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

Percentage Accurate: 43.4% → 55.7%

Time: 34.1s

Alternatives: 7

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.4% 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: 55.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m \cdot \frac{-0.5}{\sqrt[3]{y\_m}}}{{\left(\sqrt[3]{y\_m}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right|\\ \end{array} \end{array} \]

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+64)
   (/ 1.0 (cos (/ (* x_m (/ -0.5 (cbrt y_m))) (pow (cbrt y_m) 2.0))))
   (fabs (* 2.0 (pow (cbrt 0.5) 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+64) {
		tmp = 1.0 / cos(((x_m * (-0.5 / cbrt(y_m))) / pow(cbrt(y_m), 2.0)));
	} else {
		tmp = fabs((2.0 * pow(cbrt(0.5), 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+64) {
		tmp = 1.0 / Math.cos(((x_m * (-0.5 / Math.cbrt(y_m))) / Math.pow(Math.cbrt(y_m), 2.0)));
	} else {
		tmp = Math.abs((2.0 * Math.pow(Math.cbrt(0.5), 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+64)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * Float64(-0.5 / cbrt(y_m))) / (cbrt(y_m) ^ 2.0))));
	else
		tmp = abs(Float64(2.0 * (cbrt(0.5) ^ 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+64], N[(1.0 / N[Cos[N[(N[(x$95$m * N[(-0.5 / N[Power[y$95$m, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[y$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(2.0 * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000004e64

   1. Initial program 52.2%

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

      1. expm1-log1p-u 52.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      2. expm1-undefine 6.6%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)} - 1}} \]

      3. *-un-lft-identity 6.6%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)} - 1} \]

      4. *-commutative 6.6%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)} - 1} \]

      5. times-frac 6.6%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)} - 1} \]

      6. metadata-eval 6.6%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)\right)} - 1} \]

   4. Applied egg-rr 6.6%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 52.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

      2. *-commutative 52.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

      3. associate-*l/ 52.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}\right)\right)} \]

   6. Simplified 52.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]

   7. Taylor expanded in x around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*r/ 68.4%

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

   9. Simplified 68.4%

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

       1. clear-num 68.4%

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

       2. div-inv 68.4%

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

       3. metadata-eval 68.4%

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

       4. div-inv 68.3%

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

       5. associate-/r* 68.3%

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

       6. clear-num 68.5%

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

   11. Applied egg-rr 68.5%

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

       1. add-sqr-sqrt 37.2%

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

       2. sqrt-unprod 67.8%

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

       3. associate-/r/ 67.7%

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

       4. metadata-eval 67.7%

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

       5. associate-/r/ 67.6%

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

       6. metadata-eval 67.6%

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

       7. swap-sqr 67.6%

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

       8. metadata-eval 67.6%

          \[\leadsto \frac{1}{\cos \left(\sqrt{\color{blue}{0.25} \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)}\right)} \]

       9. metadata-eval 67.6%

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

       10. swap-sqr 67.6%

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

       11. associate-*r/ 67.6%

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

       12. *-commutative 67.6%

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

       13. add-cube-cbrt 67.7%

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

       14. unpow2 67.7%

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

       15. associate-/l/ 67.7%

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

       16. rem-cube-cbrt 67.4%

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

       17. associate-*r/ 67.4%

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

       18. *-commutative 67.4%

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

       19. add-cube-cbrt 67.5%

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

       20. unpow2 67.5%

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

       21. associate-/l/ 67.5%

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

   13. Applied egg-rr 69.2%

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

   if 2.00000000000000004e64 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.1%

