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

Percentage Accurate: 44.6% → 57.9%

Time: 14.3s

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: 44.6% 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: 57.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{0.5 \cdot \frac{x\_m}{y\_m}}}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{\cos \left({\left(t\_0 \cdot {t\_0}^{2}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt (* 0.5 (/ x_m y_m))))))
   (if (<= (/ x_m (* y_m 2.0)) 4e+33)
     (/ 1.0 (cos (pow (* t_0 (pow t_0 2.0)) 3.0)))
     (/ -0.5 (sqrt (cbrt 0.015625))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = cbrt(cbrt((0.5 * (x_m / y_m))));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+33) {
		tmp = 1.0 / cos(pow((t_0 * pow(t_0, 2.0)), 3.0));
	} else {
		tmp = -0.5 / sqrt(cbrt(0.015625));
	}
	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(Math.cbrt((0.5 * (x_m / y_m))));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+33) {
		tmp = 1.0 / Math.cos(Math.pow((t_0 * Math.pow(t_0, 2.0)), 3.0));
	} else {
		tmp = -0.5 / Math.sqrt(Math.cbrt(0.015625));
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = cbrt(cbrt(Float64(0.5 * Float64(x_m / y_m))))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 4e+33)
		tmp = Float64(1.0 / cos((Float64(t_0 * (t_0 ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(-0.5 / sqrt(cbrt(0.015625)));
	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[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 4e+33], N[(1.0 / N[Cos[N[Power[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 3.9999999999999998e33

   1. Initial program 52.9%

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

      1. remove-double-neg 52.9%

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

      2. distribute-frac-neg 52.9%

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

      3. tan-neg 52.9%

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

      4. distribute-frac-neg2 52.9%

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

      5. distribute-lft-neg-out 52.9%

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

      6. distribute-frac-neg2 52.9%

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

      7. distribute-lft-neg-out 52.9%

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

      8. distribute-frac-neg2 52.9%

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

      9. distribute-frac-neg 52.9%

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

      10. neg-mul-1 52.9%

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

      11. *-commutative 52.9%

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

      12. associate-/l* 52.7%

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

      13. *-commutative 52.7%

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

      14. associate-/r* 52.7%

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

      15. metadata-eval 52.7%

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

      16. sin-neg 52.7%

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

      17. distribute-frac-neg 52.7%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in x around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r/ 68.3%

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

   7. Simplified 68.3%

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

      1. add-cube-cbrt 68.1%

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

      2. pow3 68.3%

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

   9. Applied egg-rr 68.3%

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

   11. Applied egg-rr 68.6%

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

   if 3.9999999999999998e33 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.4%

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

      1. add-cbrt-cube 2.8%

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

      2. pow1/3 2.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \color{blue}{\left({\left(\left(\frac{x}{y \cdot 2} \cdot \frac{x}{y \cdot 2}\right) \cdot \frac{x}{y \cdot 2}\right)}^{0.3333333333333333}\right)}} \]

      3. pow3 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left({\color{blue}{\left({\left(\frac{x}{y \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333}\right)} \]

      4. *-un-lft-identity 1.4%

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

      5. *-commutative 1.4%

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

      6. times-frac 1.4%

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

      7. metadata-eval 1.4%

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}}} \]

      2. sqrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}}} \]

      3. cbrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}}} \]

      4. metadata-eval 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\sqrt[3]{\color{blue}{0.015625}}}} \]

   7. Applied egg-rr 13.5%

      \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{0.015625}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{\sqrt[3]{0.5 \cdot \frac{x}{y}}} \cdot {\left(\sqrt[3]{\sqrt[3]{0.5 \cdot \frac{x}{y}}}\right)}^{2}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(0.5 \cdot \frac{x\_m}{y\_m}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \end{array} \]

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)) 4e+37)
   (/ 1.0 (cos (exp (log (* 0.5 (/ x_m y_m))))))
   (/ -0.5 (sqrt (cbrt 0.015625)))))

