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

Percentage Accurate: 44.3% → 57.2%

Time: 10.9s

Alternatives: 8

Speedup: 244.0×

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: 44.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-x\_m}{\sqrt{2 \cdot y\_m}}}{\sqrt{y\_m}} \cdot {2}^{-0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+119)
   (/
    1.0
    (cos (* (/ (/ (- x_m) (sqrt (* 2.0 y_m))) (sqrt y_m)) (pow 2.0 -0.5))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / cos((((-x_m / sqrt((2.0 * y_m))) / sqrt(y_m)) * pow(2.0, -0.5)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+119) then
        tmp = 1.0d0 / cos((((-x_m / sqrt((2.0d0 * y_m))) / sqrt(y_m)) * (2.0d0 ** (-0.5d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / Math.cos((((-x_m / Math.sqrt((2.0 * y_m))) / Math.sqrt(y_m)) * Math.pow(2.0, -0.5)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+119:
		tmp = 1.0 / math.cos((((-x_m / math.sqrt((2.0 * y_m))) / math.sqrt(y_m)) * math.pow(2.0, -0.5)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+119)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(Float64(-x_m) / sqrt(Float64(2.0 * y_m))) / sqrt(y_m)) * (2.0 ^ -0.5))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+119)
		tmp = 1.0 / cos((((-x_m / sqrt((2.0 * y_m))) / sqrt(y_m)) * (2.0 ^ -0.5)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+119], N[(1.0 / N[Cos[N[(N[(N[((-x$95$m) / N[Sqrt[N[(2.0 * y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.99999999999999989e119

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.2

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

   4. Applied rewrites 46.2%

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

   6. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. times-frac N/A

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

      15. neg-mul-1 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. pow1/2 N/A

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

      18. lift-*.f64 N/A

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

      19. unpow-prod-down N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. times-frac N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 33.4%

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

   if 1.99999999999999989e119 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.6%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-x}{\sqrt{2 \cdot y}}}{\sqrt{y}} \cdot {2}^{-0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x\_m}{\sqrt{y\_m} \cdot 2}}{-\sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 4e+150)
   (/ 1.0 (cos (/ (/ x_m (* (sqrt y_m) 2.0)) (- (sqrt y_m)))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 4e+150) {
		tmp = 1.0 / cos(((x_m / (sqrt(y_m) * 2.0)) / -sqrt(y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 4d+150) then
        tmp = 1.0d0 / cos(((x_m / (sqrt(y_m) * 2.0d0)) / -sqrt(y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 4e+150) {
		tmp = 1.0 / Math.cos(((x_m / (Math.sqrt(y_m) * 2.0)) / -Math.sqrt(y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 4e+150:
		tmp = 1.0 / math.cos(((x_m / (math.sqrt(y_m) * 2.0)) / -math.sqrt(y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 4e+150)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / Float64(sqrt(y_m) * 2.0)) / Float64(-sqrt(y_m)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 4e+150)
		tmp = 1.0 / cos(((x_m / (sqrt(y_m) * 2.0)) / -sqrt(y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 4e+150], N[(1.0 / N[Cos[N[(N[(x$95$m / N[(N[Sqrt[y$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[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))) < 3.99999999999999992e150

   1. Initial program 45.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 44.8

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

   4. Applied rewrites 44.8%

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lift-*.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 32.5%

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

   if 3.99999999999999992e150 < (/.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. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.7%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x}{\sqrt{y} \cdot 2}}{-\sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-0.5}{\sqrt{y\_m}}}{\sqrt{y\_m}} \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+119)
   (/ 1.0 (cos (* (/ (/ -0.5 (sqrt y_m)) (sqrt y_m)) x_m)))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / cos((((-0.5 / sqrt(y_m)) / sqrt(y_m)) * x_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+119) then
        tmp = 1.0d0 / cos(((((-0.5d0) / sqrt(y_m)) / sqrt(y_m)) * x_m))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / Math.cos((((-0.5 / Math.sqrt(y_m)) / Math.sqrt(y_m)) * x_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+119:
		tmp = 1.0 / math.cos((((-0.5 / math.sqrt(y_m)) / math.sqrt(y_m)) * x_m))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+119)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(-0.5 / sqrt(y_m)) / sqrt(y_m)) * x_m)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+119)
		tmp = 1.0 / cos((((-0.5 / sqrt(y_m)) / sqrt(y_m)) * x_m));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+119], N[(1.0 / N[Cos[N[(N[(N[(-0.5 / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.99999999999999989e119

