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

Percentage Accurate: 44.4% → 57.3%

Time: 11.9s

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: 44.4% 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: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(\frac{x\_m}{\frac{y\_m}{0.5}}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 6e+24)
   (/ 1.0 (cos (exp (log (/ x_m (/ y_m 0.5))))))
   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)) <= 6e+24) {
		tmp = 1.0 / cos(exp(log((x_m / (y_m / 0.5)))));
	} 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)) <= 6d+24) then
        tmp = 1.0d0 / cos(exp(log((x_m / (y_m / 0.5d0)))))
    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)) <= 6e+24) {
		tmp = 1.0 / Math.cos(Math.exp(Math.log((x_m / (y_m / 0.5)))));
	} 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)) <= 6e+24:
		tmp = 1.0 / math.cos(math.exp(math.log((x_m / (y_m / 0.5)))))
	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)) <= 6e+24)
		tmp = Float64(1.0 / cos(exp(log(Float64(x_m / Float64(y_m / 0.5))))));
	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)) <= 6e+24)
		tmp = 1.0 / cos(exp(log((x_m / (y_m / 0.5)))));
	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], 6e+24], N[(1.0 / N[Cos[N[Exp[N[Log[N[(x$95$m / N[(y$95$m / 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.9999999999999999e24

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 74.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 74.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y}{x \cdot \frac{1}{2}}}\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{-1}\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right) \cdot -1}\right)\right)\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y}{x \cdot \frac{1}{2}}\right) \cdot -1\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y}{x \cdot \frac{1}{2}}\right), -1\right)\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      8. *-lowering-*.f64 33.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 33.7%

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

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y}{x \cdot \frac{1}{2}}\right) \cdot -1\right)\right)\right)\right) \]

      2. rem-log-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(e^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right) \cdot -1}\right)\right)\right)\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{-1}\right)\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left({\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)}^{-1}\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left({\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)}^{-1}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right)\right) \]

      7. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{x}{y \cdot 2}\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot 2\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot \frac{1}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 40.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 40.9%

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

   if 5.9999999999999999e24 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 11.4%

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

Alternative 2: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{0.5}{y\_m}}{\frac{1}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 6e+24)
   (/ 1.0 (cos (/ (/ 0.5 y_m) (/ 1.0 x_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)) <= 6e+24) {
		tmp = 1.0 / cos(((0.5 / y_m) / (1.0 / x_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)) <= 6d+24) then
        tmp = 1.0d0 / cos(((0.5d0 / y_m) / (1.0d0 / x_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)) <= 6e+24) {
		tmp = 1.0 / Math.cos(((0.5 / y_m) / (1.0 / x_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)) <= 6e+24:
		tmp = 1.0 / math.cos(((0.5 / y_m) / (1.0 / x_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)) <= 6e+24)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / y_m) / Float64(1.0 / x_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)) <= 6e+24)
		tmp = 1.0 / cos(((0.5 / y_m) / (1.0 / x_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], 6e+24], N[(1.0 / N[Cos[N[(N[(0.5 / y$95$m), $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.9999999999999999e24

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 74.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 74.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{y} \cdot x\right)\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{\frac{y}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{y \cdot \frac{1}{x}}\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{\frac{1}{2}}{y}}{\frac{1}{x}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{y}}{\frac{1}{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{\frac{-1}{2}}{y}\right)}{\frac{1}{x}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{-1}{2}}{y}\right)\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{y}\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{y}\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, y\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 74.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, y\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

   7. Applied egg-rr 74.6%

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

   if 5.9999999999999999e24 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 11.4%

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

Alternative 3: 55.7% 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 47.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 58.8%

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

Developer Target 1: 55.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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)))))