.5) / y)), 3.0);
	double tmp;
	if ((x / (y * 2.0)) <= 1e+85) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(Math.cbrt((t_0 * t_0))))), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = sqrt(Float64(Float64(x * 0.5) / y)) ^ 3.0
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+85)
		tmp = Float64(1.0 / cbrt(cos(cbrt(Float64(t_0 * t_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[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+85], N[Power[N[(1.0 / N[Power[N[Cos[N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 1/3], $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)) < 1e85

   1. Initial program 51.8%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

   3. Applied egg-rr 59.7%

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

      1. clear-num 59.7%

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

      2. div-inv 59.7%

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

      3. metadata-eval 59.7%

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

      4. div-inv 60.1%

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

      5. *-un-lft-identity 60.1%

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

      6. *-commutative 60.1%

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

      7. times-frac 60.1%

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

      8. metadata-eval 60.1%

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

      9. add-sqr-sqrt 33.6%

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

      10. pow2 33.6%

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

   5. Applied egg-rr 33.6%

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

      1. clear-num 33.7%

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

      2. div-inv 33.7%

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

      3. associate-/r/ 33.7%

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

      4. *-commutative 33.7%

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

      5. associate-*r/ 33.6%

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

      6. sqrt-div 11.9%

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

   7. Applied egg-rr 11.9%

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

      1. unpow2 11.9%

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

      2. add-cbrt-cube 11.9%

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

      3. add-cbrt-cube 11.7%

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

      4. cbrt-unprod 12.2%

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

      5. pow3 12.1%

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

      6. sqrt-undiv 11.8%

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

      7. pow3 12.1%

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

      8. sqrt-undiv 34.2%

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

   9. Applied egg-rr 34.2%

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

   if 1e85 < (/.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 13.9%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+85}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\sqrt[3]{{\left(\sqrt{\frac{x \cdot 0.5}{y}}\right)}^{3} \cdot {\left(\sqrt{\frac{x \cdot 0.5}{y}}\right)}^{3}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 57.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\frac{\sqrt{x \cdot 0.5}}{\sqrt{y}}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 1e+255)
   (pow (/ 1.0 (cbrt (cos (pow (/ (sqrt (* x 0.5)) (sqrt y)) 2.0)))) 3.0)
   (pow
    (/ 1.0 (expm1 (+ (log 2.0) (* (* (/ x y) (/ x y)) -0.020833333333333332))))
    3.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+255) {
		tmp = pow((1.0 / cbrt(cos(pow((sqrt((x * 0.5)) / sqrt(y)), 2.0)))), 3.0);
	} else {
		tmp = pow((1.0 / expm1((log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	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+255) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(Math.pow((Math.sqrt((x * 0.5)) / Math.sqrt(y)), 2.0)))), 3.0);
	} else {
		tmp = Math.pow((1.0 / Math.expm1((Math.log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+255)
		tmp = Float64(1.0 / cbrt(cos((Float64(sqrt(Float64(x * 0.5)) / sqrt(y)) ^ 2.0)))) ^ 3.0;
	else
		tmp = Float64(1.0 / expm1(Float64(log(2.0) + Float64(Float64(Float64(x / y) * Float64(x / y)) * -0.020833333333333332)))) ^ 3.0;
	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+255], N[Power[N[(1.0 / N[Power[N[Cos[N[Power[N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(1.0 / N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.99999999999999988e254

   1. Initial program 47.6%

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

      1. add-cube-cbrt 47.6%

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

      2. pow3 47.6%

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

   3. Applied egg-rr 54.8%

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

      1. clear-num 54.8%

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

      2. div-inv 54.8%

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

      3. metadata-eval 54.8%

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

      4. div-inv 55.1%

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

      5. *-un-lft-identity 55.1%

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

      6. *-commutative 55.1%

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

      7. times-frac 55.1%

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

      8. metadata-eval 55.1%

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

      9. add-sqr-sqrt 31.4%

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

      10. pow2 31.4%

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

   5. Applied egg-rr 31.4%

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

      1. clear-num 31.5%

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

      2. div-inv 31.5%

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

      3. associate-/r/ 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-*r/ 31.4%

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

      6. sqrt-div 11.2%

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

   7. Applied egg-rr 11.2%

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

   if 9.99999999999999988e254 < (/.f64 x (*.f64 y 2))

