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

Percentage Accurate: 44.4% → 57.0%

Time: 15.6s

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: 44.4% 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: 57.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{\frac{0.5}{y\_m}} \cdot \sqrt[3]{x\_m}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+264)
   (/ 1.0 (cos (pow (* (cbrt (/ 0.5 y_m)) (cbrt x_m)) 3.0)))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+264) {
		tmp = 1.0 / cos(pow((cbrt((0.5 / y_m)) * cbrt(x_m)), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+264) {
		tmp = 1.0 / Math.cos(Math.pow((Math.cbrt((0.5 / y_m)) * Math.cbrt(x_m)), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+264)
		tmp = Float64(1.0 / cos((Float64(cbrt(Float64(0.5 / y_m)) * cbrt(x_m)) ^ 3.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+264], N[(1.0 / N[Cos[N[Power[N[(N[Power[N[(0.5 / y$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[x$95$m, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000009e264

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf 55.4%

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

      1. metadata-eval 55.4%

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

      2. add-sqr-sqrt 34.5%

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

      3. fabs-sqr 34.5%

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

      4. add-sqr-sqrt 55.4%

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

      5. fabs-mul 55.4%

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

      6. add-cube-cbrt 55.1%

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

      7. pow3 55.3%

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

      8. fabs-mul 55.3%

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

      9. metadata-eval 55.3%

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

      10. cbrt-prod 54.8%

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

      11. add-sqr-sqrt 34.1%

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

      12. fabs-sqr 34.1%

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

      13. add-sqr-sqrt 54.8%

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

      14. cbrt-prod 55.3%

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

      15. clear-num 55.7%

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

      16. un-div-inv 55.7%

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

   5. Applied egg-rr 55.7%

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

      1. associate-/r/ 55.6%

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

      2. cbrt-prod 56.2%

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

   7. Applied egg-rr 56.2%

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

   if 2.00000000000000009e264 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.3%

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

      1. remove-double-neg 2.3%

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

      2. distribute-frac-neg 2.3%

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

      3. tan-neg 2.3%

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

      4. distribute-frac-neg2 2.3%

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

      5. distribute-lft-neg-out 2.3%

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

      6. distribute-frac-neg2 2.3%

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

      7. distribute-lft-neg-out 2.3%

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

      8. distribute-frac-neg2 2.3%

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

      9. distribute-frac-neg 2.3%

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

      10. neg-mul-1 2.3%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. *-commutative 2.3%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. associate-/l* 2.2%

          \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{-1}{y \cdot 2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. *-commutative 2.2%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-1}{\color{blue}{2 \cdot y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. associate-/r* 2.2%

          \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{\frac{-1}{2}}{y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. metadata-eval 2.2%

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

      16. sin-neg 2.2%

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

      17. distribute-frac-neg 2.2%

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

   3. Simplified 2.2%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 11.6%

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

Alternative 2: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 1.12:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x\_m \cdot \frac{0.5}{y\_m}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 1.12)
     (/ 1.0 (cos (pow (cbrt (* x_m (/ 0.5 y_m))) 3.0)))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.12) {
		tmp = 1.0 / cos(pow(cbrt((x_m * (0.5 / y_m))), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.12) {
		tmp = 1.0 / Math.cos(Math.pow(Math.cbrt((x_m * (0.5 / y_m))), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.12)
		tmp = Float64(1.0 / cos((cbrt(Float64(x_m * Float64(0.5 / y_m))) ^ 3.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.12], N[(1.0 / N[Cos[N[Power[N[Power[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 1.1200000000000001