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)) <= 4e+37) {
		tmp = 1.0 / cos(exp(log((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / sqrt(cbrt(0.015625));
	}
	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)) <= 4e+37) {
		tmp = 1.0 / Math.cos(Math.exp(Math.log((0.5 * (x_m / y_m)))));
	} else {
		tmp = -0.5 / Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 4e+37)
		tmp = Float64(1.0 / cos(exp(log(Float64(0.5 * Float64(x_m / y_m))))));
	else
		tmp = Float64(-0.5 / sqrt(cbrt(0.015625)));
	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], 4e+37], N[(1.0 / N[Cos[N[Exp[N[Log[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 3.99999999999999982e37

   1. Initial program 52.6%

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

      1. remove-double-neg 52.6%

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

      2. distribute-frac-neg 52.6%

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

      3. tan-neg 52.6%

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

      4. distribute-frac-neg2 52.6%

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

      5. distribute-lft-neg-out 52.6%

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

      6. distribute-frac-neg2 52.6%

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

      7. distribute-lft-neg-out 52.6%

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

      8. distribute-frac-neg2 52.6%

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

      9. distribute-frac-neg 52.6%

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

      10. neg-mul-1 52.6%

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

      11. *-commutative 52.6%

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

      12. associate-/l* 52.5%

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

      13. *-commutative 52.5%

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

      14. associate-/r* 52.5%

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

      15. metadata-eval 52.5%

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

      16. sin-neg 52.5%

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

      17. distribute-frac-neg 52.5%

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

   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 67.9%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r/ 67.9%

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

   7. Simplified 67.9%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 68.0%

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

   9. Applied egg-rr 68.0%

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

       1. rem-cube-cbrt 67.9%

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

       2. add-sqr-sqrt 43.0%

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

       3. sqrt-unprod 66.8%

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

       4. swap-sqr 56.5%

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

       5. frac-times 56.6%

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

       6. metadata-eval 56.6%

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

       7. unpow2 56.6%

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

       8. add-sqr-sqrt 56.5%

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

       9. swap-sqr 62.7%

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

       10. sqrt-unprod 47.6%

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

       11. add-sqr-sqrt 63.6%

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

       12. sqrt-div 64.1%

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

       13. metadata-eval 64.1%

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

       14. metadata-eval 64.1%

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

       15. sqrt-pow1 67.9%

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

       16. metadata-eval 67.9%

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

       17. pow1 67.9%

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

       18. associate-/r* 67.9%

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

       19. *-commutative 67.9%

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

       20. add-exp-log 40.2%

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

       21. div-inv 40.2%

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

   11. Applied egg-rr 40.2%

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

   if 3.99999999999999982e37 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.5%

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

      1. add-cbrt-cube 2.4%

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

      2. pow1/3 2.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \color{blue}{\left({\left(\left(\frac{x}{y \cdot 2} \cdot \frac{x}{y \cdot 2}\right) \cdot \frac{x}{y \cdot 2}\right)}^{0.3333333333333333}\right)}} \]

      3. pow3 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left({\color{blue}{\left({\left(\frac{x}{y \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333}\right)} \]

      4. *-un-lft-identity 1.4%

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

      5. *-commutative 1.4%

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

      6. times-frac 1.4%

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

      7. metadata-eval 1.4%

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in y around -inf 9.5%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}}} \]

      2. sqrt-unprod 13.2%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}}} \]

      3. cbrt-unprod 13.2%

         \[\leadsto \frac{-0.5}{\sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}}} \]

      4. metadata-eval 13.2%

         \[\leadsto \frac{-0.5}{\sqrt{\sqrt[3]{\color{blue}{0.015625}}}} \]

   7. Applied egg-rr 13.2%

      \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{0.015625}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(\frac{x\_m}{y\_m} \cdot -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \end{array} \]

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)) 2.2e+33)
   (log1p (expm1 (/ 1.0 (cos (* (/ x_m y_m) -0.5)))))
   (/ -0.5 (sqrt (cbrt 0.015625)))))