   1. Initial program 2.5%

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

      1. add-cube-cbrt 2.5%

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

      2. pow3 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. clear-num 2.5%

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

      2. div-inv 2.5%

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

      3. metadata-eval 2.5%

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

      4. div-inv 2.5%

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

      5. *-un-lft-identity 2.5%

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

      6. *-commutative 2.5%

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

      7. times-frac 2.5%

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

      8. metadata-eval 2.5%

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

      9. expm1-log1p-u 2.5%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in x around 0 8.2%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 8.2%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.020833333333333332}\right)}\right)}^{3} \]

      2. unpow2 8.2%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      3. unpow2 8.2%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      4. times-frac 8.2%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot -0.020833333333333332\right)}\right)}^{3} \]

   8. Simplified 8.2%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332}\right)}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\frac{\sqrt{x \cdot 0.5}}{\sqrt{y}}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 57.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\left(x \cdot 0.5\right) \cdot {\left({y}^{-0.5}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 5e+255)
   (pow (/ 1.0 (cbrt (cos (* (* x 0.5) (pow (pow y -0.5) 2.0))))) 3.0)
   (pow
    (/ 1.0 (expm1 (+ (log 2.0) (* (* (/ x y) (/ x y)) -0.020833333333333332))))
    3.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 5e+255) {
		tmp = pow((1.0 / cbrt(cos(((x * 0.5) * pow(pow(y, -0.5), 2.0))))), 3.0);
	} else {
		tmp = pow((1.0 / expm1((log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	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)) <= 5e+255) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(((x * 0.5) * Math.pow(Math.pow(y, -0.5), 2.0))))), 3.0);
	} else {
		tmp = Math.pow((1.0 / Math.expm1((Math.log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 5e+255)
		tmp = Float64(1.0 / cbrt(cos(Float64(Float64(x * 0.5) * ((y ^ -0.5) ^ 2.0))))) ^ 3.0;
	else
		tmp = Float64(1.0 / expm1(Float64(log(2.0) + Float64(Float64(Float64(x / y) * Float64(x / y)) * -0.020833333333333332)))) ^ 3.0;
	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], 5e+255], N[Power[N[(1.0 / N[Power[N[Cos[N[(N[(x * 0.5), $MachinePrecision] * N[Power[N[Power[y, -0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(1.0 / N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5.0000000000000002e255

   1. Initial program 47.5%

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

      1. add-cube-cbrt 47.5%

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

      2. pow3 47.5%

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

   3. Applied egg-rr 54.7%

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

      1. clear-num 54.7%

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

      2. div-inv 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 55.0%

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

      5. *-un-lft-identity 55.0%

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

      6. *-commutative 55.0%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

      9. add-sqr-sqrt 31.3%

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

      10. pow2 31.3%

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

   5. Applied egg-rr 31.3%

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

      1. clear-num 31.5%

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

      2. div-inv 31.5%

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

      3. associate-/r/ 31.3%

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

      4. *-commutative 31.3%

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

      5. associate-*r/ 31.3%

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

      6. sqrt-div 11.2%

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

   7. Applied egg-rr 11.2%

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

      1. div-inv 11.2%

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

      2. unpow-prod-down 11.2%

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

      3. pow2 11.2%

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

      4. add-sqr-sqrt 25.4%

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

      5. pow1/2 25.4%

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

      6. pow-flip 25.3%

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

      7. metadata-eval 25.3%

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

   9. Applied egg-rr 25.3%

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

   if 5.0000000000000002e255 < (/.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. Step-by-step derivation

