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

Percentage Accurate: 44.2% → 56.9%

Time: 15.7s

Alternatives: 6

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: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t\_0}{\sin t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x\_m \cdot -0.5}{{\left(\sqrt[3]{y\_m}\right)}^{2}}}{\sqrt[3]{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

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+136)
   (/ 1.0 (cos (/ (/ (* x_m -0.5) (pow (cbrt y_m) 2.0)) (cbrt y_m))))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

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+136) {
		tmp = 1.0 / cos((((x_m * -0.5) / pow(cbrt(y_m), 2.0)) / cbrt(y_m)));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	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)) <= 5e+136) {
		tmp = 1.0 / Math.cos((((x_m * -0.5) / Math.pow(Math.cbrt(y_m), 2.0)) / Math.cbrt(y_m)));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	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+136)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(x_m * -0.5) / (cbrt(y_m) ^ 2.0)) / cbrt(y_m))));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	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], 5e+136], N[(1.0 / N[Cos[N[(N[(N[(x$95$m * -0.5), $MachinePrecision] / N[Power[N[Power[y$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$95$m, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e136

   1. Initial program 41.6%

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

      1. remove-double-neg 41.6%

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

      2. distribute-frac-neg 41.6%

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

      3. tan-neg 41.6%

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

      4. distribute-frac-neg2 41.6%

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

      5. distribute-lft-neg-out 41.6%

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

      6. distribute-frac-neg2 41.6%

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

      7. distribute-lft-neg-out 41.6%

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

      8. distribute-frac-neg2 41.6%

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

      9. distribute-frac-neg 41.6%

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

      10. neg-mul-1 41.6%

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

      11. *-commutative 41.6%

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

      12. associate-/l* 41.5%

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

      13. *-commutative 41.5%

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

      14. associate-/r* 41.5%

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

      15. metadata-eval 41.5%

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

      16. sin-neg 41.5%

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

      17. distribute-frac-neg 41.5%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in x around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r/ 57.5%

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

   7. Simplified 57.5%

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

      1. associate-*r/ 57.7%

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

      2. add-cube-cbrt 58.3%

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

      3. associate-/r* 58.3%

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

      4. pow2 58.3%

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

   9. Applied egg-rr 58.3%

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

   if 5.0000000000000002e136 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.1%

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

      1. add-cube-cbrt 5.7%

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

      2. pow3 5.8%

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

   4. Applied egg-rr 5.8%

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

   5. Taylor expanded in x around 0 11.7%

      \[\leadsto \color{blue}{0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x\_m}{y\_m}\\ t_1 := {t\_0}^{0.16666666666666666}\\ \frac{1}{\cos \left(t\_1 \cdot \left(t\_1 \cdot {\left(\sqrt[3]{t\_0}\right)}^{2}\right)\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x_m y_m))) (t_1 (pow t_0 0.16666666666666666)))
   (/ 1.0 (cos (* t_1 (* t_1 (pow (cbrt t_0) 2.0)))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = 0.5 * (x_m / y_m);
	double t_1 = pow(t_0, 0.16666666666666666);
	return 1.0 / cos((t_1 * (t_1 * pow(cbrt(t_0), 2.0))));
}

Java

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

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(0.5 * Float64(x_m / y_m))
	t_1 = t_0 ^ 0.16666666666666666
	return Float64(1.0 / cos(Float64(t_1 * Float64(t_1 * (cbrt(t_0) ^ 2.0)))))
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[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]}, N[(1.0 / N[Cos[N[(t$95$1 * N[(t$95$1 * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 35.4%

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

   1. remove-double-neg 35.4%

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

   2. distribute-frac-neg 35.4%

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

   3. tan-neg 35.4%

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

   4. distribute-frac-neg2 35.4%

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

   5. distribute-lft-neg-out 35.4%

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

   6. distribute-frac-neg2 35.4%

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

   7. distribute-lft-neg-out 35.4%

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

   8. distribute-frac-neg2 35.4%

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

   9. distribute-frac-neg 35.4%

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

   10. neg-mul-1 35.4%

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

   11. *-commutative 35.4%

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

   12. associate-/l* 35.3%

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

   13. *-commutative 35.3%

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

   14. associate-/r* 35.3%

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

   15. metadata-eval 35.3%

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

   16. sin-neg 35.3%

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

   17. distribute-frac-neg 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in x around inf 48.6%