   1. Initial program 66.8%

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

   3. Taylor expanded in x around inf 66.8%

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

      1. metadata-eval 66.8%

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

      2. add-sqr-sqrt 37.0%

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

      3. fabs-sqr 37.0%

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

      4. add-sqr-sqrt 66.8%

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

      5. fabs-mul 66.8%

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

      6. add-cube-cbrt 66.6%

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

      7. pow3 67.1%

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

      8. fabs-mul 67.1%

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

      9. metadata-eval 67.1%

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

      10. cbrt-prod 67.0%

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

      11. add-sqr-sqrt 37.2%

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

      12. fabs-sqr 37.2%

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

      13. add-sqr-sqrt 67.0%

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

      14. cbrt-prod 67.1%

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

      15. clear-num 67.6%

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

      16. un-div-inv 67.6%

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

   5. Applied egg-rr 67.6%

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

      1. associate-/r/ 67.6%

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

   7. Applied egg-rr 67.6%

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

   if 1.1200000000000001 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 4.5%

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

      1. remove-double-neg 4.5%

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

      2. distribute-frac-neg 4.5%

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

      3. tan-neg 4.5%

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

      4. distribute-frac-neg2 4.5%

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

      5. distribute-lft-neg-out 4.5%

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

      6. distribute-frac-neg2 4.5%

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

      7. distribute-lft-neg-out 4.5%

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

      8. distribute-frac-neg2 4.5%

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

      9. distribute-frac-neg 4.5%

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

      10. neg-mul-1 4.5%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. *-commutative 4.5%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. associate-/l* 4.5%

          \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{-1}{y \cdot 2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. *-commutative 4.5%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-1}{\color{blue}{2 \cdot y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. associate-/r* 4.5%

          \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{\frac{-1}{2}}{y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. metadata-eval 4.5%

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

      16. sin-neg 4.5%

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

      17. distribute-frac-neg 4.5%

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

   3. Simplified 4.6%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 28.3%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.12:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 1.02:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 1.02)
     (/ 1.0 (cos (/ 0.5 (/ y_m x_m))))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.02) {
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m / (y_m * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 1.02d0) then
        tmp = 1.0d0 / cos((0.5d0 / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.02) {
		tmp = 1.0 / Math.cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 1.02:
		tmp = 1.0 / math.cos((0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.02)
		tmp = Float64(1.0 / cos(Float64(0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	t_0 = x_m / (y_m * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 1.02)
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.02], N[(1.0 / N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 1.02

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf 71.5%

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

      1. clear-num 72.0%

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

      2. un-div-inv 72.0%

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

   5. Applied egg-rr 72.0%

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

   if 1.02 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 5.2%

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

      1. remove-double-neg 5.2%

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

      2. distribute-frac-neg 5.2%

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

      3. tan-neg 5.2%

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

      4. distribute-frac-neg2 5.2%

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

      5. distribute-lft-neg-out 5.2%

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

      6. distribute-frac-neg2 5.2%

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

      7. distribute-lft-neg-out 5.2%

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

      8. distribute-frac-neg2 5.2%

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

      9. distribute-frac-neg 5.2%

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

      10. neg-mul-1 5.2%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. *-commutative 5.2%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. associate-/l* 5.3%

          \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{-1}{y \cdot 2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. *-commutative 5.3%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-1}{\color{blue}{2 \cdot y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. associate-/r* 5.3%

          \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{\frac{-1}{2}}{y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. metadata-eval 5.3%

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

      16. sin-neg 5.3%

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

      17. distribute-frac-neg 5.3%

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

   3. Simplified 5.1%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 26.4%

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

Alternative 4: 57.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{\cos \left({\left({\left(x\_m \cdot \frac{0.5}{y\_m}\right)}^{0.3333333333333333}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+274)
   (/ 1.0 (cos (pow (pow (* x_m (/ 0.5 y_m)) 0.3333333333333333) 3.0)))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+274) {
		tmp = 1.0 / cos(pow(pow((x_m * (0.5 / y_m)), 0.3333333333333333), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+274) then
        tmp = 1.0d0 / cos((((x_m * (0.5d0 / y_m)) ** 0.3333333333333333d0) ** 3.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+274) {
		tmp = 1.0 / Math.cos(Math.pow(Math.pow((x_m * (0.5 / y_m)), 0.3333333333333333), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+274:
		tmp = 1.0 / math.cos(math.pow(math.pow((x_m * (0.5 / y_m)), 0.3333333333333333), 3.0))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+274)
		tmp = Float64(1.0 / cos(((Float64(x_m * Float64(0.5 / y_m)) ^ 0.3333333333333333) ^ 3.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+274)
		tmp = 1.0 / cos((((x_m * (0.5 / y_m)) ^ 0.3333333333333333) ^ 3.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+274], N[(1.0 / N[Cos[N[Power[N[Power[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999998e274

