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

Percentage Accurate: 44.3% → 56.7%

Time: 11.0s

Alternatives: 4

Speedup: 244.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m} \cdot 0.5\\ \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\cos \left(\left(-{\left(e^{0.5}\right)}^{\log t\_0}\right) \cdot {t\_0}^{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (/ x_m y_m) 0.5)))
   (if (<= (/ x_m (* 2.0 y_m)) 2e+284)
     (/ 1.0 (cos (* (- (pow (exp 0.5) (log t_0))) (pow t_0 0.5))))
     1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (x_m / y_m) * 0.5;
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+284) {
		tmp = 1.0 / cos((-pow(exp(0.5), log(t_0)) * pow(t_0, 0.5)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / y_m) * 0.5d0
    if ((x_m / (2.0d0 * y_m)) <= 2d+284) then
        tmp = 1.0d0 / cos((-(exp(0.5d0) ** log(t_0)) * (t_0 ** 0.5d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double t_0 = (x_m / y_m) * 0.5;
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+284) {
		tmp = 1.0 / Math.cos((-Math.pow(Math.exp(0.5), Math.log(t_0)) * Math.pow(t_0, 0.5)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	t_0 = (x_m / y_m) * 0.5
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+284:
		tmp = 1.0 / math.cos((-math.pow(math.exp(0.5), math.log(t_0)) * math.pow(t_0, 0.5)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(x_m / y_m) * 0.5)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+284)
		tmp = Float64(1.0 / cos(Float64(Float64(-(exp(0.5) ^ log(t_0))) * (t_0 ^ 0.5))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	t_0 = (x_m / y_m) * 0.5;
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+284)
		tmp = 1.0 / cos((-(exp(0.5) ^ log(t_0)) * (t_0 ^ 0.5)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(x$95$m / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+284], N[(1.0 / N[Cos[N[((-N[Power[N[Exp[0.5], $MachinePrecision], N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]) * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000016e284

   1. Initial program 41.0%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. 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 54.0%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}} \]
   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. metadata-eval N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. unpow-prod-down N/A

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

      11. associate-/l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. sqr-pow N/A

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

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

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 33.2%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. *-commutative N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-log.f64 34.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 34.3

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

   8. Applied rewrites 34.3%

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

   if 2.00000000000000016e284 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 3.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 12.4%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+284}:\\ \;\;\;\;\frac{1}{\cos \left(\left(-{\left(e^{0.5}\right)}^{\log \left(\frac{x}{y} \cdot 0.5\right)}\right) \cdot {\left(\frac{x}{y} \cdot 0.5\right)}^{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 10^{+173}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 1e+173) (/ 1.0 (cos (/ -0.5 (/ y_m x_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+173) {
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 1d+173) then
        tmp = 1.0d0 / cos(((-0.5d0) / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+173) {
		tmp = 1.0 / Math.cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 1e+173:
		tmp = 1.0 / math.cos((-0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 1e+173)
		tmp = Float64(1.0 / cos(Float64(-0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 1e+173)
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 1e+173], N[(1.0 / N[Cos[N[(-0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e173

   1. Initial program 43.0%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. 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 56.7%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}} \]
   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. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1e173 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 12.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 10^{+173}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\cos \left(-0.5 \cdot \frac{x\_m}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 5e+16) (/ 1.0 (cos (* -0.5 (/ x_m y_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+16) {
		tmp = 1.0 / cos((-0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 5d+16) then
        tmp = 1.0d0 / cos(((-0.5d0) * (x_m / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+16) {
		tmp = 1.0 / Math.cos((-0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 5e+16:
		tmp = 1.0 / math.cos((-0.5 * (x_m / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 5e+16)
		tmp = Float64(1.0 / cos(Float64(-0.5 * Float64(x_m / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 5e+16)
		tmp = 1.0 / cos((-0.5 * (x_m / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+16], N[(1.0 / N[Cos[N[(-0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5e16

   1. Initial program 48.7%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 64.8%

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

   if 5e16 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 12.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% 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 37.5%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 50.1%

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

Developer Target 1: 55.4% 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 2024248 
(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)))))