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

   1. associate-*r/ 48.6%

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

   2. *-commutative 48.6%

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

   3. associate-*r/ 48.5%

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

   4. cos-neg 48.5%

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

   5. associate-*r/ 48.6%

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

   6. distribute-frac-neg 48.6%

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

   7. distribute-rgt-neg-in 48.6%

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

   8. metadata-eval 48.6%

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

7. Simplified 48.6%

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

   1. add-sqr-sqrt 32.4%

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

   2. pow2 32.4%

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

   3. associate-/l* 32.4%

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

9. Applied egg-rr 32.4%

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

    1. unpow2 32.4%

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

    2. add-sqr-sqrt 48.5%

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

    3. rem-3cbrt-rft 49.1%

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

    4. add-sqr-sqrt 32.5%

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

    5. unpow2 32.5%

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

    6. associate-*l* 32.6%

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

    7. pow1/3 32.4%

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

    8. sqrt-pow1 32.5%

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

    9. associate-*r/ 32.6%

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

    10. *-commutative 32.6%

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

    11. associate-/l* 32.6%

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

    12. metadata-eval 32.6%

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

    13. pow1/3 32.9%

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

    14. sqrt-pow1 32.9%

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

    15. associate-*r/ 32.9%

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

    16. *-commutative 32.9%

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

    17. associate-/l* 32.9%

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

    18. metadata-eval 32.9%

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

    19. add-sqr-sqrt 32.9%

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

11. Applied egg-rr 33.0%

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

Alternative 3: 56.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{1}{\sqrt{y\_m \cdot \frac{2}{x\_m}}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

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+167)
   (/ 1.0 (cos (pow (/ 1.0 (sqrt (* y_m (/ 2.0 x_m)))) 2.0)))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

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+167) {
		tmp = 1.0 / cos(pow((1.0 / sqrt((y_m * (2.0 / x_m)))), 2.0));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	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+167) {
		tmp = 1.0 / Math.cos(Math.pow((1.0 / Math.sqrt((y_m * (2.0 / x_m)))), 2.0));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	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+167)
		tmp = Float64(1.0 / cos((Float64(1.0 / sqrt(Float64(y_m * Float64(2.0 / x_m)))) ^ 2.0)));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	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+167], N[(1.0 / N[Cos[N[Power[N[(1.0 / N[Sqrt[N[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e167

   1. Initial program 40.8%

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

      1. remove-double-neg 40.8%

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

      2. distribute-frac-neg 40.8%

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

      3. tan-neg 40.8%

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

      4. distribute-frac-neg2 40.8%

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

      5. distribute-lft-neg-out 40.8%

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

      6. distribute-frac-neg2 40.8%

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

      7. distribute-lft-neg-out 40.8%

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

      8. distribute-frac-neg2 40.8%

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

      9. distribute-frac-neg 40.8%

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

      10. neg-mul-1 40.8%

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

      11. *-commutative 40.8%

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

      12. associate-/l* 40.6%

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

      13. *-commutative 40.6%

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

      14. associate-/r* 40.6%

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

      15. metadata-eval 40.6%

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

      16. sin-neg 40.6%

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

      17. distribute-frac-neg 40.6%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in x around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. *-commutative 56.4%

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

      3. associate-*r/ 56.1%

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

      4. cos-neg 56.1%

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

      5. associate-*r/ 56.4%

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

      6. distribute-frac-neg 56.4%

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

      7. distribute-rgt-neg-in 56.4%

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

      8. metadata-eval 56.4%

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

   7. Simplified 56.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow2 37.2%

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

      3. associate-/l* 37.2%

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

   9. Applied egg-rr 37.2%

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

       1. sqrt-prod 15.9%

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

       2. metadata-eval 15.9%

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

       3. associate-/r* 15.9%

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

       4. *-commutative 15.9%

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

       5. sqrt-prod 37.2%

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

       6. div-inv 37.2%

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

       7. clear-num 37.6%

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

       8. sqrt-div 29.5%

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

       9. metadata-eval 29.5%

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

       10. associate-/l* 29.4%

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

   11. Applied egg-rr 29.4%

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

   if 2.0000000000000001e167 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.3%

