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

Percentage Accurate: 43.7% → 57.1%

Time: 13.6s

Alternatives: 3

Speedup: 211.0×

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: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{\frac{-2}{x\_m}}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+24)
   (/ 1.0 (cos (/ (/ 1.0 (/ -2.0 x_m)) y_m)))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+24) {
		tmp = 1.0 / cos(((1.0 / (-2.0 / x_m)) / y_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+24) then
        tmp = 1.0d0 / cos(((1.0d0 / ((-2.0d0) / x_m)) / y_m))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+24) {
		tmp = 1.0 / Math.cos(((1.0 / (-2.0 / x_m)) / y_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+24:
		tmp = 1.0 / math.cos(((1.0 / (-2.0 / x_m)) / y_m))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+24)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / Float64(-2.0 / x_m)) / y_m)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+24)
		tmp = 1.0 / cos(((1.0 / (-2.0 / x_m)) / y_m));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+24], N[(1.0 / N[Cos[N[(N[(1.0 / N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000045e24

   1. Initial program 55.1%

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

      1. remove-double-neg 55.1%

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

      2. distribute-frac-neg 55.1%

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

      3. tan-neg 55.1%

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

      4. distribute-frac-neg2 55.1%

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

      5. distribute-lft-neg-out 55.1%

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

      6. distribute-frac-neg2 55.1%

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

      7. distribute-lft-neg-out 55.1%

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

      8. distribute-frac-neg2 55.1%

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

      9. distribute-frac-neg 55.1%

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

      10. neg-mul-1 55.1%

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

      11. *-commutative 55.1%

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

      12. associate-/l* 55.0%

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

      13. *-commutative 55.0%

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

      14. associate-/r* 55.0%

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

      15. metadata-eval 55.0%

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

      16. sin-neg 55.0%

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

      17. distribute-frac-neg 55.0%

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

   3. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-*r/ 69.5%

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

      4. cos-neg 69.5%

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

      5. associate-*r/ 70.0%

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

      6. distribute-frac-neg 70.0%

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

      7. distribute-rgt-neg-in 70.0%

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

      8. metadata-eval 70.0%

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

   7. Simplified 70.0%

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

      1. add-sqr-sqrt 33.7%

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

      2. sqrt-unprod 64.8%

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

      3. swap-sqr 64.8%

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

      4. metadata-eval 64.8%

         \[\leadsto \frac{1}{\cos \left(\frac{\sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}}{y}\right)} \]

      5. metadata-eval 64.8%

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

      6. swap-sqr 64.8%

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

      7. metadata-eval 64.8%

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

      8. div-inv 64.8%

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

      9. metadata-eval 64.8%

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

      10. div-inv 64.8%

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

      11. sqrt-unprod 36.1%

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

      12. add-sqr-sqrt 70.0%

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

      13. clear-num 70.1%

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

   9. Applied egg-rr 70.1%

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

   if 5.00000000000000045e24 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 9.0%

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

      1. remove-double-neg 9.0%

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

      2. distribute-frac-neg 9.0%

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

      3. tan-neg 9.0%

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

      4. distribute-frac-neg2 9.0%

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

      5. distribute-lft-neg-out 9.0%

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

      6. distribute-frac-neg2 9.0%

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

      7. distribute-lft-neg-out 9.0%

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

      8. distribute-frac-neg2 9.0%

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

      9. distribute-frac-neg 9.0%

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

      10. neg-mul-1 9.0%

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

      11. *-commutative 9.0%

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

      12. associate-/l* 8.9%

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

      13. *-commutative 8.9%

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

      14. associate-/r* 8.9%

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

      15. metadata-eval 8.9%

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

      16. sin-neg 8.9%

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

      17. distribute-frac-neg 8.9%

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

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 11.2%

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

Alternative 2: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{1}{e^{\log \left(y\_m \cdot 2\right) - \log x\_m}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ 1.0 (exp (- (log (* y_m 2.0)) (log x_m)))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos((1.0 / exp((log((y_m * 2.0)) - log(x_m)))));
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = 1.0d0 / cos((1.0d0 / exp((log((y_m * 2.0d0)) - log(x_m)))))
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	return 1.0 / Math.cos((1.0 / Math.exp((Math.log((y_m * 2.0)) - Math.log(x_m)))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos((1.0 / math.exp((math.log((y_m * 2.0)) - math.log(x_m)))))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(1.0 / exp(Float64(log(Float64(y_m * 2.0)) - log(x_m))))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((1.0 / exp((log((y_m * 2.0)) - log(x_m)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(1.0 / N[Exp[N[(N[Log[N[(y$95$m * 2.0), $MachinePrecision]], $MachinePrecision] - N[Log[x$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1%