   1. Initial program 46.3%

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

   3. Taylor expanded in x around inf 54.6%

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

      1. metadata-eval 54.6%

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

      2. add-sqr-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. add-sqr-sqrt 54.6%

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

      5. fabs-mul 54.6%

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

      6. add-cube-cbrt 54.2%

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

      7. pow3 54.4%

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

      8. fabs-mul 54.4%

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

      9. metadata-eval 54.4%

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

      10. cbrt-prod 53.8%

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

      11. add-sqr-sqrt 33.6%

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

      12. fabs-sqr 33.6%

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

      13. add-sqr-sqrt 53.8%

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

      14. cbrt-prod 54.4%

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

      15. clear-num 54.8%

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

      16. un-div-inv 54.8%

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

   5. Applied egg-rr 54.8%

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

      1. cbrt-div 53.7%

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

      2. div-inv 54.5%

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

   7. Applied egg-rr 54.5%

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

      1. un-div-inv 53.7%

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

      2. rem-cube-cbrt 53.7%

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

      3. cbrt-div 54.4%

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

      4. rem-cube-cbrt 54.8%

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

      5. pow1/3 35.4%

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

      6. rem-cube-cbrt 35.1%

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

      7. associate-/r/ 34.0%

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

      8. rem-cube-cbrt 34.7%

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

   9. Applied egg-rr 34.7%

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

   if 4.9999999999999998e274 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 1.8%

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

      1. remove-double-neg 1.8%

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

      2. distribute-frac-neg 1.8%

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

      3. tan-neg 1.8%

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

      4. distribute-frac-neg2 1.8%

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

      5. distribute-lft-neg-out 1.8%

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

      6. distribute-frac-neg2 1.8%

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

      7. distribute-lft-neg-out 1.8%

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

      8. distribute-frac-neg2 1.8%

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

      9. distribute-frac-neg 1.8%

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

      10. neg-mul-1 1.8%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. *-commutative 1.8%

          \[\leadsto \frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. associate-/l* 1.8%

          \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{-1}{y \cdot 2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. *-commutative 1.8%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-1}{\color{blue}{2 \cdot y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. associate-/r* 1.8%

          \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{\frac{-1}{2}}{y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. metadata-eval 1.8%

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

      16. sin-neg 1.8%

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

      17. distribute-frac-neg 1.8%

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

   3. Simplified 2.3%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 11.8%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{\cos \left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{0.3333333333333333}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 211.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 1.0)

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := 1.0

Derivation

1. Initial program 42.0%

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

   1. remove-double-neg 42.0%

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

   2. distribute-frac-neg 42.0%

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

   3. tan-neg 42.0%

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

   4. distribute-frac-neg2 42.0%

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

   5. distribute-lft-neg-out 42.0%

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

   6. distribute-frac-neg2 42.0%

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

   7. distribute-lft-neg-out 42.0%

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

   8. distribute-frac-neg2 42.0%

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

   9. distribute-frac-neg 42.0%

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

   10. neg-mul-1 42.0%

       \[\leadsto \frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   11. *-commutative 42.0%

       \[\leadsto \frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{y \cdot 2}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   12. associate-/l* 41.8%

       \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{-1}{y \cdot 2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   13. *-commutative 41.8%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-1}{\color{blue}{2 \cdot y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   14. associate-/r* 41.8%

       \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{\frac{-1}{2}}{y}}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   15. metadata-eval 41.8%

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

   16. sin-neg 41.8%

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

   17. distribute-frac-neg 41.8%

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

3. Simplified 42.1%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.1%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer target: 55.6% 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 2024100 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (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)))))