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

Percentage Accurate: 44.7% → 57.1%

Time: 20.1s

Alternatives: 7

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.7% 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: 57.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{y\_m \cdot 2}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x\_m}{t\_0}}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (* y_m 2.0))))
   (if (<= (/ x_m (* y_m 2.0)) 5e+259) (/ 1.0 (cos (/ (/ x_m t_0) t_0))) 1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = sqrt((y_m * 2.0));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+259) {
		tmp = 1.0 / cos(((x_m / t_0) / t_0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((y_m * 2.0d0))
    if ((x_m / (y_m * 2.0d0)) <= 5d+259) then
        tmp = 1.0d0 / cos(((x_m / t_0) / t_0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = Math.sqrt((y_m * 2.0));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+259) {
		tmp = 1.0 / Math.cos(((x_m / t_0) / t_0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.sqrt((y_m * 2.0))
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+259:
		tmp = 1.0 / math.cos(((x_m / t_0) / t_0))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = sqrt(Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+259)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / t_0) / t_0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	t_0 = sqrt((y_m * 2.0));
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+259)
		tmp = 1.0 / cos(((x_m / t_0) / t_0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[Sqrt[N[(y$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+259], N[(1.0 / N[Cos[N[(N[(x$95$m / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000033e259

   1. Initial program 44.2%

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

      1. add-log-exp 8.5%

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

      2. *-un-lft-identity 8.5%

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

      3. *-commutative 8.5%

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

      4. times-frac 8.5%

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

      5. metadata-eval 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. *-commutative 58.2%

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

      3. associate-*r/ 58.2%

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

   7. Simplified 58.2%

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

      1. metadata-eval 58.2%

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

      2. associate-/r* 58.2%

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

      3. *-commutative 58.2%

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

      4. div-inv 58.2%

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

      5. add-sqr-sqrt 23.9%

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

      6. associate-/r* 23.8%

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

   9. Applied egg-rr 23.8%

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

   if 5.00000000000000033e259 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.8%

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

      1. remove-double-neg 2.8%

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

      2. distribute-frac-neg 2.8%

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

      3. tan-neg 2.8%

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

      4. distribute-frac-neg2 2.8%

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

      5. distribute-lft-neg-out 2.8%

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

      6. distribute-frac-neg2 2.8%

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

      7. distribute-lft-neg-out 2.8%

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

      8. distribute-frac-neg2 2.8%

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

      9. distribute-frac-neg 2.8%

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

      10. neg-mul-1 2.8%

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

      11. *-commutative 2.8%

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

      12. associate-/l* 1.2%

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

      13. *-commutative 1.2%

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

      14. associate-/r* 1.2%

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

      15. metadata-eval 1.2%

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

      16. sin-neg 1.2%

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

      17. distribute-frac-neg 1.2%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in x around 0 15.8%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x}{\sqrt{y \cdot 2}}}{\sqrt{y \cdot 2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{{\left(\frac{-0.5}{\frac{y\_m}{x\_m}}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+54)
   (/ 1.0 (cos (cbrt (pow (/ -0.5 (/ y_m x_m)) 3.0))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+54) {
		tmp = 1.0 / cos(cbrt(pow((-0.5 / (y_m / x_m)), 3.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+54) {
		tmp = 1.0 / Math.cos(Math.cbrt(Math.pow((-0.5 / (y_m / x_m)), 3.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+54)
		tmp = Float64(1.0 / cos(cbrt((Float64(-0.5 / Float64(y_m / x_m)) ^ 3.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+54], N[(1.0 / N[Cos[N[Power[N[Power[N[(-0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000005e54

   1. Initial program 49.7%

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

      1. add-log-exp 8.4%

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

      2. *-un-lft-identity 8.4%

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

      3. *-commutative 8.4%

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

      4. times-frac 8.4%

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

      5. metadata-eval 8.4%

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

   4. Applied egg-rr 8.4%

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

   5. Taylor expanded in x around inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*r/ 65.5%

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

   7. Simplified 65.5%

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

      1. associate-*r/ 65.9%

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

      2. associate-*l/ 65.9%

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

      3. *-commutative 65.9%

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

      4. add-cbrt-cube 64.0%

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

      5. pow3 63.8%

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

      6. add-sqr-sqrt 42.1%

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

      7. sqrt-unprod 63.8%

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

      8. swap-sqr 63.8%

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

      9. metadata-eval 63.8%

         \[\leadsto \frac{1}{\cos \left(\sqrt[3]{{\left(\sqrt{\color{blue}{0.25} \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)}\right)}^{3}}\right)} \]