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

      1. remove-double-neg 8.1%

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

      2. distribute-frac-neg 8.1%

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

      3. tan-neg 8.1%

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

      4. distribute-frac-neg2 8.1%

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

      5. distribute-lft-neg-out 8.1%

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

      6. distribute-frac-neg2 8.1%

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

      7. distribute-lft-neg-out 8.1%

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

      8. distribute-frac-neg2 8.1%

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

      9. distribute-frac-neg 8.1%

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

      10. neg-mul-1 8.1%

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

      11. *-commutative 8.1%

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

      12. associate-/l* 8.0%

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

      13. *-commutative 8.0%

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

      14. associate-/r* 8.0%

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

      15. metadata-eval 8.0%

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

      16. sin-neg 8.0%

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

      17. distribute-frac-neg 8.0%

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

   3. Simplified 8.2%

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

      1. add-sqr-sqrt 5.0%

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

      2. sqrt-unprod 8.8%

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

      3. pow2 8.8%

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

   6. Applied egg-rr 8.8%

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

      1. unpow2 8.8%

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

      2. rem-sqrt-square 8.8%

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

      3. remove-double-neg 8.8%

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

      4. distribute-frac-neg 8.8%

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

      5. distribute-neg-frac2 8.8%

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

      6. tan-neg 8.8%

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

      7. associate-*r/ 8.6%

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

      8. distribute-frac-neg 8.6%

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

      9. distribute-rgt-neg-in 8.6%

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

      10. metadata-eval 8.6%

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

      11. *-commutative 8.6%

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

      12. associate-*r/ 8.6%

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

      13. sin-neg 8.6%

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

      14. associate-*r/ 8.8%

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

      15. distribute-frac-neg 8.8%

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

   8. Simplified 8.8%

      \[\leadsto \color{blue}{\left|\frac{\tan \left(x \cdot \frac{0.5}{y}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right|} \]
   9. Step-by-step derivation

      1. associate-*r/ 8.6%

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

      2. metadata-eval 8.6%

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

      3. distribute-rgt-neg-in 8.6%

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

      4. *-commutative 8.6%

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

      5. distribute-neg-frac 8.6%

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

      6. neg-sub0 8.6%

         \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5 \cdot x}{y}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      7. associate-*r/ 8.6%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{-0.5 \cdot \frac{x}{y}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      8. clear-num 7.9%

         \[\leadsto \left|\frac{\tan \left(0 - -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      9. un-div-inv 7.9%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{\frac{-0.5}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

   10. Applied egg-rr 7.9%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   11. Step-by-step derivation

       1. neg-sub0 7.9%

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

       2. distribute-neg-frac 7.9%

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

       3. metadata-eval 7.9%

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

   12. Simplified 7.9%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   13. Step-by-step derivation

       1. add-cube-cbrt 7.9%

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

       2. pow3 7.9%

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

       3. associate-/r/ 7.6%

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

       4. *-commutative 7.6%

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

       5. associate-*r/ 7.7%

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

   14. Applied egg-rr 7.7%

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

   15. Taylor expanded in x around 0 11.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot \frac{-0.5}{\sqrt[3]{y}}}{{\left(\sqrt[3]{y}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 100000000000:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\cos \left(-0.5 \cdot \frac{x\_m}{y\_m}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right|\\ \end{array} \end{array} \]

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)) 100000000000.0)
   (/ 1.0 (cbrt (pow (cos (* -0.5 (/ x_m y_m))) 3.0)))
   (fabs (* 2.0 (pow (cbrt 0.5) 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)) <= 100000000000.0) {
		tmp = 1.0 / cbrt(pow(cos((-0.5 * (x_m / y_m))), 3.0));
	} else {
		tmp = fabs((2.0 * pow(cbrt(0.5), 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)) <= 100000000000.0) {
		tmp = 1.0 / Math.cbrt(Math.pow(Math.cos((-0.5 * (x_m / y_m))), 3.0));
	} else {
		tmp = Math.abs((2.0 * Math.pow(Math.cbrt(0.5), 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)) <= 100000000000.0)
		tmp = Float64(1.0 / cbrt((cos(Float64(-0.5 * Float64(x_m / y_m))) ^ 3.0)));
	else
		tmp = abs(Float64(2.0 * (cbrt(0.5) ^ 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], 100000000000.0], N[(1.0 / N[Power[N[Power[N[Cos[N[(-0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Abs[N[(2.0 * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e11

