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

Percentage Accurate: 44.4% → 57.2%

Time: 16.7s

Alternatives: 10

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := {\left({\left(x\_m \cdot \frac{0.5}{y\_m}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\frac{1}{\cos \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (pow (pow (* x_m (/ 0.5 y_m)) 1.5) 0.3333333333333333)))
   (if (<= (/ x_m (* y_m 2.0)) 5e+170)
     (/ 1.0 (cos (* t_0 t_0)))
     (* 0.5 (pow (cbrt 2.0) 3.0)))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = pow(pow((x_m * (0.5 / y_m)), 1.5), 0.3333333333333333);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+170) {
		tmp = 1.0 / cos((t_0 * t_0));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = Math.pow(Math.pow((x_m * (0.5 / y_m)), 1.5), 0.3333333333333333);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+170) {
		tmp = 1.0 / Math.cos((t_0 * t_0));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = (Float64(x_m * Float64(0.5 / y_m)) ^ 1.5) ^ 0.3333333333333333
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+170)
		tmp = Float64(1.0 / cos(Float64(t_0 * t_0)));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[Power[N[Power[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+170], N[(1.0 / N[Cos[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.99999999999999977e170

   1. Initial program 52.5%

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

      1. remove-double-neg 52.5%

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

      2. distribute-frac-neg 52.5%

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

      3. tan-neg 52.5%

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

      4. distribute-frac-neg2 52.5%

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

      5. distribute-lft-neg-out 52.5%

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

      6. distribute-frac-neg2 52.5%

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

      7. distribute-lft-neg-out 52.5%

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

      8. distribute-frac-neg2 52.5%

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

      9. distribute-frac-neg 52.5%

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

      10. neg-mul-1 52.5%

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

      11. *-commutative 52.5%

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

      12. associate-/l* 52.2%

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

      13. *-commutative 52.2%

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

      14. associate-/r* 52.2%

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

      15. metadata-eval 52.2%

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

      16. sin-neg 52.2%

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

      17. distribute-frac-neg 52.2%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around inf 61.3%

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

      1. associate-*r/ 61.3%

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

      2. *-commutative 61.3%

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

      3. associate-*r/ 61.5%

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

   7. Simplified 61.5%

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

      1. associate-*r/ 61.3%

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

      2. add-sqr-sqrt 33.2%

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

      3. associate-/r* 33.2%

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

   9. Applied egg-rr 33.2%

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

       1. associate-/l/ 33.2%

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

       2. add-sqr-sqrt 61.3%

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

       3. add-sqr-sqrt 35.1%

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

       4. sqrt-unprod 51.5%

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

       5. swap-sqr 51.5%

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

       6. metadata-eval 51.5%

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

       7. metadata-eval 51.5%

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

       8. swap-sqr 51.5%

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

       9. sqrt-unprod 26.1%

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

       10. add-sqr-sqrt 61.3%

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

       11. *-commutative 61.3%

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

       12. associate-*r/ 61.3%

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

       13. rem-cbrt-cube 59.4%

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

       14. unpow1/3 47.3%

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

       15. add-sqr-sqrt 47.4%

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

       16. unpow-prod-down 47.6%

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

       17. sqrt-pow1 36.5%

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

       18. associate-*r/ 36.5%

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

       19. *-commutative 36.5%

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

       20. associate-/l* 36.5%

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

       21. metadata-eval 36.5%

           \[\leadsto \frac{1}{\cos \left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(0.5 \cdot \frac{x}{y}\right)}^{3}}\right)}^{0.3333333333333333}\right)} \]

   11. Applied egg-rr 37.1%

       \[\leadsto \frac{1}{\cos \color{blue}{\left({\left({\left(x \cdot \frac{0.5}{y}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(x \cdot \frac{0.5}{y}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

   if 4.99999999999999977e170 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.4%

