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

Percentage Accurate: 43.7% → 57.0%

Time: 12.8s

Alternatives: 6

Speedup: 244.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m}{y\_m} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 8e+31) (/ 1.0 (cos (* (/ x_m y_m) -0.5))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 8e+31) {
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 8d+31) then
        tmp = 1.0d0 / cos(((x_m / y_m) * (-0.5d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 8e+31) {
		tmp = 1.0 / Math.cos(((x_m / y_m) * -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 8e+31:
		tmp = 1.0 / math.cos(((x_m / y_m) * -0.5))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 8e+31)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / y_m) * -0.5)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 8e+31)
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 8e+31], N[(1.0 / N[Cos[N[(N[(x$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 7.9999999999999997e31

   1. Initial program 73.3%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 96.6%

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

   if 7.9999999999999997e31 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \frac{1}{\cos \left(\left(\sqrt{x\_m} \cdot e^{\mathsf{fma}\left(\log x\_m, 0.5, -\log y\_m\right)}\right) \cdot -0.5\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (* (* (sqrt x_m) (exp (fma (log x_m) 0.5 (- (log y_m))))) -0.5))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos(((sqrt(x_m) * exp(fma(log(x_m), 0.5, -log(y_m)))) * -0.5));
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(Float64(sqrt(x_m) * exp(fma(log(x_m), 0.5, Float64(-log(y_m))))) * -0.5)))
end

Wolfram

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

Derivation

1. Initial program 43.7%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
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 55.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. div-inv N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-/.f64 55.2

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

6. Applied rewrites 55.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. inv-pow N/A

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

   12. metadata-eval N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. unpow1/2 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. inv-pow N/A

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

   19. metadata-eval N/A

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

   20. pow-pow N/A

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

   21. metadata-eval N/A

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

   22. unpow1/2 N/A

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

   23. lower-sqrt.f64 55.2

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

8. Applied rewrites 55.2%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. inv-pow N/A

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

   7. exp-to-pow N/A

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

   8. lift-log.f64 N/A

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

   9. prod-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 54.8

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

10. Applied rewrites 54.8%

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

11. Final simplification 54.8%

    \[\leadsto \frac{1}{\cos \left(\left(\sqrt{x} \cdot e^{\mathsf{fma}\left(\log x, 0.5, -\log y\right)}\right) \cdot -0.5\right)} \]
12. Add Preprocessing

Developer Target 1: 55.2% 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 2024230 
(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)))))