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.2

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

   4. Applied rewrites 46.2%

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

   6. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lift-*.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 33.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. /-rgt-identity N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 33.5

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

   10. Applied rewrites 33.5%

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

   if 1.99999999999999989e119 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.6%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-0.5}{\sqrt{y}}}{\sqrt{y}} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot y\_m \leq 2 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\sqrt{y\_m}} \cdot \frac{x\_m}{\sqrt{y\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* 2.0 y_m) 2e-134)
   1.0
   (/ 1.0 (cos (* (/ -0.5 (sqrt y_m)) (/ x_m (sqrt y_m)))))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((2.0 * y_m) <= 2e-134) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / cos(((-0.5 / sqrt(y_m)) * (x_m / sqrt(y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((2.0d0 * y_m) <= 2d-134) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / cos((((-0.5d0) / sqrt(y_m)) * (x_m / sqrt(y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((2.0 * y_m) <= 2e-134) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / Math.cos(((-0.5 / Math.sqrt(y_m)) * (x_m / Math.sqrt(y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (2.0 * y_m) <= 2e-134:
		tmp = 1.0
	else:
		tmp = 1.0 / math.cos(((-0.5 / math.sqrt(y_m)) * (x_m / math.sqrt(y_m))))
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(2.0 * y_m) <= 2e-134)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / cos(Float64(Float64(-0.5 / sqrt(y_m)) * Float64(x_m / sqrt(y_m)))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((2.0 * y_m) <= 2e-134)
		tmp = 1.0;
	else
		tmp = 1.0 / cos(((-0.5 / sqrt(y_m)) * (x_m / sqrt(y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(2.0 * y$95$m), $MachinePrecision], 2e-134], 1.0, N[(1.0 / N[Cos[N[(N[(-0.5 / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 2 binary64)) < 2.00000000000000008e-134

   1. Initial program 35.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 43.8%

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

   if 2.00000000000000008e-134 < (*.f64 y #s(literal 2 binary64))

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 45.6

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

   4. Applied rewrites 45.6%

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

   6. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lift-*.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 67.5

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

   10. Applied rewrites 67.5%

       \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{x}{\sqrt{y}} \cdot \frac{-0.5}{\sqrt{y}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot y \leq 2 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\sqrt{y}} \cdot \frac{x}{\sqrt{y}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+119) (/ 1.0 (cos (/ -0.5 (/ y_m x_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+119) then
        tmp = 1.0d0 / cos(((-0.5d0) / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / Math.cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+119:
		tmp = 1.0 / math.cos((-0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+119)
		tmp = Float64(1.0 / cos(Float64(-0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+119)
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+119], N[(1.0 / N[Cos[N[(-0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.99999999999999989e119

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.2

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

   4. Applied rewrites 46.2%

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

   6. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if 1.99999999999999989e119 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.6%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m}{y\_m} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+119) (/ 1.0 (cos (* (/ x_m y_m) -0.5))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+119) then
        tmp = 1.0d0 / cos(((x_m / y_m) * (-0.5d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / Math.cos(((x_m / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+119:
		tmp = 1.0 / math.cos(((x_m / y_m) * -0.5))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+119)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / y_m) * -0.5)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+119)
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+119], N[(1.0 / N[Cos[N[(N[(x$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.99999999999999989e119

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. cos-neg N/A

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

      16. lift-/.f64 N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lower-cos.f64 N/A

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

      19. distribute-frac-neg2 N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 62.1%

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

   if 1.99999999999999989e119 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y\_m} \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+119) (/ 1.0 (cos (* (/ 0.5 y_m) x_m))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / cos(((0.5 / y_m) * x_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+119) then
        tmp = 1.0d0 / cos(((0.5d0 / y_m) * x_m))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+119) {
		tmp = 1.0 / Math.cos(((0.5 / y_m) * x_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+119:
		tmp = 1.0 / math.cos(((0.5 / y_m) * x_m))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+119)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / y_m) * x_m)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+119)
		tmp = 1.0 / cos(((0.5 / y_m) * x_m));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+119], N[(1.0 / N[Cos[N[(N[(0.5 / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.99999999999999989e119

   1. Initial program 47.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 1.99999999999999989e119 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.6% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

y_m = abs(y)
x_m = abs(x)
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 52.2%

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

Developer Target 1: 55.6% 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 2024268 
(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)))))