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

      1. add-cube-cbrt 4.8%

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

      2. pow3 5.3%

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

   4. Applied egg-rr 5.3%

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

   5. Taylor expanded in x around 0 10.6%

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

Alternative 2: 56.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt[3]{x\_m \cdot \frac{0.5}{y\_m}}\\ t_1 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_1}{\sin t\_1} \leq 1.05:\\ \;\;\;\;\frac{1}{\cos \left(t\_0 \cdot {t\_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (cbrt (* x_m (/ 0.5 y_m)))) (t_1 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 1.05)
     (/ 1.0 (cos (* t_0 (pow t_0 2.0))))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = cbrt((x_m * (0.5 / y_m)));
	double t_1 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 1.05) {
		tmp = 1.0 / cos((t_0 * pow(t_0, 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = Math.cbrt((x_m * (0.5 / y_m)));
	double t_1 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 1.05) {
		tmp = 1.0 / Math.cos((t_0 * Math.pow(t_0, 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = cbrt(Float64(x_m * Float64(0.5 / y_m)))
	t_1 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 1.05)
		tmp = Float64(1.0 / cos(Float64(t_0 * (t_0 ^ 2.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 1.05], N[(1.0 / N[Cos[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 1.05000000000000004

   1. Initial program 66.7%

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

      1. remove-double-neg 66.7%

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

      2. distribute-frac-neg 66.7%

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

      3. tan-neg 66.7%

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

      4. distribute-frac-neg2 66.7%

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

      5. distribute-lft-neg-out 66.7%

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

      6. distribute-frac-neg2 66.7%

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

      7. distribute-lft-neg-out 66.7%

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

      8. distribute-frac-neg2 66.7%

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

      9. distribute-frac-neg 66.7%

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

      10. neg-mul-1 66.7%

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

      11. *-commutative 66.7%

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

      12. associate-/l* 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. sin-neg 66.3%

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

      17. distribute-frac-neg 66.3%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*r/ 66.8%

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

   7. Simplified 66.8%

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

      1. associate-*r/ 66.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. associate-/r* 36.4%

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

   9. Applied egg-rr 36.4%

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

   11. Applied egg-rr 67.5%

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

   if 1.05000000000000004 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 4.1%

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

      1. remove-double-neg 4.1%

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

      2. distribute-frac-neg 4.1%

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

      3. tan-neg 4.1%

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

      4. distribute-frac-neg2 4.1%

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

      5. distribute-lft-neg-out 4.1%

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

      6. distribute-frac-neg2 4.1%

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

      7. distribute-lft-neg-out 4.1%

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

      8. distribute-frac-neg2 4.1%

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

      9. distribute-frac-neg 4.1%

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

      10. neg-mul-1 4.1%

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

      11. *-commutative 4.1%

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

      12. associate-/l* 3.5%

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

      13. *-commutative 3.5%

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

      14. associate-/r* 3.5%

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

      15. metadata-eval 3.5%

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

      16. sin-neg 3.5%

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

      17. distribute-frac-neg 3.5%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in x around 0 30.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.05:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{x \cdot \frac{0.5}{y}} \cdot {\left(\sqrt[3]{x \cdot \frac{0.5}{y}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{1}{{y\_m}^{0.25}} \cdot \frac{x\_m \cdot \frac{-0.5}{\sqrt{y\_m}}}{{y\_m}^{0.25}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/
  1.0
  (cos
   (* (/ 1.0 (pow y_m 0.25)) (/ (* x_m (/ -0.5 (sqrt y_m))) (pow y_m 0.25))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(((1.0 / pow(y_m, 0.25)) * ((x_m * (-0.5 / sqrt(y_m))) / pow(y_m, 0.25))));
}

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 / cos(((1.0d0 / (y_m ** 0.25d0)) * ((x_m * ((-0.5d0) / sqrt(y_m))) / (y_m ** 0.25d0))))
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 / Math.cos(((1.0 / Math.pow(y_m, 0.25)) * ((x_m * (-0.5 / Math.sqrt(y_m))) / Math.pow(y_m, 0.25))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos(((1.0 / math.pow(y_m, 0.25)) * ((x_m * (-0.5 / math.sqrt(y_m))) / math.pow(y_m, 0.25))))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(Float64(1.0 / (y_m ^ 0.25)) * Float64(Float64(x_m * Float64(-0.5 / sqrt(y_m))) / (y_m ^ 0.25)))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((1.0 / (y_m ^ 0.25)) * ((x_m * (-0.5 / sqrt(y_m))) / (y_m ^ 0.25))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(N[(1.0 / N[Power[y$95$m, 0.25], $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * N[(-0.5 / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$95$m, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.1%