      10. metadata-eval 63.8%

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

      11. swap-sqr 63.8%

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

      12. sqrt-unprod 38.0%

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

      13. add-sqr-sqrt 63.8%

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

      14. clear-num 63.9%

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

      15. un-div-inv 63.9%

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

   9. Applied egg-rr 63.9%

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

   if 5.00000000000000005e54 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.7%

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

      1. remove-double-neg 6.7%

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

      2. distribute-frac-neg 6.7%

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

      3. tan-neg 6.7%

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

      4. distribute-frac-neg2 6.7%

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

      5. distribute-lft-neg-out 6.7%

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

      6. distribute-frac-neg2 6.7%

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

      7. distribute-lft-neg-out 6.7%

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

      8. distribute-frac-neg2 6.7%

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

      9. distribute-frac-neg 6.7%

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

      10. neg-mul-1 6.7%

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

      11. *-commutative 6.7%

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

      12. associate-/l* 6.6%

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

      13. *-commutative 6.6%

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

      14. associate-/r* 6.6%

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

      15. metadata-eval 6.6%

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

      16. sin-neg 6.6%

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

      17. distribute-frac-neg 6.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in x around 0 13.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{{\left(\frac{-0.5}{\frac{y}{x}}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+135}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+135)
   (/ 1.0 (cos (exp (log (* x_m (/ 0.5 y_m))))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+135) {
		tmp = 1.0 / cos(exp(log((x_m * (0.5 / y_m)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+135) then
        tmp = 1.0d0 / cos(exp(log((x_m * (0.5d0 / y_m)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+135) {
		tmp = 1.0 / Math.cos(Math.exp(Math.log((x_m * (0.5 / y_m)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+135:
		tmp = 1.0 / math.cos(math.exp(math.log((x_m * (0.5 / y_m)))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+135)
		tmp = Float64(1.0 / cos(exp(log(Float64(x_m * Float64(0.5 / y_m))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+135)
		tmp = 1.0 / cos(exp(log((x_m * (0.5 / y_m)))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+135], N[(1.0 / N[Cos[N[Exp[N[Log[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999962e134

   1. Initial program 46.4%

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

      1. add-log-exp 8.3%

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

      2. *-un-lft-identity 8.3%

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

      3. *-commutative 8.3%

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

      4. times-frac 8.3%

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

      5. metadata-eval 8.3%

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in x around inf 61.4%

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

      1. associate-*r/ 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-*r/ 61.3%

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

   7. Simplified 61.3%

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

      1. metadata-eval 61.3%

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

      2. associate-/r* 61.3%

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

      3. *-commutative 61.3%

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

      4. div-inv 61.4%

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

      5. add-sqr-sqrt 25.1%

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

      6. associate-/r* 25.1%

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

   9. Applied egg-rr 25.1%

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

       1. associate-/l/ 25.1%

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

       2. add-sqr-sqrt 61.4%

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

       3. *-un-lft-identity 61.4%

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

       4. *-commutative 61.4%

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

       5. times-frac 61.4%

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

       6. metadata-eval 61.4%

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

       7. clear-num 61.2%

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

       8. div-inv 61.2%

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

       9. add-exp-log 40.5%

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

       10. associate-/r/ 40.1%

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

   11. Applied egg-rr 40.1%

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

   if 9.99999999999999962e134 < (/.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. Step-by-step derivation