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

   1. remove-double-neg 44.1%

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

   2. distribute-frac-neg 44.1%

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

   3. tan-neg 44.1%

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

   4. distribute-frac-neg2 44.1%

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

   5. distribute-lft-neg-out 44.1%

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

   6. distribute-frac-neg2 44.1%

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

   7. distribute-lft-neg-out 44.1%

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

   8. distribute-frac-neg2 44.1%

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

   9. distribute-frac-neg 44.1%

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

   10. neg-mul-1 44.1%

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

   11. *-commutative 44.1%

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

   12. associate-/l* 44.0%

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

   13. *-commutative 44.0%

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

   14. associate-/r* 44.0%

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

   15. metadata-eval 44.0%

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

   16. sin-neg 44.0%

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

   17. distribute-frac-neg 44.0%

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

3. Simplified 43.6%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 55.5%

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

   1. associate-*r/ 55.5%

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

   2. *-commutative 55.5%

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

   3. associate-*r/ 55.0%

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

7. Simplified 55.0%

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

   1. add-cube-cbrt 55.9%

      \[\leadsto \frac{1}{\cos \left(x \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{-0.5}{y}} \cdot \sqrt[3]{\frac{-0.5}{y}}\right) \cdot \sqrt[3]{\frac{-0.5}{y}}\right)}\right)} \]

   2. pow3 56.0%

      \[\leadsto \frac{1}{\cos \left(x \cdot \color{blue}{{\left(\sqrt[3]{\frac{-0.5}{y}}\right)}^{3}}\right)} \]

9. Applied egg-rr 56.0%

   \[\leadsto \frac{1}{\cos \left(x \cdot \color{blue}{{\left(\sqrt[3]{\frac{-0.5}{y}}\right)}^{3}}\right)} \]
10. Step-by-step derivation

    1. unpow3 55.9%

       \[\leadsto \frac{1}{\cos \left(x \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{-0.5}{y}} \cdot \sqrt[3]{\frac{-0.5}{y}}\right) \cdot \sqrt[3]{\frac{-0.5}{y}}\right)}\right)} \]

    2. add-cube-cbrt 55.0%

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

    3. associate-*r/ 55.5%

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

    4. metadata-eval 55.5%

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

    5. div-inv 55.5%

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

    6. div-inv 55.0%

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

    7. clear-num 55.2%

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

    8. frac-times 55.9%

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

    9. metadata-eval 55.9%

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

11. Applied egg-rr 55.9%

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

    1. /-rgt-identity 55.9%

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

    2. clear-num 55.9%

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

    3. add-sqr-sqrt 33.5%

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

    4. sqrt-unprod 54.0%

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

    5. frac-2neg 54.0%

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

    6. metadata-eval 54.0%

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

    7. frac-2neg 54.0%

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

    8. metadata-eval 54.0%

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

    9. frac-times 54.1%

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

    10. metadata-eval 54.1%

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

    11. metadata-eval 54.1%

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

    12. associate-*l/ 53.9%

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

    13. metadata-eval 53.9%

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

    14. distribute-lft-neg-in 53.9%

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

    15. *-commutative 53.9%

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

    16. distribute-neg-frac2 53.9%

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

    17. frac-2neg 53.9%

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

    18. associate-*l/ 53.6%

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

    19. metadata-eval 53.6%

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

    20. distribute-lft-neg-in 53.6%

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

    21. *-commutative 53.6%

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

13. Applied egg-rr 12.7%

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

Alternative 3: 55.4% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

   1. remove-double-neg 44.1%

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

   2. distribute-frac-neg 44.1%

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

   3. tan-neg 44.1%

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

   4. distribute-frac-neg2 44.1%

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

   5. distribute-lft-neg-out 44.1%

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

   6. distribute-frac-neg2 44.1%

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

   7. distribute-lft-neg-out 44.1%

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

   8. distribute-frac-neg2 44.1%

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

   9. distribute-frac-neg 44.1%

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

   10. neg-mul-1 44.1%

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

   11. *-commutative 44.1%

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

   12. associate-/l* 44.0%

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

   13. *-commutative 44.0%

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

   14. associate-/r* 44.0%

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

   15. metadata-eval 44.0%

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

   16. sin-neg 44.0%

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

   17. distribute-frac-neg 44.0%

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

3. Simplified 43.6%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 56.4%

   \[\leadsto \color{blue}{1} \]
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 2024137 
(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)))))