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

Percentage Accurate: 43.5% → 56.1%

Time: 10.5s

Alternatives: 7

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: 43.5% 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: 56.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x}{2 \cdot y\_m}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-1}{\sqrt{2 \cdot y\_m}} \cdot \frac{{2}^{-0.5}}{\frac{\sqrt{y\_m}}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ x (* 2.0 y_m))))
   (if (<= (/ (tan t_0) (sin t_0)) 20.0)
     (/
      1.0
      (cos
       (* (/ -1.0 (sqrt (* 2.0 y_m))) (/ (pow 2.0 -0.5) (/ (sqrt y_m) x)))))
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = x / (2.0 * y_m);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 20.0) {
		tmp = 1.0 / cos(((-1.0 / sqrt((2.0 * y_m))) * (pow(2.0, -0.5) / (sqrt(y_m) / x))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 * y_m)
    if ((tan(t_0) / sin(t_0)) <= 20.0d0) then
        tmp = 1.0d0 / cos((((-1.0d0) / sqrt((2.0d0 * y_m))) * ((2.0d0 ** (-0.5d0)) / (sqrt(y_m) / x))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = x / (2.0 * y_m);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 20.0) {
		tmp = 1.0 / Math.cos(((-1.0 / Math.sqrt((2.0 * y_m))) * (Math.pow(2.0, -0.5) / (Math.sqrt(y_m) / x))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = x / (2.0 * y_m)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 20.0:
		tmp = 1.0 / math.cos(((-1.0 / math.sqrt((2.0 * y_m))) * (math.pow(2.0, -0.5) / (math.sqrt(y_m) / x))))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(x / Float64(2.0 * y_m))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 20.0)
		tmp = Float64(1.0 / cos(Float64(Float64(-1.0 / sqrt(Float64(2.0 * y_m))) * Float64((2.0 ^ -0.5) / Float64(sqrt(y_m) / x)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = x / (2.0 * y_m);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 20.0)
		tmp = 1.0 / cos(((-1.0 / sqrt((2.0 * y_m))) * ((2.0 ^ -0.5) / (sqrt(y_m) / x))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 20.0], N[(1.0 / N[Cos[N[(N[(-1.0 / N[Sqrt[N[(2.0 * y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 20

   1. Initial program 54.7%

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

      1. lift-/.f64 N/A

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

      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 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. inv-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-div N/A

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

      17. lift-pow.f64 N/A

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

      18. pow-flip N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. associate-*l/ N/A

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

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

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

   6. Applied rewrites 27.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-/l* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. pow-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 20 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 0.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 43.3%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \leq 20:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-1}{\sqrt{2 \cdot y}} \cdot \frac{{2}^{-0.5}}{\frac{\sqrt{y}}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{2 \cdot y\_m}\\ t_1 := \frac{x}{2 \cdot y\_m}\\ \mathbf{if}\;\frac{\tan t\_1}{\sin t\_1} \leq 400:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-1}{t\_0}}{t\_0} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 y_m))) (t_1 (/ x (* 2.0 y_m))))
   (if (<= (/ (tan t_1) (sin t_1)) 400.0)
     (/ 1.0 (cos (* (/ (/ -1.0 t_0) t_0) x)))
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = sqrt((2.0 * y_m));
	double t_1 = x / (2.0 * y_m);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 400.0) {
		tmp = 1.0 / cos((((-1.0 / t_0) / t_0) * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((2.0d0 * y_m))
    t_1 = x / (2.0d0 * y_m)
    if ((tan(t_1) / sin(t_1)) <= 400.0d0) then
        tmp = 1.0d0 / cos(((((-1.0d0) / t_0) / t_0) * x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = Math.sqrt((2.0 * y_m));
	double t_1 = x / (2.0 * y_m);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 400.0) {
		tmp = 1.0 / Math.cos((((-1.0 / t_0) / t_0) * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = math.sqrt((2.0 * y_m))
	t_1 = x / (2.0 * y_m)
	tmp = 0
	if (math.tan(t_1) / math.sin(t_1)) <= 400.0:
		tmp = 1.0 / math.cos((((-1.0 / t_0) / t_0) * x))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = sqrt(Float64(2.0 * y_m))
	t_1 = Float64(x / Float64(2.0 * y_m))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 400.0)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(-1.0 / t_0) / t_0) * x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = sqrt((2.0 * y_m));
	t_1 = x / (2.0 * y_m);
	tmp = 0.0;
	if ((tan(t_1) / sin(t_1)) <= 400.0)
		tmp = 1.0 / cos((((-1.0 / t_0) / t_0) * x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 400.0], N[(1.0 / N[Cos[N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 400

   1. Initial program 54.3%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. inv-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-div N/A

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

      17. lift-pow.f64 N/A

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

      18. pow-flip N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. associate-*l/ N/A