      1. remove-double-neg 6.4%

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

      2. distribute-frac-neg 6.4%

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

      3. tan-neg 6.4%

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

      4. distribute-frac-neg2 6.4%

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

      5. distribute-lft-neg-out 6.4%

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

      6. distribute-frac-neg2 6.4%

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

      7. distribute-lft-neg-out 6.4%

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

      8. distribute-frac-neg2 6.4%

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

      9. distribute-frac-neg 6.4%

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

      10. neg-mul-1 6.4%

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

      11. *-commutative 6.4%

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

      12. associate-/l* 5.8%

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

      13. *-commutative 5.8%

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

      14. associate-/r* 5.8%

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

      15. metadata-eval 5.8%

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

      16. sin-neg 5.8%

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

      17. distribute-frac-neg 5.8%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0 13.6%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+135}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(x \cdot \frac{0.5}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+183}:\\ \;\;\;\;\frac{1}{\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x\_m \cdot 0.5}{y\_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+183)
   (/ 1.0 (cos (expm1 (log1p (/ (* x_m 0.5) y_m)))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+183) {
		tmp = 1.0 / cos(expm1(log1p(((x_m * 0.5) / y_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+183) {
		tmp = 1.0 / Math.cos(Math.expm1(Math.log1p(((x_m * 0.5) / y_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+183:
		tmp = 1.0 / math.cos(math.expm1(math.log1p(((x_m * 0.5) / y_m))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+183)
		tmp = Float64(1.0 / cos(expm1(log1p(Float64(Float64(x_m * 0.5) / y_m)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+183], N[(1.0 / N[Cos[N[(Exp[N[Log[1 + N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999947e182

   1. Initial program 45.9%

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

      1. add-log-exp 8.5%

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

      2. *-un-lft-identity 8.5%

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

      3. *-commutative 8.5%

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

      4. times-frac 8.5%

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

      5. metadata-eval 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*r/ 60.5%

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

   7. Simplified 60.5%

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

      1. metadata-eval 60.5%

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

      2. associate-/r* 60.5%

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

      3. *-commutative 60.5%

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

      4. div-inv 60.5%

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

      5. add-sqr-sqrt 24.8%

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

      6. associate-/r* 24.7%

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

   9. Applied egg-rr 24.7%

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

       1. associate-/l/ 24.8%

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

       2. add-sqr-sqrt 60.5%

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

       3. expm1-log1p-u 58.2%

          \[\leadsto \frac{1}{\cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

       4. expm1-undefine 58.1%

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

       5. *-un-lft-identity 58.1%

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

       6. *-commutative 58.1%

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

       7. times-frac 58.1%

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

       8. metadata-eval 58.1%

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

       9. clear-num 58.1%

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

       10. div-inv 58.1%

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

       11. associate-/r/ 58.1%

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

   11. Applied egg-rr 58.1%

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

       1. expm1-define 58.2%

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

       2. associate-*l/ 58.2%

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

       3. *-commutative 58.2%

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

   13. Simplified 58.2%

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

   if 9.99999999999999947e182 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.1%

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

      1. remove-double-neg 5.1%

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

      2. distribute-frac-neg 5.1%

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

      3. tan-neg 5.1%

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

      4. distribute-frac-neg2 5.1%

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

      5. distribute-lft-neg-out 5.1%

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

      6. distribute-frac-neg2 5.1%

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

      7. distribute-lft-neg-out 5.1%

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

      8. distribute-frac-neg2 5.1%

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

      9. distribute-frac-neg 5.1%

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

      10. neg-mul-1 5.1%

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

      11. *-commutative 5.1%

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

      12. associate-/l* 4.5%

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

      13. *-commutative 4.5%

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

      14. associate-/r* 4.5%

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

      15. metadata-eval 4.5%

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

      16. sin-neg 4.5%

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

      17. distribute-frac-neg 4.5%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around 0 14.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+183}:\\ \;\;\;\;\frac{1}{\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 0.5}{y}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x\_m}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 8e+60) (/ 1.0 (cos (* 0.5 (/ x_m y_m)))) 1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 8e+60) {
		tmp = 1.0 / cos((0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 8d+60) then
        tmp = 1.0d0 / cos((0.5d0 * (x_m / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 8e+60) {
		tmp = 1.0 / Math.cos((0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 8e+60:
		tmp = 1.0 / math.cos((0.5 * (x_m / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 8e+60)
		tmp = Float64(1.0 / cos(Float64(0.5 * Float64(x_m / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 8e+60)
		tmp = 1.0 / cos((0.5 * (x_m / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 8e+60], N[(1.0 / N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 7.9999999999999996e60