      1. add-cube-cbrt 1.0%

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

      2. pow3 1.0%

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

   3. Applied egg-rr 1.0%

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

      1. clear-num 1.0%

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

      2. div-inv 1.0%

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

      3. metadata-eval 1.0%

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

      4. div-inv 1.0%

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

      5. *-un-lft-identity 1.0%

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

      6. *-commutative 1.0%

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

      7. times-frac 1.0%

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

      8. metadata-eval 1.0%

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

      9. expm1-log1p-u 1.0%

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

   5. Applied egg-rr 1.0%

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

   6. Taylor expanded in x around 0 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.020833333333333332}\right)}\right)}^{3} \]

      2. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      3. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      4. times-frac 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot -0.020833333333333332\right)}\right)}^{3} \]

   8. Simplified 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332}\right)}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\left(x \cdot 0.5\right) \cdot {\left({y}^{-0.5}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternative 4: 57.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 5e+255)
   (pow (/ 1.0 (cbrt (cos (pow (sqrt (* 0.5 (/ x y))) 2.0)))) 3.0)
   (pow
    (/ 1.0 (expm1 (+ (log 2.0) (* (* (/ x y) (/ x y)) -0.020833333333333332))))
    3.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 5e+255) {
		tmp = pow((1.0 / cbrt(cos(pow(sqrt((0.5 * (x / y))), 2.0)))), 3.0);
	} else {
		tmp = pow((1.0 / expm1((log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	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)) <= 5e+255) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos(Math.pow(Math.sqrt((0.5 * (x / y))), 2.0)))), 3.0);
	} else {
		tmp = Math.pow((1.0 / Math.expm1((Math.log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 5e+255)
		tmp = Float64(1.0 / cbrt(cos((sqrt(Float64(0.5 * Float64(x / y))) ^ 2.0)))) ^ 3.0;
	else
		tmp = Float64(1.0 / expm1(Float64(log(2.0) + Float64(Float64(Float64(x / y) * Float64(x / y)) * -0.020833333333333332)))) ^ 3.0;
	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], 5e+255], N[Power[N[(1.0 / N[Power[N[Cos[N[Power[N[Sqrt[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(1.0 / N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5.0000000000000002e255

   1. Initial program 47.5%

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

      1. add-cube-cbrt 47.5%

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

      2. pow3 47.5%

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

   3. Applied egg-rr 54.7%

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

      1. clear-num 54.7%

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

      2. div-inv 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 55.0%

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

      5. *-un-lft-identity 55.0%

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

      6. *-commutative 55.0%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

      9. add-sqr-sqrt 31.3%

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

      10. pow2 31.3%

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

   5. Applied egg-rr 31.3%

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

   if 5.0000000000000002e255 < (/.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. Step-by-step derivation

      1. add-cube-cbrt 1.0%

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

      2. pow3 1.0%

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

   3. Applied egg-rr 1.0%

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

      1. clear-num 1.0%

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

      2. div-inv 1.0%

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

      3. metadata-eval 1.0%

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

      4. div-inv 1.0%

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

      5. *-un-lft-identity 1.0%

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

      6. *-commutative 1.0%

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

      7. times-frac 1.0%

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

      8. metadata-eval 1.0%

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

      9. expm1-log1p-u 1.0%

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

   5. Applied egg-rr 1.0%

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

   6. Taylor expanded in x around 0 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.020833333333333332}\right)}\right)}^{3} \]

      2. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      3. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      4. times-frac 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot -0.020833333333333332\right)}\right)}^{3} \]