   1. Initial program 54.9%

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

      1. remove-double-neg 54.9%

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

      2. distribute-frac-neg 54.9%

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

      3. tan-neg 54.9%

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

      4. distribute-frac-neg2 54.9%

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

      5. distribute-lft-neg-out 54.9%

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

      6. distribute-frac-neg2 54.9%

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

      7. distribute-lft-neg-out 54.9%

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

      8. distribute-frac-neg2 54.9%

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

      9. distribute-frac-neg 54.9%

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

      10. neg-mul-1 54.9%

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

      11. *-commutative 54.9%

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

      12. associate-/l* 54.8%

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

      13. *-commutative 54.8%

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

      14. associate-/r* 54.8%

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

      15. metadata-eval 54.8%

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

      16. sin-neg 54.8%

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

      17. distribute-frac-neg 54.8%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 72.1%

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

      1. associate-*r/ 72.1%

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

   7. Simplified 72.1%

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

      1. add-cbrt-cube 72.1%

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

      2. pow3 72.1%

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

      3. associate-*r/ 72.1%

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

   9. Applied egg-rr 72.1%

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

   if 1e11 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.8%

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

      1. remove-double-neg 8.8%

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

      2. distribute-frac-neg 8.8%

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

      3. tan-neg 8.8%

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

      4. distribute-frac-neg2 8.8%

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

      5. distribute-lft-neg-out 8.8%

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

      6. distribute-frac-neg2 8.8%

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

      7. distribute-lft-neg-out 8.8%

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

      8. distribute-frac-neg2 8.8%

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

      9. distribute-frac-neg 8.8%

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

      10. neg-mul-1 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-/l* 8.3%

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

      13. *-commutative 8.3%

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

      14. associate-/r* 8.3%

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

      15. metadata-eval 8.3%

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

      16. sin-neg 8.3%

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

      17. distribute-frac-neg 8.3%

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

   3. Simplified 8.9%

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

      1. add-sqr-sqrt 5.1%

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

      2. sqrt-unprod 9.1%

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

      3. pow2 9.1%

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

   6. Applied egg-rr 9.1%

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

      1. unpow2 9.1%

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

      2. rem-sqrt-square 9.1%

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

      3. remove-double-neg 9.1%

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

      4. distribute-frac-neg 9.1%

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

      5. distribute-neg-frac2 9.1%

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

      6. tan-neg 9.1%

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

      7. associate-*r/ 8.8%

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

      8. distribute-frac-neg 8.8%

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

      9. distribute-rgt-neg-in 8.8%

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

      10. metadata-eval 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-*r/ 8.8%

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

      13. sin-neg 8.8%

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

      14. associate-*r/ 9.0%

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

      15. distribute-frac-neg 9.0%

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

   8. Simplified 9.1%

      \[\leadsto \color{blue}{\left|\frac{\tan \left(x \cdot \frac{0.5}{y}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right|} \]
   9. Step-by-step derivation

      1. associate-*r/ 8.8%

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

      2. metadata-eval 8.8%

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

      3. distribute-rgt-neg-in 8.8%

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

      4. *-commutative 8.8%

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

      5. distribute-neg-frac 8.8%

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

      6. neg-sub0 8.8%

         \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5 \cdot x}{y}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      7. associate-*r/ 8.8%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{-0.5 \cdot \frac{x}{y}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      8. clear-num 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      9. un-div-inv 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{\frac{-0.5}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