   1. Initial program 49.6%

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

   3. Taylor expanded in x around inf 65.7%

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

   if 7.9999999999999996e60 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.3%

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

      1. remove-double-neg 6.3%

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

      2. distribute-frac-neg 6.3%

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

      3. tan-neg 6.3%

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

      4. distribute-frac-neg2 6.3%

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

      5. distribute-lft-neg-out 6.3%

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

      6. distribute-frac-neg2 6.3%

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

      7. distribute-lft-neg-out 6.3%

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

      8. distribute-frac-neg2 6.3%

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

      9. distribute-frac-neg 6.3%

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

      10. neg-mul-1 6.3%

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

      11. *-commutative 6.3%

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

      12. associate-/l* 6.2%

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

      13. *-commutative 6.2%

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

      14. associate-/r* 6.2%

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

      15. metadata-eval 6.2%

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

      16. sin-neg 6.2%

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

      17. distribute-frac-neg 6.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in x around 0 13.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 40.9%

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

   1. add-cube-cbrt 40.3%

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

   2. pow3 40.3%

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

   3. *-un-lft-identity 40.3%

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

   4. *-commutative 40.3%

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

   5. times-frac 40.3%

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

   6. metadata-eval 40.3%

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

4. Applied egg-rr 40.3%

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

6. Applied egg-rr 4.4%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin \left(\frac{-0.5}{\frac{y}{x}}\right)}\right)} - 1\right)}}^{3}} \]
7. Step-by-step derivation

   1. expm1-define 4.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \left(\frac{-0.5}{\frac{y}{x}}\right)}\right)\right)\right)}}^{3}} \]

   2. associate-/r/ 4.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \color{blue}{\left(\frac{-0.5}{y} \cdot x\right)}}\right)\right)\right)}^{3}} \]

   3. associate-*l/ 4.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \color{blue}{\left(\frac{-0.5 \cdot x}{y}\right)}}\right)\right)\right)}^{3}} \]

   4. associate-*r/ 4.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \color{blue}{\left(-0.5 \cdot \frac{x}{y}\right)}}\right)\right)\right)}^{3}} \]

   5. associate-*r/ 4.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \color{blue}{\left(\frac{-0.5 \cdot x}{y}\right)}}\right)\right)\right)}^{3}} \]

   6. *-commutative 4.3%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \left(\frac{\color{blue}{x \cdot -0.5}}{y}\right)}\right)\right)\right)}^{3}} \]

   7. associate-/l* 4.2%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \color{blue}{\left(x \cdot \frac{-0.5}{y}\right)}}\right)\right)\right)}^{3}} \]

8. Simplified 4.2%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \left(x \cdot \frac{-0.5}{y}\right)}\right)\right)\right)}}^{3}} \]

9. Taylor expanded in x around 0 6.3%

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

10. Final simplification 6.3%

    \[\leadsto -1 \]
11. Add Preprocessing

Alternative 7: 55.5% 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 40.9%

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

   1. remove-double-neg 40.9%

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

   2. distribute-frac-neg 40.9%

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

   3. tan-neg 40.9%

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

   4. distribute-frac-neg2 40.9%

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

   5. distribute-lft-neg-out 40.9%

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

   6. distribute-frac-neg2 40.9%

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

   7. distribute-lft-neg-out 40.9%

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

   8. distribute-frac-neg2 40.9%

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

   9. distribute-frac-neg 40.9%

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

   10. neg-mul-1 40.9%

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

   11. *-commutative 40.9%

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

   12. associate-/l* 40.6%

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

   13. *-commutative 40.6%

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

   14. associate-/r* 40.6%

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

   15. metadata-eval 40.6%

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

   16. sin-neg 40.6%

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

   17. distribute-frac-neg 40.6%

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

3. Simplified 40.9%

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

5. Taylor expanded in x around 0 54.0%

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

6. Final simplification 54.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 55.5% 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 2024075 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

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