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

   1. remove-double-neg 45.1%

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

   2. distribute-frac-neg 45.1%

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

   3. tan-neg 45.1%

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

   4. distribute-frac-neg2 45.1%

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

   5. distribute-lft-neg-out 45.1%

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

   6. distribute-frac-neg2 45.1%

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

   7. distribute-lft-neg-out 45.1%

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

   8. distribute-frac-neg2 45.1%

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

   9. distribute-frac-neg 45.1%

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

   10. neg-mul-1 45.1%

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

   11. *-commutative 45.1%

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

   12. associate-/l* 44.7%

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

   13. *-commutative 44.7%

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

   14. associate-/r* 44.7%

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

   15. metadata-eval 44.7%

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

   16. sin-neg 44.7%

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

   17. distribute-frac-neg 44.7%

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

3. Simplified 45.3%

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

5. Taylor expanded in x around inf 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*r/ 52.7%

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

7. Simplified 52.7%

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

   1. associate-*r/ 52.6%

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

   2. add-sqr-sqrt 28.5%

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

   3. associate-/r* 28.5%

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

9. Applied egg-rr 28.5%

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

    1. *-un-lft-identity 28.5%

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

    2. add-sqr-sqrt 28.4%

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

    3. times-frac 28.4%

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

    4. pow1/2 28.4%

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

    5. sqrt-pow1 28.6%

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

    6. metadata-eval 28.6%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{{y}^{\color{blue}{0.25}}} \cdot \frac{\frac{x \cdot -0.5}{\sqrt{y}}}{\sqrt{\sqrt{y}}}\right)} \]

    7. associate-/l* 28.3%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{{y}^{0.25}} \cdot \frac{\color{blue}{x \cdot \frac{-0.5}{\sqrt{y}}}}{\sqrt{\sqrt{y}}}\right)} \]

    8. pow1/2 28.3%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{{y}^{0.25}} \cdot \frac{x \cdot \frac{-0.5}{\sqrt{y}}}{\sqrt{\color{blue}{{y}^{0.5}}}}\right)} \]

    9. sqrt-pow1 28.4%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{{y}^{0.25}} \cdot \frac{x \cdot \frac{-0.5}{\sqrt{y}}}{\color{blue}{{y}^{\left(\frac{0.5}{2}\right)}}}\right)} \]

    10. metadata-eval 28.4%

        \[\leadsto \frac{1}{\cos \left(\frac{1}{{y}^{0.25}} \cdot \frac{x \cdot \frac{-0.5}{\sqrt{y}}}{{y}^{\color{blue}{0.25}}}\right)} \]

11. Applied egg-rr 28.4%

    \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1}{{y}^{0.25}} \cdot \frac{x \cdot \frac{-0.5}{\sqrt{y}}}{{y}^{0.25}}\right)}} \]
12. Add Preprocessing

Alternative 4: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+187}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m \cdot \frac{0.5}{\sqrt{y\_m}}}{\sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+187)
   (/ 1.0 (cos (/ (* x_m (/ 0.5 (sqrt y_m))) (sqrt y_m))))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / cos(((x_m * (0.5 / sqrt(y_m))) / sqrt(y_m)));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / Math.cos(((x_m * (0.5 / Math.sqrt(y_m))) / Math.sqrt(y_m)));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+187)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * Float64(0.5 / sqrt(y_m))) / sqrt(y_m))));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+187], N[(1.0 / N[Cos[N[(N[(x$95$m * N[(0.5 / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999907e186

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-frac-neg 51.9%

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

      3. tan-neg 51.9%

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

      4. distribute-frac-neg2 51.9%

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

      5. distribute-lft-neg-out 51.9%

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

      6. distribute-frac-neg2 51.9%

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

      7. distribute-lft-neg-out 51.9%

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

      8. distribute-frac-neg2 51.9%

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

      9. distribute-frac-neg 51.9%

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

      10. neg-mul-1 51.9%

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

      11. *-commutative 51.9%

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

      12. associate-/l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. metadata-eval 51.6%

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

      16. sin-neg 51.6%

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

      17. distribute-frac-neg 51.6%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in x around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*r/ 60.8%