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

      1. add-cube-cbrt 4.5%

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

      2. pow3 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around 0 11.7%

      \[\leadsto \color{blue}{0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{\frac{2}{x\_m}}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

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+165)
   (/ 1.0 (cos (/ (/ 1.0 (/ 2.0 x_m)) y_m)))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

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+165) {
		tmp = 1.0 / cos(((1.0 / (2.0 / x_m)) / y_m));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	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)) <= 5e+165) {
		tmp = 1.0 / Math.cos(((1.0 / (2.0 / x_m)) / y_m));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	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+165)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / Float64(2.0 / x_m)) / y_m)));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	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], 5e+165], N[(1.0 / N[Cos[N[(N[(1.0 / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999997e165

   1. Initial program 40.9%

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

      1. remove-double-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

      3. tan-neg 40.9%

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

      4. distribute-frac-neg2 40.9%

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

      5. distribute-lft-neg-out 40.9%

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

      6. distribute-frac-neg2 40.9%

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

      7. distribute-lft-neg-out 40.9%

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

      8. distribute-frac-neg2 40.9%

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

      9. distribute-frac-neg 40.9%

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

      10. neg-mul-1 40.9%

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

      11. *-commutative 40.9%

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

      12. associate-/l* 40.8%

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

      13. *-commutative 40.8%

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

      14. associate-/r* 40.8%

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

      15. metadata-eval 40.8%

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

      16. sin-neg 40.8%

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

      17. distribute-frac-neg 40.8%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in x around inf 56.6%

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

      1. associate-*r/ 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-*r/ 56.2%

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

      4. cos-neg 56.2%

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

      5. associate-*r/ 56.6%

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

      6. distribute-frac-neg 56.6%

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

      7. distribute-rgt-neg-in 56.6%

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

      8. metadata-eval 56.6%

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

   7. Simplified 56.6%

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

      1. add-sqr-sqrt 37.4%

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

      2. pow2 37.4%

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

      3. associate-/l* 37.3%

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

   9. Applied egg-rr 37.3%

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

       1. unpow2 37.3%

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

       2. add-sqr-sqrt 56.2%

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

       3. associate-*r/ 56.6%

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

       4. clear-num 56.0%

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

       5. inv-pow 56.0%

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

       6. div-inv 56.3%

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

       7. unpow-prod-down 56.4%

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

       8. inv-pow 56.4%

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

       9. *-commutative 56.4%

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

       10. associate-/r* 56.4%

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

       11. metadata-eval 56.4%

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

   11. Applied egg-rr 56.4%

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

       1. associate-*l/ 57.0%

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

       2. *-lft-identity 57.0%

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

       3. unpow-1 57.0%

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

   13. Simplified 57.0%

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

   if 4.9999999999999997e165 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.3%

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

      1. add-cube-cbrt 4.5%

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

      2. pow3 4.3%

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

   4. Applied egg-rr 4.3%

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

   5. Taylor expanded in x around 0 11.8%

      \[\leadsto \color{blue}{0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.05 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{\frac{2}{x\_m}}}{y\_m}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= y_m 2.05e-98) 1.0 (/ 1.0 (cos (/ (/ 1.0 (/ 2.0 x_m)) y_m)))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if (y_m <= 2.05e-98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / cos(((1.0 / (2.0 / x_m)) / y_m));
	}
	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 (y_m <= 2.05d-98) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / cos(((1.0d0 / (2.0d0 / x_m)) / y_m))
    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 (y_m <= 2.05e-98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / Math.cos(((1.0 / (2.0 / x_m)) / y_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if y_m <= 2.05e-98:
		tmp = 1.0
	else:
		tmp = 1.0 / math.cos(((1.0 / (2.0 / x_m)) / y_m))
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (y_m <= 2.05e-98)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / Float64(2.0 / x_m)) / y_m)));
	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 (y_m <= 2.05e-98)
		tmp = 1.0;
	else
		tmp = 1.0 / cos(((1.0 / (2.0 / x_m)) / y_m));
	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[y$95$m, 2.05e-98], 1.0, N[(1.0 / N[Cos[N[(N[(1.0 / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0499999999999999e-98