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)) <= 2.2e+33) {
		tmp = log1p(expm1((1.0 / cos(((x_m / y_m) * -0.5)))));
	} else {
		tmp = -0.5 / sqrt(cbrt(0.015625));
	}
	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)) <= 2.2e+33) {
		tmp = Math.log1p(Math.expm1((1.0 / Math.cos(((x_m / y_m) * -0.5)))));
	} else {
		tmp = -0.5 / Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 2.2e+33)
		tmp = log1p(expm1(Float64(1.0 / cos(Float64(Float64(x_m / y_m) * -0.5)))));
	else
		tmp = Float64(-0.5 / sqrt(cbrt(0.015625)));
	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], 2.2e+33], N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(N[(x$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.5 / N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.19999999999999994e33

   1. Initial program 52.9%

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

      1. remove-double-neg 52.9%

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

      2. distribute-frac-neg 52.9%

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

      3. tan-neg 52.9%

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

      4. distribute-frac-neg2 52.9%

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

      5. distribute-lft-neg-out 52.9%

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

      6. distribute-frac-neg2 52.9%

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

      7. distribute-lft-neg-out 52.9%

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

      8. distribute-frac-neg2 52.9%

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

      9. distribute-frac-neg 52.9%

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

      10. neg-mul-1 52.9%

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

      11. *-commutative 52.9%

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

      12. associate-/l* 52.7%

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

      13. *-commutative 52.7%

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

      14. associate-/r* 52.7%

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

      15. metadata-eval 52.7%

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

      16. sin-neg 52.7%

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

      17. distribute-frac-neg 52.7%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in x around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r/ 68.3%

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

   7. Simplified 68.3%

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

   9. Applied egg-rr 68.2%

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

   if 2.19999999999999994e33 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.4%

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

      1. add-cbrt-cube 2.8%

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

      2. pow1/3 2.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \color{blue}{\left({\left(\left(\frac{x}{y \cdot 2} \cdot \frac{x}{y \cdot 2}\right) \cdot \frac{x}{y \cdot 2}\right)}^{0.3333333333333333}\right)}} \]

      3. pow3 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left({\color{blue}{\left({\left(\frac{x}{y \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333}\right)} \]

      4. *-un-lft-identity 1.4%

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

      5. *-commutative 1.4%

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

      6. times-frac 1.4%

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

      7. metadata-eval 1.4%

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}}} \]

      2. sqrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}}} \]

      3. cbrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}}} \]

      4. metadata-eval 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\sqrt[3]{\color{blue}{0.015625}}}} \]

   7. Applied egg-rr 13.5%

      \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{0.015625}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{x\_m \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}}{y\_m}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ (* x_m (pow (cbrt -0.5) 3.0)) 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 * pow(cbrt(-0.5), 3.0)) / y_m));
}

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 * Math.pow(Math.cbrt(-0.5), 3.0)) / y_m));
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(Float64(x_m * (cbrt(-0.5) ^ 3.0)) / 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[(N[(x$95$m * N[Power[N[Power[-0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

   1. remove-double-neg 44.3%

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

   2. distribute-frac-neg 44.3%

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

   3. tan-neg 44.3%

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

   4. distribute-frac-neg2 44.3%

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

   5. distribute-lft-neg-out 44.3%

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

   6. distribute-frac-neg2 44.3%

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

   7. distribute-lft-neg-out 44.3%

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

   8. distribute-frac-neg2 44.3%

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

   9. distribute-frac-neg 44.3%

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

   10. neg-mul-1 44.3%

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

   11. *-commutative 44.3%

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

   12. associate-/l* 44.5%

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

   13. *-commutative 44.5%

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

   14. associate-/r* 44.5%

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

   15. metadata-eval 44.5%

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

   16. sin-neg 44.5%

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

   17. distribute-frac-neg 44.5%

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

3. Simplified 44.7%

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

5. Taylor expanded in x around inf 56.8%

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

   1. associate-*r/ 56.8%

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

   2. *-commutative 56.8%

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

   3. associate-*r/ 57.2%

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

7. Simplified 57.2%

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

   1. add-cube-cbrt 57.2%

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

   2. pow3 57.4%

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

9. Applied egg-rr 57.4%

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

10. Taylor expanded in x around inf 57.9%

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

Alternative 5: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x\_m}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\sqrt{\sqrt[3]{0.015625}}}\\ \end{array} \end{array} \]