   10. Applied egg-rr 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   11. Step-by-step derivation

       1. neg-sub0 8.4%

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

       2. distribute-neg-frac 8.4%

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

       3. metadata-eval 8.4%

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

   12. Simplified 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   13. Step-by-step derivation

       1. add-cube-cbrt 8.3%

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

       2. pow3 8.3%

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

       3. associate-/r/ 8.1%

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

       4. *-commutative 8.1%

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

       5. associate-*r/ 8.1%

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

   14. Applied egg-rr 8.1%

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

   15. Taylor expanded in x around 0 11.9%

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

Alternative 3: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x\_m \cdot \frac{-0.5}{y\_m}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right|\\ \end{array} \end{array} \]

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)) 4000000000.0)
   (log1p (expm1 (/ 1.0 (cos (* x_m (/ -0.5 y_m))))))
   (fabs (* 2.0 (pow (cbrt 0.5) 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)) <= 4000000000.0) {
		tmp = log1p(expm1((1.0 / cos((x_m * (-0.5 / y_m))))));
	} else {
		tmp = fabs((2.0 * pow(cbrt(0.5), 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)) <= 4000000000.0) {
		tmp = Math.log1p(Math.expm1((1.0 / Math.cos((x_m * (-0.5 / y_m))))));
	} else {
		tmp = Math.abs((2.0 * Math.pow(Math.cbrt(0.5), 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)) <= 4000000000.0)
		tmp = log1p(expm1(Float64(1.0 / cos(Float64(x_m * Float64(-0.5 / y_m))))));
	else
		tmp = abs(Float64(2.0 * (cbrt(0.5) ^ 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], 4000000000.0], N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4e9

   1. Initial program 54.9%

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

      1. expm1-log1p-u 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      2. expm1-undefine 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)} - 1}} \]

      3. *-un-lft-identity 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)} - 1} \]

      4. *-commutative 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)} - 1} \]

      5. times-frac 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)} - 1} \]

      6. metadata-eval 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)\right)} - 1} \]

   4. Applied egg-rr 6.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

      2. *-commutative 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

      3. associate-*l/ 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}\right)\right)} \]

   6. Simplified 54.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]

   7. Taylor expanded in x around inf 72.1%

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

      1. associate-*r/ 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-*r/ 72.1%

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

   9. Simplified 72.1%

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

       1. clear-num 72.1%

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

       2. div-inv 72.1%

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

       3. metadata-eval 72.1%

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

       4. div-inv 72.1%

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

       5. associate-/r* 72.1%

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

       6. clear-num 72.2%

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

   11. Applied egg-rr 72.2%

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

   13. Applied egg-rr 72.1%

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

   if 4e9 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.8%

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

      1. remove-double-neg 8.8%

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

      2. distribute-frac-neg 8.8%

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

      3. tan-neg 8.8%

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

      4. distribute-frac-neg2 8.8%

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

      5. distribute-lft-neg-out 8.8%

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

      6. distribute-frac-neg2 8.8%

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

      7. distribute-lft-neg-out 8.8%

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

      8. distribute-frac-neg2 8.8%

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

      9. distribute-frac-neg 8.8%

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

      10. neg-mul-1 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-/l* 8.3%

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

      13. *-commutative 8.3%

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

      14. associate-/r* 8.3%

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

      15. metadata-eval 8.3%

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

      16. sin-neg 8.3%

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

      17. distribute-frac-neg 8.3%

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

   3. Simplified 8.9%

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

      1. add-sqr-sqrt 5.1%

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

      2. sqrt-unprod 9.1%

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

      3. pow2 9.1%

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

   6. Applied egg-rr 9.1%

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

      1. unpow2 9.1%

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

      2. rem-sqrt-square 9.1%

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

      3. remove-double-neg 9.1%

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

      4. distribute-frac-neg 9.1%

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

      5. distribute-neg-frac2 9.1%

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

      6. tan-neg 9.1%

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

      7. associate-*r/ 8.8%

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

      8. distribute-frac-neg 8.8%

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

      9. distribute-rgt-neg-in 8.8%

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

      10. metadata-eval 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-*r/ 8.8%

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

      13. sin-neg 8.8%

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

      14. associate-*r/ 9.0%

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

      15. distribute-frac-neg 9.0%

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

   8. Simplified 9.1%

      \[\leadsto \color{blue}{\left|\frac{\tan \left(x \cdot \frac{0.5}{y}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right|} \]
   9. Step-by-step derivation