   8. Simplified 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332}\right)}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left({\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)}^{2}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternative 5: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 5e+255)
   (pow (/ 1.0 (cbrt (cos (/ 0.5 (/ y x))))) 3.0)
   (pow
    (/ 1.0 (expm1 (+ (log 2.0) (* (* (/ x y) (/ x y)) -0.020833333333333332))))
    3.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 5e+255) {
		tmp = pow((1.0 / cbrt(cos((0.5 / (y / x))))), 3.0);
	} else {
		tmp = pow((1.0 / expm1((log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	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)) <= 5e+255) {
		tmp = Math.pow((1.0 / Math.cbrt(Math.cos((0.5 / (y / x))))), 3.0);
	} else {
		tmp = Math.pow((1.0 / Math.expm1((Math.log(2.0) + (((x / y) * (x / y)) * -0.020833333333333332)))), 3.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 5e+255)
		tmp = Float64(1.0 / cbrt(cos(Float64(0.5 / Float64(y / x))))) ^ 3.0;
	else
		tmp = Float64(1.0 / expm1(Float64(log(2.0) + Float64(Float64(Float64(x / y) * Float64(x / y)) * -0.020833333333333332)))) ^ 3.0;
	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], 5e+255], N[Power[N[(1.0 / N[Power[N[Cos[N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(1.0 / N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5.0000000000000002e255

   1. Initial program 47.5%

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

      1. add-cube-cbrt 47.5%

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

      2. pow3 47.5%

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

   3. Applied egg-rr 54.7%

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

      1. clear-num 54.7%

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

      2. div-inv 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 55.0%

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

      5. *-un-lft-identity 55.0%

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

      6. *-commutative 55.0%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

      9. clear-num 55.2%

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

      10. un-div-inv 55.2%

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

   5. Applied egg-rr 55.2%

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

   if 5.0000000000000002e255 < (/.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. Step-by-step derivation

      1. add-cube-cbrt 1.0%

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

      2. pow3 1.0%

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

   3. Applied egg-rr 1.0%

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

      1. clear-num 1.0%

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

      2. div-inv 1.0%

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

      3. metadata-eval 1.0%

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

      4. div-inv 1.0%

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

      5. *-un-lft-identity 1.0%

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

      6. *-commutative 1.0%

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

      7. times-frac 1.0%

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

      8. metadata-eval 1.0%

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

      9. expm1-log1p-u 1.0%

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

   5. Applied egg-rr 1.0%

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

   6. Taylor expanded in x around 0 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.020833333333333332 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.020833333333333332}\right)}\right)}^{3} \]

      2. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      3. unpow2 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.020833333333333332\right)}\right)}^{3} \]

      4. times-frac 8.6%

         \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot -0.020833333333333332\right)}\right)}^{3} \]

   8. Simplified 8.6%

      \[\leadsto {\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332}\right)}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{expm1}\left(\log 2 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot -0.020833333333333332\right)}\right)}^{3}\\ \end{array} \]

Alternative 6: 57.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+36}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+36) (/ 1.0 (log (exp (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+36) {
		tmp = 1.0 / log(exp(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+36) then
        tmp = 1.0d0 / log(exp(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+36) {
		tmp = 1.0 / Math.log(Math.exp(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+36:
		tmp = 1.0 / math.log(math.exp(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+36)
		tmp = Float64(1.0 / log(exp(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+36)
		tmp = 1.0 / log(exp(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+36], N[(1.0 / N[Log[N[Exp[N[Cos[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)) < 1.00000000000000004e36

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 65.2%

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

      1. add-log-exp 65.2%

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

   4. Applied egg-rr 65.2%

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

   if 1.00000000000000004e36 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.4%

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

   2. Taylor expanded in x around 0 12.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+36}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+36}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+36) (/ 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+36) {
		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+36) 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+36) {
		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+36:
		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+36)
		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+36)
		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+36], 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)) < 1.00000000000000004e36

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 65.2%

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

   if 1.00000000000000004e36 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.4%

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

   2. Taylor expanded in x around 0 12.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+36}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 56.1% 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 44.6%

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

2. Taylor expanded in x around 0 51.7%

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

3. Final simplification 51.7%

   \[\leadsto 1 \]

Developer target: 56.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(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)))))