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

Percentage Accurate: 44.1% → 57.0%

Time: 27.7s

Alternatives: 7

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: 44.1% 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, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{y\_m}{x\_m}}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{0.5}{t\_0}}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ y_m x_m))))
   (if (<= (/ x_m (* y_m 2.0)) 2e+259)
     (/ 1.0 (cos (/ (/ 0.5 t_0) t_0)))
     -1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = sqrt((y_m / x_m));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+259) {
		tmp = 1.0 / cos(((0.5 / t_0) / t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((y_m / x_m))
    if ((x_m / (y_m * 2.0d0)) <= 2d+259) then
        tmp = 1.0d0 / cos(((0.5d0 / t_0) / t_0))
    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 t_0 = Math.sqrt((y_m / x_m));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+259) {
		tmp = 1.0 / Math.cos(((0.5 / t_0) / t_0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.sqrt((y_m / x_m))
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+259:
		tmp = 1.0 / math.cos(((0.5 / t_0) / t_0))
	else:
		tmp = -1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = sqrt(Float64(y_m / x_m))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+259)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / t_0) / t_0)));
	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)
	t_0 = sqrt((y_m / x_m));
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+259)
		tmp = 1.0 / cos(((0.5 / t_0) / t_0));
	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_] := Block[{t$95$0 = N[Sqrt[N[(y$95$m / x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+259], N[(1.0 / N[Cos[N[(N[(0.5 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2e259

   1. Initial program 54.3%

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

   3. Taylor expanded in x around inf 65.7%

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

      1. *-un-lft-identity 65.7%

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

      2. *-commutative 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. *-rgt-identity 65.7%

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

      2. *-commutative 65.7%

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

      3. associate-*l/ 65.7%

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

      4. associate-/l* 65.1%

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

   7. Simplified 65.1%

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

      1. add-sqr-sqrt 38.3%

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

      2. pow2 38.3%

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

      3. *-commutative 38.3%

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

      4. associate-*l/ 38.6%

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

      5. associate-*r/ 38.6%

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

      6. clear-num 38.7%

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

      7. un-div-inv 38.7%

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

   9. Applied egg-rr 38.7%

      \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(\sqrt{\frac{0.5}{\frac{y}{x}}}\right)}^{2}\right)}} \]
   10. Step-by-step derivation

       1. unpow2 38.7%

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

       2. add-sqr-sqrt 65.2%

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

       3. add-sqr-sqrt 33.2%

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

       4. associate-/r* 33.7%

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

   11. Applied egg-rr 33.7%

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

   if 2e259 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.9%

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

      1. remove-double-neg 0.9%

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

      2. distribute-frac-neg 0.9%

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

      3. tan-neg 0.9%

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

      4. distribute-frac-neg2 0.9%

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

      5. distribute-lft-neg-out 0.9%

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

      6. distribute-frac-neg2 0.9%

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

      7. distribute-lft-neg-out 0.9%

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

      8. distribute-frac-neg2 0.9%

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

      9. distribute-frac-neg 0.9%

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

      10. neg-mul-1 0.9%

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

      11. *-commutative 0.9%

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

      12. associate-/l* 0.9%

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

      13. *-commutative 0.9%

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

      14. associate-/r* 0.9%

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

      15. metadata-eval 0.9%

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

      16. sin-neg 0.9%

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

      17. distribute-frac-neg 0.9%

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

   3. Simplified 0.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. pow2 0.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \color{blue}{\left(\sqrt{{\left(x \cdot \frac{-0.5}{y}\right)}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. unpow2 0.0%

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

      2. rem-sqrt-square 1.6%

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

      3. associate-*r/ 1.6%

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

      4. *-commutative 1.6%

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

      5. associate-*r/ 1.6%

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

   8. Simplified 1.6%

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

      1. *-commutative 1.6%

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

      2. associate-*l/ 1.6%

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

      3. associate-*r/ 1.6%

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

      4. rem-cube-cbrt 0.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 1.6%

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

      8. rem-cube-cbrt 0.9%

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

      9. log1p-expm1-u 0.9%

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

      10. log1p-undefine 0.9%

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

      11. add-sqr-sqrt 0.7%

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

      12. add-sqr-sqrt 0.9%

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

      13. rem-cube-cbrt 1.6%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}\right)}\right)\right)} \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}} \cdot \sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right)}\right)\right)} \]

      15. fabs-sqr 0.0%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\left|\sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}} \cdot \sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right|\right)}\right)\right)} \]

      16. add-sqr-sqrt 0.9%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\left|\color{blue}{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right|\right)\right)\right)} \]

      17. unpow3 1.6%

          \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\left|\color{blue}{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}} \cdot \sqrt[3]{x \cdot \frac{-0.5}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{-0.5}{y}}}\right|\right)\right)\right)} \]

   10. Applied egg-rr 1.6%

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

   11. Taylor expanded in x around 0 12.6%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{0.5}{\sqrt{\frac{y}{x}}}}{\sqrt{\frac{y}{x}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left({\left(\sqrt{\frac{x\_m}{y\_m}} \cdot \sqrt{0.5}\right)}^{2}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (pow (* (sqrt (/ x_m y_m)) (sqrt 0.5)) 2.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(pow((sqrt((x_m / y_m)) * sqrt(0.5)), 2.0));
}

