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

Percentage Accurate: 43.7% → 55.9%

Time: 12.3s

Alternatives: 5

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: 43.7% 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: 55.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{x\_m \cdot 0.5}}{\sqrt{y\_m}}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{1}{\cos \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* x_m 0.5)) (sqrt y_m))))
   (if (<= (/ x_m (* y_m 2.0)) 5e+240) (/ 1.0 (cos (* t_0 t_0))) 1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = sqrt((x_m * 0.5)) / sqrt(y_m);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+240) {
		tmp = 1.0 / cos((t_0 * t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x_m * 0.5d0)) / sqrt(y_m)
    if ((x_m / (y_m * 2.0d0)) <= 5d+240) then
        tmp = 1.0d0 / cos((t_0 * t_0))
    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 t_0 = Math.sqrt((x_m * 0.5)) / Math.sqrt(y_m);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+240) {
		tmp = 1.0 / Math.cos((t_0 * t_0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	t_0 = math.sqrt((x_m * 0.5)) / math.sqrt(y_m)
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+240:
		tmp = 1.0 / math.cos((t_0 * t_0))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(sqrt(Float64(x_m * 0.5)) / sqrt(y_m))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+240)
		tmp = Float64(1.0 / cos(Float64(t_0 * t_0)));
	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)
	t_0 = sqrt((x_m * 0.5)) / sqrt(y_m);
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+240)
		tmp = 1.0 / cos((t_0 * t_0));
	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_] := Block[{t$95$0 = N[(N[Sqrt[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+240], N[(1.0 / N[Cos[N[(t$95$0 * 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))) < 5.0000000000000003e240

   1. Initial program 45.8%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. lower-cos.f64 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. sqrt-undiv N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. distribute-frac-neg N/A

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

      19. distribute-frac-neg N/A

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

      20. sqr-neg N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 18.6%

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

   if 5.0000000000000003e240 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 12.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt{x \cdot 0.5}}{\sqrt{y}} \cdot \frac{\sqrt{x \cdot 0.5}}{\sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{x\_m \cdot 0.5}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{t\_0 \cdot t\_0}{\sqrt{y\_m} \cdot \sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (* x_m 0.5))))
   (if (<= (/ x_m (* y_m 2.0)) 5e+189)
     (/ 1.0 (cos (/ (* t_0 t_0) (* (sqrt y_m) (sqrt y_m)))))
     1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = sqrt((x_m * 0.5));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+189) {
		tmp = 1.0 / cos(((t_0 * t_0) / (sqrt(y_m) * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x_m * 0.5d0))
    if ((x_m / (y_m * 2.0d0)) <= 5d+189) then
        tmp = 1.0d0 / cos(((t_0 * t_0) / (sqrt(y_m) * 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 t_0 = Math.sqrt((x_m * 0.5));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+189) {
		tmp = 1.0 / Math.cos(((t_0 * t_0) / (Math.sqrt(y_m) * 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):
	t_0 = math.sqrt((x_m * 0.5))
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+189:
		tmp = 1.0 / math.cos(((t_0 * t_0) / (math.sqrt(y_m) * 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)
	t_0 = sqrt(Float64(x_m * 0.5))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+189)
		tmp = Float64(1.0 / cos(Float64(Float64(t_0 * t_0) / Float64(sqrt(y_m) * 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)
	t_0 = sqrt((x_m * 0.5));
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+189)
		tmp = 1.0 / cos(((t_0 * t_0) / (sqrt(y_m) * 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_] := Block[{t$95$0 = N[Sqrt[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+189], N[(1.0 / N[Cos[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.0000000000000004e189

   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

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

      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. lower-cos.f64 62.6

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. sqrt-undiv N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-times N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 19.1%

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

   if 5.0000000000000004e189 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.7%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt{x \cdot 0.5} \cdot \sqrt{x \cdot 0.5}}{\sqrt{y} \cdot \sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{y\_m \cdot 2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+240)
   (/ 1.0 (cos (/ 1.0 (/ (* y_m 2.0) 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 / (y_m * 2.0)) <= 5e+240) {
		tmp = 1.0 / cos((1.0 / ((y_m * 2.0) / 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 / (y_m * 2.0d0)) <= 5d+240) then
        tmp = 1.0d0 / cos((1.0d0 / ((y_m * 2.0d0) / 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 / (y_m * 2.0)) <= 5e+240) {
		tmp = 1.0 / Math.cos((1.0 / ((y_m * 2.0) / 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 / (y_m * 2.0)) <= 5e+240:
		tmp = 1.0 / math.cos((1.0 / ((y_m * 2.0) / 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(y_m * 2.0)) <= 5e+240)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(Float64(y_m * 2.0) / 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 / (y_m * 2.0)) <= 5e+240)
		tmp = 1.0 / cos((1.0 / ((y_m * 2.0) / 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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+240], N[(1.0 / N[Cos[N[(1.0 / N[(N[(y$95$m * 2.0), $MachinePrecision] / 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))) < 5.0000000000000003e240

   1. Initial program 45.8%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. lower-cos.f64 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.8

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

   6. Applied rewrites 61.8%

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

   if 5.0000000000000003e240 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 12.6%

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

Alternative 4: 56.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+189) (/ 1.0 (cos (* x_m (/ 0.5 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 / (y_m * 2.0)) <= 5e+189) {
		tmp = 1.0 / cos((x_m * (0.5 / 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 / (y_m * 2.0d0)) <= 5d+189) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / 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 / (y_m * 2.0)) <= 5e+189) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / 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 / (y_m * 2.0)) <= 5e+189:
		tmp = 1.0 / math.cos((x_m * (0.5 / 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(y_m * 2.0)) <= 5e+189)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / 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 / (y_m * 2.0)) <= 5e+189)
		tmp = 1.0 / cos((x_m * (0.5 / 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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+189], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.0000000000000004e189

   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

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

      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. lower-cos.f64 62.6

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 62.9

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

   6. Applied rewrites 62.9%

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

   if 5.0000000000000004e189 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.7%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.6% 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 41.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.5%

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

Developer Target 1: 54.6% 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 2024232 
(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)))))