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)) 2.2e+33)
   (/ 1.0 (cos (* 0.5 (/ x_m y_m))))
   (/ -0.5 (sqrt (cbrt 0.015625)))))

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)) <= 2.2e+33) {
		tmp = 1.0 / cos((0.5 * (x_m / y_m)));
	} else {
		tmp = -0.5 / sqrt(cbrt(0.015625));
	}
	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)) <= 2.2e+33) {
		tmp = 1.0 / Math.cos((0.5 * (x_m / y_m)));
	} else {
		tmp = -0.5 / Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 2.2e+33)
		tmp = Float64(1.0 / cos(Float64(0.5 * Float64(x_m / y_m))));
	else
		tmp = Float64(-0.5 / sqrt(cbrt(0.015625)));
	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], 2.2e+33], N[(1.0 / N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.19999999999999994e33

   1. Initial program 52.9%

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

   3. Taylor expanded in x around inf 68.2%

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

   if 2.19999999999999994e33 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.4%

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

      1. add-cbrt-cube 2.8%

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

      2. pow1/3 2.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \color{blue}{\left({\left(\left(\frac{x}{y \cdot 2} \cdot \frac{x}{y \cdot 2}\right) \cdot \frac{x}{y \cdot 2}\right)}^{0.3333333333333333}\right)}} \]

      3. pow3 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left({\color{blue}{\left({\left(\frac{x}{y \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333}\right)} \]

      4. *-un-lft-identity 1.4%

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

      5. *-commutative 1.4%

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

      6. times-frac 1.4%

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

      7. metadata-eval 1.4%

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in y around -inf 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\sqrt[3]{-0.125}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}}} \]

      2. sqrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}}} \]

      3. cbrt-unprod 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}}} \]

      4. metadata-eval 13.5%

         \[\leadsto \frac{-0.5}{\sqrt{\sqrt[3]{\color{blue}{0.015625}}}} \]

   7. Applied egg-rr 13.5%

      \[\leadsto \frac{-0.5}{\color{blue}{\sqrt{\sqrt[3]{0.015625}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{2}{x\_m}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ 1.0 (* y_m (/ 2.0 x_m))))))

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 * (2.0 / x_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((1.0d0 / (y_m * (2.0d0 / x_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((1.0 / (y_m * (2.0 / x_m))));
}

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 * (2.0 / x_m))))

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(2.0 / x_m)))))
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 * (2.0 / x_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[(1.0 / N[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