      1. associate-*r/ 8.8%

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

      2. metadata-eval 8.8%

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

      3. distribute-rgt-neg-in 8.8%

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

      4. *-commutative 8.8%

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

      5. distribute-neg-frac 8.8%

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

      6. neg-sub0 8.8%

         \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5 \cdot x}{y}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      7. associate-*r/ 8.8%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{-0.5 \cdot \frac{x}{y}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      8. clear-num 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      9. un-div-inv 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{\frac{-0.5}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

   10. Applied egg-rr 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   11. Step-by-step derivation

       1. neg-sub0 8.4%

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

       2. distribute-neg-frac 8.4%

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

       3. metadata-eval 8.4%

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

   12. Simplified 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   13. Step-by-step derivation

       1. add-cube-cbrt 8.3%

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

       2. pow3 8.3%

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

       3. associate-/r/ 8.1%

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

       4. *-commutative 8.1%

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

       5. associate-*r/ 8.1%

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

   14. Applied egg-rr 8.1%

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

   15. Taylor expanded in x around 0 11.9%

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

Alternative 4: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4000000000:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right|\\ \end{array} \end{array} \]

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)) 4000000000.0)
   (/ 1.0 (cos (* x_m (/ 0.5 y_m))))
   (fabs (* 2.0 (pow (cbrt 0.5) 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)) <= 4000000000.0) {
		tmp = 1.0 / cos((x_m * (0.5 / y_m)));
	} else {
		tmp = fabs((2.0 * pow(cbrt(0.5), 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)) <= 4000000000.0) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / y_m)));
	} else {
		tmp = Math.abs((2.0 * Math.pow(Math.cbrt(0.5), 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)) <= 4000000000.0)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_m))));
	else
		tmp = abs(Float64(2.0 * (cbrt(0.5) ^ 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], 4000000000.0], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(2.0 * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4e9

   1. Initial program 54.9%

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

      1. expm1-log1p-u 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      2. expm1-undefine 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)} - 1}} \]

      3. *-un-lft-identity 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)} - 1} \]

      4. *-commutative 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)} - 1} \]

      5. times-frac 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)} - 1} \]

      6. metadata-eval 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)\right)} - 1} \]

   4. Applied egg-rr 6.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

      2. *-commutative 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

      3. associate-*l/ 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}\right)\right)} \]