   1. Initial program 26.8%

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

      1. remove-double-neg 26.8%

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

      2. distribute-frac-neg 26.8%

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

      3. tan-neg 26.8%

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

      4. distribute-frac-neg2 26.8%

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

      5. distribute-lft-neg-out 26.8%

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

      6. distribute-frac-neg2 26.8%

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

      7. distribute-lft-neg-out 26.8%

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

      8. distribute-frac-neg2 26.8%

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

      9. distribute-frac-neg 26.8%

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

      10. neg-mul-1 26.8%

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

      11. *-commutative 26.8%

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

      12. associate-/l* 26.8%

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

      13. *-commutative 26.8%

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

      14. associate-/r* 26.8%

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

      15. metadata-eval 26.8%

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

      16. sin-neg 26.8%

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

      17. distribute-frac-neg 26.8%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in x around 0 39.9%

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

   if 2.0499999999999999e-98 < y

   1. Initial program 52.9%

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

      1. remove-double-neg 52.9%

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

      2. distribute-frac-neg 52.9%

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

      3. tan-neg 52.9%

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

      4. distribute-frac-neg2 52.9%

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

      5. distribute-lft-neg-out 52.9%

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

      6. distribute-frac-neg2 52.9%

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

      7. distribute-lft-neg-out 52.9%

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

      8. distribute-frac-neg2 52.9%

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

      9. distribute-frac-neg 52.9%

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

      10. neg-mul-1 52.9%

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

      11. *-commutative 52.9%

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

      12. associate-/l* 52.9%

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

      13. *-commutative 52.9%

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

      14. associate-/r* 52.9%

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

      15. metadata-eval 52.9%

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

      16. sin-neg 52.9%

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

      17. distribute-frac-neg 52.9%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in x around inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r/ 70.2%

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

      4. cos-neg 70.2%

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

      5. associate-*r/ 70.7%

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

      6. distribute-frac-neg 70.7%

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

      7. distribute-rgt-neg-in 70.7%

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

      8. metadata-eval 70.7%

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

   7. Simplified 70.7%

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

      1. add-sqr-sqrt 49.5%

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

      2. pow2 49.5%

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

      3. associate-/l* 49.5%

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

   9. Applied egg-rr 49.5%

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

       1. unpow2 49.5%

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

       2. add-sqr-sqrt 70.2%

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

       3. associate-*r/ 70.7%

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

       4. clear-num 70.5%

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

       5. inv-pow 70.5%

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

       6. div-inv 70.9%

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

       7. unpow-prod-down 70.5%

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

       8. inv-pow 70.5%

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

       9. *-commutative 70.5%

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

       10. associate-/r* 70.5%

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

       11. metadata-eval 70.5%

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

   11. Applied egg-rr 70.5%

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

       1. associate-*l/ 71.0%

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

       2. *-lft-identity 71.0%

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

       3. unpow-1 71.0%

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

   13. Simplified 71.0%

       \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\frac{1}{\frac{2}{x}}}{y}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.3% 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 35.4%

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

   1. remove-double-neg 35.4%

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

   2. distribute-frac-neg 35.4%

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

   3. tan-neg 35.4%

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

   4. distribute-frac-neg2 35.4%

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

   5. distribute-lft-neg-out 35.4%

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

   6. distribute-frac-neg2 35.4%

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

   7. distribute-lft-neg-out 35.4%

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

   8. distribute-frac-neg2 35.4%

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

   9. distribute-frac-neg 35.4%

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

   10. neg-mul-1 35.4%

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

   11. *-commutative 35.4%

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

   12. associate-/l* 35.3%

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

   13. *-commutative 35.3%

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

   14. associate-/r* 35.3%

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

   15. metadata-eval 35.3%

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

   16. sin-neg 35.3%

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

   17. distribute-frac-neg 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in x around 0 49.0%

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

Developer Target 1: 55.3% 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 2024131 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))