   1. remove-double-neg 44.3%

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

   2. distribute-frac-neg 44.3%

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

   3. tan-neg 44.3%

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

   4. distribute-frac-neg2 44.3%

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

   5. distribute-lft-neg-out 44.3%

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

   6. distribute-frac-neg2 44.3%

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

   7. distribute-lft-neg-out 44.3%

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

   8. distribute-frac-neg2 44.3%

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

   9. distribute-frac-neg 44.3%

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

   10. neg-mul-1 44.3%

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

   11. *-commutative 44.3%

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

   12. associate-/l* 44.5%

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

   13. *-commutative 44.5%

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

   14. associate-/r* 44.5%

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

   15. metadata-eval 44.5%

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

   16. sin-neg 44.5%

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

   17. distribute-frac-neg 44.5%

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

3. Simplified 44.7%

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

5. Taylor expanded in x around inf 56.8%

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

   1. associate-*r/ 56.8%

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

   2. *-commutative 56.8%

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

   3. associate-*r/ 57.2%

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

7. Simplified 57.2%

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

   1. add-cube-cbrt 57.2%

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

   2. pow3 57.4%

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

9. Applied egg-rr 57.4%

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

    1. rem-cube-cbrt 57.2%

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

    2. add-sqr-sqrt 35.3%

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

    3. sqrt-unprod 55.3%

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

    4. swap-sqr 46.5%

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

    5. frac-times 46.6%

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

    6. metadata-eval 46.6%

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

    7. unpow2 46.6%

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

    8. add-sqr-sqrt 46.5%

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

    9. swap-sqr 51.7%

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

    10. sqrt-unprod 39.2%

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

    11. add-sqr-sqrt 52.9%

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

    12. sqrt-div 53.3%

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

    13. metadata-eval 53.3%

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

    14. metadata-eval 53.3%

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

    15. sqrt-pow1 57.2%

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

    16. metadata-eval 57.2%

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

    17. pow1 57.2%

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

    18. associate-/r* 57.2%

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

    19. *-commutative 57.2%

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

    20. div-inv 56.8%

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

    21. clear-num 56.9%

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

    22. associate-/l* 57.2%

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

11. Applied egg-rr 57.2%

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

Alternative 7: 56.5% 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 44.3%

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

   1. remove-double-neg 44.3%

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

   2. distribute-frac-neg 44.3%

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

   3. tan-neg 44.3%

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

   4. distribute-frac-neg2 44.3%

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

   5. distribute-lft-neg-out 44.3%

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

   6. distribute-frac-neg2 44.3%

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

   7. distribute-lft-neg-out 44.3%

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

   8. distribute-frac-neg2 44.3%

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

   9. distribute-frac-neg 44.3%

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

   10. neg-mul-1 44.3%

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

   11. *-commutative 44.3%

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

   12. associate-/l* 44.5%

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

   13. *-commutative 44.5%

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

   14. associate-/r* 44.5%

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

   15. metadata-eval 44.5%

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

   16. sin-neg 44.5%

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

   17. distribute-frac-neg 44.5%

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

3. Simplified 44.7%

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

5. Taylor expanded in x around inf 56.8%

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

   1. associate-*r/ 56.8%

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

   2. *-commutative 56.8%

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

   3. associate-*r/ 57.2%

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

7. Simplified 57.2%

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

Alternative 8: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((0.5 * (x_m / 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((0.5d0 * (x_m / 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((0.5 * (x_m / y_m)));
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((0.5 * (x_m / 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[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

3. Taylor expanded in x around inf 56.8%

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

Alternative 9: 56.3% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.3%

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

   1. remove-double-neg 44.3%

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

   2. distribute-frac-neg 44.3%

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

   3. tan-neg 44.3%

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

   4. distribute-frac-neg2 44.3%

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

   5. distribute-lft-neg-out 44.3%

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

   6. distribute-frac-neg2 44.3%

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

   7. distribute-lft-neg-out 44.3%

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

   8. distribute-frac-neg2 44.3%

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

   9. distribute-frac-neg 44.3%

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

   10. neg-mul-1 44.3%

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

   11. *-commutative 44.3%

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

   12. associate-/l* 44.5%

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

   13. *-commutative 44.5%

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

   14. associate-/r* 44.5%

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

   15. metadata-eval 44.5%

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

   16. sin-neg 44.5%

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

   17. distribute-frac-neg 44.5%

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

3. Simplified 44.7%

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

5. Taylor expanded in x around 0 56.7%

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

Alternative 10: 6.5% 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 44.3%

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

3. Taylor expanded in x around inf 44.1%

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

   1. *-commutative 44.1%

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

   2. associate-*l/ 44.3%

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

   3. associate-*r/ 44.4%

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

   4. *-commutative 44.4%

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

   5. associate-*l/ 44.4%

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

   6. associate-*r/ 44.5%

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

5. Simplified 44.5%

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

   1. metadata-eval 44.5%

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

   2. associate-/r* 44.5%

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

   3. *-commutative 44.5%

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

   4. div-inv 44.7%

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

   5. add-cube-cbrt 43.5%

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

   6. pow3 44.0%

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

7. Applied egg-rr 3.8%

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

8. Taylor expanded in x around 0 6.8%

   \[\leadsto \color{blue}{2 \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 6.8%

      \[\leadsto 2 \cdot \color{blue}{-0.5} \]

   2. metadata-eval 6.8%

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

10. Simplified 6.8%

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

Developer Target 1: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t\_1}{t\_1 \cdot \log \left(e^{\cos t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024133 
(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)))))