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

Percentage Accurate: 43.7% → 56.6%

Time: 32.1s

Alternatives: 5

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: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{e^{\mathsf{log1p}\left(\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+263)
   (/ 1.0 (+ (exp (log1p (cos (* x_m (/ 0.5 y_m))))) -1.0))
   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)) <= 2e+263) {
		tmp = 1.0 / (exp(log1p(cos((x_m * (0.5 / y_m))))) + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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)) <= 2e+263) {
		tmp = 1.0 / (Math.exp(Math.log1p(Math.cos((x_m * (0.5 / y_m))))) + -1.0);
	} 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)) <= 2e+263:
		tmp = 1.0 / (math.exp(math.log1p(math.cos((x_m * (0.5 / y_m))))) + -1.0)
	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)) <= 2e+263)
		tmp = Float64(1.0 / Float64(exp(log1p(cos(Float64(x_m * Float64(0.5 / y_m))))) + -1.0));
	else
		tmp = 1.0;
	end
	return 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], 2e+263], N[(1.0 / N[(N[Exp[N[Log[1 + N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000003e263

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. metadata-eval 60.4%

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

      3. div-inv 60.4%

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

      4. associate-/r* 60.4%

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

      5. div-inv 60.6%

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

      6. metadata-eval 60.6%

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

      7. div-inv 60.6%

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

      8. clear-num 60.6%

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

      9. *-commutative 60.6%

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

   5. Applied egg-rr 60.6%

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

      1. associate-*l/ 60.4%

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

      2. associate-*r/ 60.4%

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

      3. expm1-log1p-u 60.4%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

      4. expm1-undefine 60.4%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]

      5. associate-*r/ 60.4%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{0.5 \cdot x}{y}\right)}\right)} - 1} \]

      6. associate-*l/ 60.6%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}\right)} - 1} \]

   7. Applied egg-rr 60.6%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(\frac{0.5}{y} \cdot x\right)\right)} - 1}} \]

   if 2.00000000000000003e263 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0 13.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{e^{\mathsf{log1p}\left(\cos \left(x \cdot \frac{0.5}{y}\right)\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+263)
   (/ 1.0 (log (exp (cos (* x_m (/ 0.5 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)) <= 2e+263) {
		tmp = 1.0 / log(exp(cos((x_m * (0.5 / 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)) <= 2d+263) then
        tmp = 1.0d0 / log(exp(cos((x_m * (0.5d0 / 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)) <= 2e+263) {
		tmp = 1.0 / Math.log(Math.exp(Math.cos((x_m * (0.5 / 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)) <= 2e+263:
		tmp = 1.0 / math.log(math.exp(math.cos((x_m * (0.5 / 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)) <= 2e+263)
		tmp = Float64(1.0 / log(exp(cos(Float64(x_m * Float64(0.5 / 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)) <= 2e+263)
		tmp = 1.0 / log(exp(cos((x_m * (0.5 / 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], 2e+263], N[(1.0 / N[Log[N[Exp[N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $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))) < 2.00000000000000003e263

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. metadata-eval 60.4%

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

      3. div-inv 60.4%

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

      4. associate-/r* 60.4%

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

      5. div-inv 60.6%

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

      6. metadata-eval 60.6%

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

      7. div-inv 60.6%

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

      8. clear-num 60.6%

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

      9. *-commutative 60.6%

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

   5. Applied egg-rr 60.6%

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

      1. add-log-exp 60.6%

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

   7. Applied egg-rr 60.6%

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

   if 2.00000000000000003e263 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0 13.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\sqrt[3]{{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}^{-3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+263)
   (cbrt (pow (cos (* x_m (/ 0.5 y_m))) -3.0))
   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)) <= 2e+263) {
		tmp = cbrt(pow(cos((x_m * (0.5 / y_m))), -3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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)) <= 2e+263) {
		tmp = Math.cbrt(Math.pow(Math.cos((x_m * (0.5 / y_m))), -3.0));
	} 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)) <= 2e+263)
		tmp = cbrt((cos(Float64(x_m * Float64(0.5 / y_m))) ^ -3.0));
	else
		tmp = 1.0;
	end
	return 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], 2e+263], N[Power[N[Power[N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000003e263

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. metadata-eval 60.4%

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

      3. div-inv 60.4%

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

      4. associate-/r* 60.4%

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

      5. div-inv 60.6%

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

      6. metadata-eval 60.6%

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

      7. div-inv 60.6%

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

      8. clear-num 60.6%

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

      9. *-commutative 60.6%

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

   5. Applied egg-rr 60.6%

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

      1. associate-*l/ 60.4%

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

      2. associate-*r/ 60.4%

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

      3. expm1-log1p-u 57.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)} \]

      4. expm1-define 57.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)} - 1} \]

      5. log1p-undefine 57.6%

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

      6. rem-exp-log 60.3%

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

      7. associate-*r/ 60.3%

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

      8. associate-*l/ 60.6%

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

   7. Applied egg-rr 60.6%

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

      1. add-exp-log 58.0%

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

      2. expm1-define 58.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\right)\right)} \]

      3. log1p-define 58.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\right)}\right) \]

      4. add-cbrt-cube 58.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)} \cdot \frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\right) \cdot \frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}}}\right)\right) \]

      5. expm1-log1p-u 60.6%

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

      6. pow3 60.6%

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

      7. inv-pow 60.6%

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

      8. pow-pow 60.6%

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

      9. *-commutative 60.6%

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

      10. metadata-eval 60.6%

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

   9. Applied egg-rr 60.6%

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

   if 2.00000000000000003e263 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0 13.9%

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

Alternative 4: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+263) (/ 1.0 (cos (* x_m (/ 0.5 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)) <= 2e+263) {
		tmp = 1.0 / cos((x_m * (0.5 / 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)) <= 2d+263) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / 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)) <= 2e+263) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / 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)) <= 2e+263:
		tmp = 1.0 / math.cos((x_m * (0.5 / 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)) <= 2e+263)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / 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)) <= 2e+263)
		tmp = 1.0 / cos((x_m * (0.5 / 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], 2e+263], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / 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))) < 2.00000000000000003e263

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. metadata-eval 60.4%

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

      3. div-inv 60.4%

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

      4. associate-/r* 60.4%

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

      5. div-inv 60.6%

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

      6. metadata-eval 60.6%

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

      7. div-inv 60.6%

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

      8. clear-num 60.6%

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

      9. *-commutative 60.6%

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

   5. Applied egg-rr 60.6%

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

   if 2.00000000000000003e263 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0 13.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% 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 43.1%

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

3. Taylor expanded in x around 0 54.0%

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

Developer target: 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 2024096 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))