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

   7. Simplified 60.8%

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

      1. associate-*r/ 60.6%

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

      2. add-sqr-sqrt 32.9%

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

      3. associate-/r* 32.9%

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

   9. Applied egg-rr 32.9%

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

       1. add-sqr-sqrt 18.4%

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

       2. sqrt-unprod 27.8%

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

       3. swap-sqr 27.8%

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

       4. metadata-eval 27.8%

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

       5. metadata-eval 27.8%

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

       6. swap-sqr 27.8%

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

       7. sqrt-unprod 13.7%

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

       8. add-sqr-sqrt 32.9%

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

       9. metadata-eval 32.9%

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

       10. distribute-rgt-neg-in 32.9%

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

       11. distribute-neg-frac 32.9%

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

       12. neg-sub0 32.9%

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

       13. associate-/l* 32.8%

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

   11. Applied egg-rr 32.8%

       \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{0 - x \cdot \frac{-0.5}{\sqrt{y}}}}{\sqrt{y}}\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 32.8%

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

       2. distribute-rgt-neg-in 32.8%

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

       3. distribute-neg-frac 32.8%

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

       4. metadata-eval 32.8%

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

   13. Simplified 32.8%

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

   if 9.99999999999999907e186 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.2%

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

      1. add-cube-cbrt 5.1%

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

      2. pow3 5.6%

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

   4. Applied egg-rr 5.6%

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

   5. Taylor expanded in x around 0 10.3%

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

Alternative 5: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+187}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\sqrt{y\_m}} \cdot \frac{x\_m}{\sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+187)
   (/ 1.0 (cos (* (/ 0.5 (sqrt y_m)) (/ x_m (sqrt y_m)))))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / cos(((0.5 / sqrt(y_m)) * (x_m / sqrt(y_m))));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / Math.cos(((0.5 / Math.sqrt(y_m)) * (x_m / Math.sqrt(y_m))));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+187)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / sqrt(y_m)) * Float64(x_m / sqrt(y_m)))));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+187], N[(1.0 / N[Cos[N[(N[(0.5 / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999907e186

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-frac-neg 51.9%

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

      3. tan-neg 51.9%

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

      4. distribute-frac-neg2 51.9%

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

      5. distribute-lft-neg-out 51.9%

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

      6. distribute-frac-neg2 51.9%

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

      7. distribute-lft-neg-out 51.9%

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

      8. distribute-frac-neg2 51.9%

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

      9. distribute-frac-neg 51.9%

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

      10. neg-mul-1 51.9%

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

      11. *-commutative 51.9%

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

      12. associate-/l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. metadata-eval 51.6%

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

      16. sin-neg 51.6%

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

      17. distribute-frac-neg 51.6%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in x around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*r/ 60.8%

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

   7. Simplified 60.8%

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

      1. associate-*r/ 60.6%

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

      2. add-sqr-sqrt 32.9%

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

      3. associate-/r* 32.9%

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

   9. Applied egg-rr 32.9%

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

       1. associate-/l/ 32.9%

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

       2. add-sqr-sqrt 18.5%

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

       3. sqrt-unprod 27.8%

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

       4. swap-sqr 27.8%

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

       5. metadata-eval 27.8%

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

       6. metadata-eval 27.8%

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

       7. swap-sqr 27.8%

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

       8. sqrt-unprod 13.7%

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

       9. add-sqr-sqrt 32.9%

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

       10. times-frac 32.9%

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

   11. Applied egg-rr 32.9%

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

   if 9.99999999999999907e186 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.2%

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

      1. add-cube-cbrt 5.1%

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

      2. pow3 5.6%

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

   4. Applied egg-rr 5.6%

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

   5. Taylor expanded in x around 0 10.3%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+187}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\sqrt{y}} \cdot \frac{x}{\sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ x_m y_m)))) (/ 1.0 (cos (* t_0 (* 0.5 t_0))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = sqrt((x_m / y_m));
	return 1.0 / cos((t_0 * (0.5 * t_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
    real(8) :: t_0
    t_0 = sqrt((x_m / y_m))
    code = 1.0d0 / cos((t_0 * (0.5d0 * t_0)))
end function