   6. Simplified 54.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]

   7. Taylor expanded in x around inf 72.1%

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

      1. associate-*r/ 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-*r/ 72.1%

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

   9. Simplified 72.1%

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

   if 4e9 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.8%

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

      1. remove-double-neg 8.8%

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

      2. distribute-frac-neg 8.8%

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

      3. tan-neg 8.8%

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

      4. distribute-frac-neg2 8.8%

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

      5. distribute-lft-neg-out 8.8%

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

      6. distribute-frac-neg2 8.8%

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

      7. distribute-lft-neg-out 8.8%

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

      8. distribute-frac-neg2 8.8%

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

      9. distribute-frac-neg 8.8%

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

      10. neg-mul-1 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-/l* 8.3%

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

      13. *-commutative 8.3%

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

      14. associate-/r* 8.3%

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

      15. metadata-eval 8.3%

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

      16. sin-neg 8.3%

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

      17. distribute-frac-neg 8.3%

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

   3. Simplified 8.9%

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

      1. add-sqr-sqrt 5.1%

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

      2. sqrt-unprod 9.1%

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

      3. pow2 9.1%

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

   6. Applied egg-rr 9.1%

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

      1. unpow2 9.1%

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

      2. rem-sqrt-square 9.1%

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

      3. remove-double-neg 9.1%

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

      4. distribute-frac-neg 9.1%

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

      5. distribute-neg-frac2 9.1%

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

      6. tan-neg 9.1%

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

      7. associate-*r/ 8.8%

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

      8. distribute-frac-neg 8.8%

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

      9. distribute-rgt-neg-in 8.8%

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

      10. metadata-eval 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-*r/ 8.8%

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

      13. sin-neg 8.8%

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

      14. associate-*r/ 9.0%

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

      15. distribute-frac-neg 9.0%

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

   8. Simplified 9.1%

      \[\leadsto \color{blue}{\left|\frac{\tan \left(x \cdot \frac{0.5}{y}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right|} \]
   9. Step-by-step derivation

      1. associate-*r/ 8.8%

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

      2. metadata-eval 8.8%

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

      3. distribute-rgt-neg-in 8.8%

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

      4. *-commutative 8.8%

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

      5. distribute-neg-frac 8.8%

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

      6. neg-sub0 8.8%

         \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5 \cdot x}{y}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      7. associate-*r/ 8.8%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{-0.5 \cdot \frac{x}{y}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      8. clear-num 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

      9. un-div-inv 8.4%

         \[\leadsto \left|\frac{\tan \left(0 - \color{blue}{\frac{-0.5}{\frac{y}{x}}}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]

   10. Applied egg-rr 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(0 - \frac{-0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   11. Step-by-step derivation

       1. neg-sub0 8.4%

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

       2. distribute-neg-frac 8.4%

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

       3. metadata-eval 8.4%

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

   12. Simplified 8.4%

       \[\leadsto \left|\frac{\tan \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right| \]
   13. Step-by-step derivation

       1. add-cube-cbrt 8.3%

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

       2. pow3 8.3%

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

       3. associate-/r/ 8.1%

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

       4. *-commutative 8.1%

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

       5. associate-*r/ 8.1%

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

   14. Applied egg-rr 8.1%

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

   15. Taylor expanded in x around 0 11.9%

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

Alternative 5: 55.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4000000000:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 4000000000.0)
   (/ 1.0 (cos (* x_m (/ 0.5 y_m))))
   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)) <= 4000000000.0) {
		tmp = 1.0 / cos((x_m * (0.5 / y_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) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 4000000000.0d0) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / y_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 tmp;
	if ((x_m / (y_m * 2.0)) <= 4000000000.0) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / y_m)));
	} 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)) <= 4000000000.0:
		tmp = 1.0 / math.cos((x_m * (0.5 / y_m)))
	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)) <= 4000000000.0)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_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)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 4000000000.0)
		tmp = 1.0 / cos((x_m * (0.5 / y_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_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 4000000000.0], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4e9

   1. Initial program 54.9%

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

      1. expm1-log1p-u 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      2. expm1-undefine 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)} - 1}} \]

      3. *-un-lft-identity 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)} - 1} \]

      4. *-commutative 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)} - 1} \]

      5. times-frac 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)} - 1} \]

      6. metadata-eval 6.2%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)\right)} - 1} \]

   4. Applied egg-rr 6.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-define 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

      2. *-commutative 54.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

      3. associate-*l/ 54.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}\right)\right)} \]