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

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

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

   6. Applied rewrites 27.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-neg.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 400 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \leq 400:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-1}{\sqrt{2 \cdot y}}}{\sqrt{2 \cdot y}} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{2 \cdot y\_m}\\ t_1 := \frac{x}{2 \cdot y\_m}\\ \mathbf{if}\;\frac{\tan t\_1}{\sin t\_1} \leq 400:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{t\_0} \cdot \frac{-1}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 y_m))) (t_1 (/ x (* 2.0 y_m))))
   (if (<= (/ (tan t_1) (sin t_1)) 400.0)
     (/ 1.0 (cos (* (/ x t_0) (/ -1.0 t_0))))
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = sqrt((2.0 * y_m));
	double t_1 = x / (2.0 * y_m);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 400.0) {
		tmp = 1.0 / cos(((x / t_0) * (-1.0 / t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((2.0d0 * y_m))
    t_1 = x / (2.0d0 * y_m)
    if ((tan(t_1) / sin(t_1)) <= 400.0d0) then
        tmp = 1.0d0 / cos(((x / t_0) * ((-1.0d0) / t_0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = Math.sqrt((2.0 * y_m));
	double t_1 = x / (2.0 * y_m);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 400.0) {
		tmp = 1.0 / Math.cos(((x / t_0) * (-1.0 / t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = math.sqrt((2.0 * y_m))
	t_1 = x / (2.0 * y_m)
	tmp = 0
	if (math.tan(t_1) / math.sin(t_1)) <= 400.0:
		tmp = 1.0 / math.cos(((x / t_0) * (-1.0 / t_0)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = sqrt(Float64(2.0 * y_m))
	t_1 = Float64(x / Float64(2.0 * y_m))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 400.0)
		tmp = Float64(1.0 / cos(Float64(Float64(x / t_0) * Float64(-1.0 / t_0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = sqrt((2.0 * y_m));
	t_1 = x / (2.0 * y_m);
	tmp = 0.0;
	if ((tan(t_1) / sin(t_1)) <= 400.0)
		tmp = 1.0 / cos(((x / t_0) * (-1.0 / t_0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 400.0], N[(1.0 / N[Cos[N[(N[(x / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 400

   1. Initial program 54.3%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. inv-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-div N/A

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

      17. lift-pow.f64 N/A

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

      18. pow-flip N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. associate-*l/ N/A

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

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

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

   6. Applied rewrites 27.3%

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

   if 400 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.3%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \leq 400:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{\sqrt{2 \cdot y}} \cdot \frac{-1}{\sqrt{2 \cdot y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x}{2 \cdot y\_m}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ x (* 2.0 y_m))))
   (if (<= (/ (tan t_0) (sin t_0)) -1.0) -1.0 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = x / (2.0 * y_m);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (2.0d0 * y_m)
    if ((tan(t_0) / sin(t_0)) <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = x / (2.0 * y_m);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = x / (2.0 * y_m)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= -1.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(x / Float64(2.0 * y_m))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= -1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = x / (2.0 * y_m);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= -1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], -1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < -1

   1. Initial program 16.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 8.9

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

   4. Applied rewrites 8.9%

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

   5. Taylor expanded in y around -inf

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

   7. Applied rewrites 14.9%

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

   if -1 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 50.1%

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

Alternative 5: 55.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y\_m} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y\_m} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (/ x (* 2.0 y_m)) 5e+277) (/ 1.0 (cos (* (/ x y_m) -0.5))) 1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x / (2.0 * y_m)) <= 5e+277) {
		tmp = 1.0 / cos(((x / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x / (2.0d0 * y_m)) <= 5d+277) then
        tmp = 1.0d0 / cos(((x / y_m) * (-0.5d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((x / (2.0 * y_m)) <= 5e+277) {
		tmp = 1.0 / Math.cos(((x / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (x / (2.0 * y_m)) <= 5e+277:
		tmp = 1.0 / math.cos(((x / y_m) * -0.5))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x / Float64(2.0 * y_m)) <= 5e+277)
		tmp = Float64(1.0 / cos(Float64(Float64(x / y_m) * -0.5)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((x / (2.0 * y_m)) <= 5e+277)
		tmp = 1.0 / cos(((x / y_m) * -0.5));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+277], N[(1.0 / N[Cos[N[(N[(x / 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))) < 4.99999999999999982e277

   1. Initial program 46.6%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 55.7%

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

   if 4.99999999999999982e277 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y\_m} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y\_m} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (/ x (* 2.0 y_m)) 5e+178) (/ 1.0 (cos (* (/ 0.5 y_m) x))) 1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x / (2.0 * y_m)) <= 5e+178) {
		tmp = 1.0 / cos(((0.5 / y_m) * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x / (2.0d0 * y_m)) <= 5d+178) then
        tmp = 1.0d0 / cos(((0.5d0 / y_m) * x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((x / (2.0 * y_m)) <= 5e+178) {
		tmp = 1.0 / Math.cos(((0.5 / y_m) * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (x / (2.0 * y_m)) <= 5e+178:
		tmp = 1.0 / math.cos(((0.5 / y_m) * x))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x / Float64(2.0 * y_m)) <= 5e+178)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / y_m) * x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((x / (2.0 * y_m)) <= 5e+178)
		tmp = 1.0 / cos(((0.5 / y_m) * x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+178], N[(1.0 / N[Cos[N[(N[(0.5 / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e178

   1. Initial program 49.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   if 4.9999999999999999e178 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.7%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.4%

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

Developer Target 1: 54.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024294 
(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)))))