Java

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

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.sqrt((x_m / y_m))
	return 1.0 / math.cos((t_0 * (0.5 * t_0)))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = sqrt(Float64(x_m / y_m))
	return Float64(1.0 / cos(Float64(t_0 * Float64(0.5 * t_0))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	t_0 = sqrt((x_m / y_m));
	tmp = 1.0 / cos((t_0 * (0.5 * t_0)));
end

Wolfram

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

Derivation

1. Initial program 45.1%

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

   1. remove-double-neg 45.1%

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

   2. distribute-frac-neg 45.1%

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

   3. tan-neg 45.1%

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

   4. distribute-frac-neg2 45.1%

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

   5. distribute-lft-neg-out 45.1%

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

   6. distribute-frac-neg2 45.1%

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

   7. distribute-lft-neg-out 45.1%

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

   8. distribute-frac-neg2 45.1%

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

   9. distribute-frac-neg 45.1%

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

   10. neg-mul-1 45.1%

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

   11. *-commutative 45.1%

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

   12. associate-/l* 44.7%

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

   13. *-commutative 44.7%

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

   14. associate-/r* 44.7%

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

   15. metadata-eval 44.7%

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

   16. sin-neg 44.7%

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

   17. distribute-frac-neg 44.7%

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

3. Simplified 45.3%

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

5. Taylor expanded in x around inf 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*r/ 52.7%

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

7. Simplified 52.7%

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

   1. associate-*r/ 52.6%

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

   2. add-sqr-sqrt 28.5%

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

   3. associate-/r* 28.5%

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

9. Applied egg-rr 28.5%

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

    1. associate-/l/ 28.5%

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

    2. add-sqr-sqrt 15.8%

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

    3. sqrt-unprod 24.0%

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

    4. swap-sqr 24.0%

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

    5. metadata-eval 24.0%

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

    6. metadata-eval 24.0%

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

    7. swap-sqr 24.0%

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

    8. sqrt-unprod 12.1%

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

    9. add-sqr-sqrt 28.5%

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

    10. *-commutative 28.5%

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

    11. add-sqr-sqrt 52.6%

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

    12. associate-*r/ 52.6%

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

    13. add-sqr-sqrt 31.9%

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

    14. associate-*r* 31.9%

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

11. Applied egg-rr 31.9%

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

12. Final simplification 31.9%

    \[\leadsto \frac{1}{\cos \left(\sqrt{\frac{x}{y}} \cdot \left(0.5 \cdot \sqrt{\frac{x}{y}}\right)\right)} \]
13. Add Preprocessing

Alternative 7: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+187}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{1}{x\_m \cdot -0.5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+187)
   (/ 1.0 (cos (/ 1.0 (* y_m (/ 1.0 (* x_m -0.5))))))
   (* 0.5 (pow (cbrt 2.0) 3.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))));
	} else {
		tmp = 0.5 * pow(cbrt(2.0), 3.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+187) {
		tmp = 1.0 / Math.cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt(2.0), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+187)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(y_m * Float64(1.0 / Float64(x_m * -0.5))))));
	else
		tmp = Float64(0.5 * (cbrt(2.0) ^ 3.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+187], N[(1.0 / N[Cos[N[(1.0 / N[(y$95$m * N[(1.0 / N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[2.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999907e186

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-frac-neg 51.9%

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

      3. tan-neg 51.9%

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

      4. distribute-frac-neg2 51.9%

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

      5. distribute-lft-neg-out 51.9%

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

      6. distribute-frac-neg2 51.9%

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

      7. distribute-lft-neg-out 51.9%

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

      8. distribute-frac-neg2 51.9%

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

      9. distribute-frac-neg 51.9%

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

      10. neg-mul-1 51.9%

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

      11. *-commutative 51.9%

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

      12. associate-/l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. metadata-eval 51.6%

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

      16. sin-neg 51.6%

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

      17. distribute-frac-neg 51.6%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in x around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*r/ 60.8%

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

   7. Simplified 60.8%

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

      1. associate-*r/ 60.6%

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

      2. clear-num 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. div-inv 60.9%

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

   11. Applied egg-rr 60.9%

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

   if 9.99999999999999907e186 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.2%