   6. Simplified 54.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]

   7. Taylor expanded in x around inf 72.1%

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

      1. associate-*r/ 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-*r/ 72.1%

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

   9. Simplified 72.1%

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

   if 4e9 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.8%

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

      1. remove-double-neg 8.8%

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

      2. distribute-frac-neg 8.8%

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

      3. tan-neg 8.8%

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

      4. distribute-frac-neg2 8.8%

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

      5. distribute-lft-neg-out 8.8%

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

      6. distribute-frac-neg2 8.8%

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

      7. distribute-lft-neg-out 8.8%

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

      8. distribute-frac-neg2 8.8%

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

      9. distribute-frac-neg 8.8%

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

      10. neg-mul-1 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-/l* 8.3%

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

      13. *-commutative 8.3%

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

      14. associate-/r* 8.3%

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

      15. metadata-eval 8.3%

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

      16. sin-neg 8.3%

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

      17. distribute-frac-neg 8.3%

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

   3. Simplified 8.9%

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

   5. Taylor expanded in x around 0 11.9%

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

Alternative 6: 54.0% 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 41.2%

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

   1. remove-double-neg 41.2%

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

   2. distribute-frac-neg 41.2%

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

   3. tan-neg 41.2%

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

   4. distribute-frac-neg2 41.2%

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

   5. distribute-lft-neg-out 41.2%

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

   6. distribute-frac-neg2 41.2%

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

   7. distribute-lft-neg-out 41.2%

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

   8. distribute-frac-neg2 41.2%

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

   9. distribute-frac-neg 41.2%

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

   10. neg-mul-1 41.2%

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

   11. *-commutative 41.2%

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

   12. associate-/l* 41.0%

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

   13. *-commutative 41.0%

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

   14. associate-/r* 41.0%

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

   15. metadata-eval 41.0%

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

   16. sin-neg 41.0%

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

   17. distribute-frac-neg 41.0%

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

3. Simplified 41.2%

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

5. Taylor expanded in x around 0 53.5%

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

Alternative 7: 6.8% 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 41.2%

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

   1. expm1-log1p-u 41.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

   2. expm1-undefine 7.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)} - 1}} \]

   3. *-un-lft-identity 7.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)} - 1} \]

   4. *-commutative 7.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)} - 1} \]

   5. times-frac 7.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)} - 1} \]

   6. metadata-eval 7.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{e^{\mathsf{log1p}\left(\sin \left(\color{blue}{0.5} \cdot \frac{x}{y}\right)\right)} - 1} \]

4. Applied egg-rr 7.0%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]
5. Step-by-step derivation

   1. expm1-define 41.1%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

   2. *-commutative 41.1%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

   3. associate-*l/ 41.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}\right)\right)} \]

6. Simplified 41.2%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{x \cdot 0.5}{y}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative 41.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\color{blue}{0.5 \cdot x}}{y}\right)\right)\right)} \]

   2. associate-*r/ 41.1%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

   3. add-sqr-sqrt 16.6%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}\right)}\right)\right)\right)} \]

   4. metadata-eval 16.6%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\color{blue}{\left|-0.5\right|} \cdot \left(\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}\right)\right)\right)\right)} \]

   5. fabs-sqr 16.6%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left|-0.5\right| \cdot \color{blue}{\left|\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}\right|}\right)\right)\right)} \]

   6. add-sqr-sqrt 19.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left|-0.5\right| \cdot \left|\color{blue}{\frac{x}{y}}\right|\right)\right)\right)} \]

   7. fabs-mul 19.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\left|-0.5 \cdot \frac{x}{y}\right|\right)}\right)\right)} \]

   8. log1p-undefine 5.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \sin \left(\left|-0.5 \cdot \frac{x}{y}\right|\right)\right)}\right)} \]

   9. add-sqr-sqrt 1.4%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\log \left(1 + \sin \left(\left|\color{blue}{\sqrt{-0.5 \cdot \frac{x}{y}} \cdot \sqrt{-0.5 \cdot \frac{x}{y}}}\right|\right)\right)\right)} \]

   10. fabs-sqr 1.4%

       \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\log \left(1 + \sin \color{blue}{\left(\sqrt{-0.5 \cdot \frac{x}{y}} \cdot \sqrt{-0.5 \cdot \frac{x}{y}}\right)}\right)\right)} \]

   11. add-sqr-sqrt 4.3%

       \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\log \left(1 + \sin \color{blue}{\left(-0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

8. Applied egg-rr 4.3%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \sin \left(-0.5 \cdot \frac{x}{y}\right)\right)}\right)} \]

9. Taylor expanded in x around 0 7.0%

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

Developer target: 54.0% 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 2024107 
(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)))))