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

      1. add-cube-cbrt 5.1%

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

      2. pow3 5.6%

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

   4. Applied egg-rr 5.6%

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

   5. Taylor expanded in x around 0 10.3%

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

Alternative 8: 55.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{1}{x\_m \cdot -0.5}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ 1.0 (* y_m (/ 1.0 (* x_m -0.5)))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))));
}

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 / cos((1.0d0 / (y_m * (1.0d0 / (x_m * (-0.5d0))))))
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 / Math.cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(1.0 / Float64(y_m * Float64(1.0 / Float64(x_m * -0.5))))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((1.0 / (y_m * (1.0 / (x_m * -0.5)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(1.0 / N[(y$95$m * N[(1.0 / N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.1%

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

   1. remove-double-neg 45.1%

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

   2. distribute-frac-neg 45.1%

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

   3. tan-neg 45.1%

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

   4. distribute-frac-neg2 45.1%

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

   5. distribute-lft-neg-out 45.1%

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

   6. distribute-frac-neg2 45.1%

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

   7. distribute-lft-neg-out 45.1%

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

   8. distribute-frac-neg2 45.1%

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

   9. distribute-frac-neg 45.1%

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

   10. neg-mul-1 45.1%

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

   11. *-commutative 45.1%

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

   12. associate-/l* 44.7%

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

   13. *-commutative 44.7%

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

   14. associate-/r* 44.7%

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

   15. metadata-eval 44.7%

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

   16. sin-neg 44.7%

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

   17. distribute-frac-neg 44.7%

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

3. Simplified 45.3%

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

5. Taylor expanded in x around inf 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*r/ 52.7%

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

7. Simplified 52.7%

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

   1. associate-*r/ 52.6%

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

   2. clear-num 52.7%

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

9. Applied egg-rr 52.7%

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

    1. div-inv 52.9%

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

11. Applied egg-rr 52.9%

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

Alternative 9: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(x\_m \cdot \frac{-0.5}{y\_m}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* x_m (/ -0.5 y_m)))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((x_m * (-0.5 / y_m)));
}

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 / cos((x_m * ((-0.5d0) / y_m)))
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 / Math.cos((x_m * (-0.5 / y_m)));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos((x_m * (-0.5 / y_m)))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(x_m * Float64(-0.5 / y_m))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((x_m * (-0.5 / y_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.1%

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

   1. remove-double-neg 45.1%

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

   2. distribute-frac-neg 45.1%

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

   3. tan-neg 45.1%

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

   4. distribute-frac-neg2 45.1%

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

   5. distribute-lft-neg-out 45.1%

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

   6. distribute-frac-neg2 45.1%

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

   7. distribute-lft-neg-out 45.1%

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

   8. distribute-frac-neg2 45.1%

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

   9. distribute-frac-neg 45.1%

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

   10. neg-mul-1 45.1%

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

   11. *-commutative 45.1%

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

   12. associate-/l* 44.7%

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

   13. *-commutative 44.7%

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

   14. associate-/r* 44.7%

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

   15. metadata-eval 44.7%

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

   16. sin-neg 44.7%

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

   17. distribute-frac-neg 44.7%

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

3. Simplified 45.3%

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

5. Taylor expanded in x around inf 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*r/ 52.7%

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

7. Simplified 52.7%

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

Alternative 10: 55.7% 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 45.1%

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

   1. remove-double-neg 45.1%

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

   2. distribute-frac-neg 45.1%

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

   3. tan-neg 45.1%

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

   4. distribute-frac-neg2 45.1%

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

   5. distribute-lft-neg-out 45.1%

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

   6. distribute-frac-neg2 45.1%

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

   7. distribute-lft-neg-out 45.1%

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

   8. distribute-frac-neg2 45.1%

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

   9. distribute-frac-neg 45.1%

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

   10. neg-mul-1 45.1%

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

   11. *-commutative 45.1%

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

   12. associate-/l* 44.7%

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

   13. *-commutative 44.7%

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

   14. associate-/r* 44.7%

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

   15. metadata-eval 44.7%

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

   16. sin-neg 44.7%

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

   17. distribute-frac-neg 44.7%

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

3. Simplified 45.3%

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

5. Taylor expanded in x around 0 52.2%

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

Developer Target 1: 55.7% 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 2024116 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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