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 / cos(((sqrt((x_m / y_m)) * sqrt(0.5d0)) ** 2.0d0))
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 / Math.cos(Math.pow((Math.sqrt((x_m / y_m)) * Math.sqrt(0.5)), 2.0));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos(math.pow((math.sqrt((x_m / y_m)) * math.sqrt(0.5)), 2.0))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos((Float64(sqrt(Float64(x_m / y_m)) * sqrt(0.5)) ^ 2.0)))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((sqrt((x_m / y_m)) * sqrt(0.5)) ^ 2.0));
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[Power[N[(N[Sqrt[N[(x$95$m / y$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around inf 60.6%

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

   1. *-un-lft-identity 60.6%

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

   2. *-commutative 60.6%

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

5. Applied egg-rr 60.6%

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

   1. *-rgt-identity 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*l/ 60.6%

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

   4. associate-/l* 60.1%

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

7. Simplified 60.1%

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

   1. add-sqr-sqrt 35.4%

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

   2. pow2 35.4%

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

   3. *-commutative 35.4%

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

   4. associate-*l/ 35.7%

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

   5. associate-*r/ 35.7%

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

   6. clear-num 35.7%

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

   7. un-div-inv 35.7%

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

9. Applied egg-rr 35.7%

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

10. Taylor expanded in y around 0 36.2%

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

11. Final simplification 36.2%

    \[\leadsto \frac{1}{\cos \left({\left(\sqrt{\frac{x}{y}} \cdot \sqrt{0.5}\right)}^{2}\right)} \]
12. Add Preprocessing

Alternative 3: 55.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left({\left(\sqrt{\frac{0.5}{\frac{y\_m}{x\_m}}}\right)}^{2}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (pow (sqrt (/ 0.5 (/ y_m x_m))) 2.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(pow(sqrt((0.5 / (y_m / x_m))), 2.0));
}

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 / cos((sqrt((0.5d0 / (y_m / x_m))) ** 2.0d0))
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 / Math.cos(Math.pow(Math.sqrt((0.5 / (y_m / x_m))), 2.0));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos(math.pow(math.sqrt((0.5 / (y_m / x_m))), 2.0))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos((sqrt(Float64(0.5 / Float64(y_m / x_m))) ^ 2.0)))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((sqrt((0.5 / (y_m / x_m))) ^ 2.0));
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[Power[N[Sqrt[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around inf 60.6%

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

   1. *-un-lft-identity 60.6%

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

   2. *-commutative 60.6%

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

5. Applied egg-rr 60.6%

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

   1. *-rgt-identity 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*l/ 60.6%

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

   4. associate-/l* 60.1%

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

7. Simplified 60.1%

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

   1. add-sqr-sqrt 35.4%

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

   2. pow2 35.4%

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

   3. *-commutative 35.4%

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

   4. associate-*l/ 35.7%

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

   5. associate-*r/ 35.7%

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

   6. clear-num 35.7%

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

   7. un-div-inv 35.7%

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

9. Applied egg-rr 35.7%

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

10. Final simplification 35.7%

    \[\leadsto \frac{1}{\cos \left({\left(\sqrt{\frac{0.5}{\frac{y}{x}}}\right)}^{2}\right)} \]
11. Add Preprocessing

Alternative 4: 55.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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 / log(exp(cos(((x_m / y_m) * 0.5d0))))
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.log(Math.exp(Math.cos(((x_m / y_m) * 0.5))));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.log(math.exp(math.cos(((x_m / y_m) * 0.5))))

Julia

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / log(exp(cos(((x_m / y_m) * 0.5))));
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[Log[N[Exp[N[Cos[N[(N[(x$95$m / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around inf 60.6%

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

   1. add-log-exp 60.6%

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

5. Applied egg-rr 60.6%

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

6. Final simplification 60.6%

   \[\leadsto \frac{1}{\log \left(e^{\cos \left(\frac{x}{y} \cdot 0.5\right)}\right)} \]
7. Add Preprocessing

Alternative 5: 55.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{x\_m}{y\_m} \cdot 0.5\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* (/ x_m y_m) 0.5))))

C

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

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(((x_m / y_m) * 0.5d0))
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(((x_m / y_m) * 0.5));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return 1.0 / math.cos(((x_m / y_m) * 0.5))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(Float64(x_m / y_m) * 0.5)))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((x_m / y_m) * 0.5));
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[(N[(x$95$m / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around inf 60.6%

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

4. Final simplification 60.6%

   \[\leadsto \frac{1}{\cos \left(\frac{x}{y} \cdot 0.5\right)} \]
5. Add Preprocessing

Alternative 6: 6.6% 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 50.1%

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

   1. remove-double-neg 50.1%

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

   2. distribute-frac-neg 50.1%

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

   3. tan-neg 50.1%

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

   4. distribute-frac-neg2 50.1%

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

   5. distribute-lft-neg-out 50.1%

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

   6. distribute-frac-neg2 50.1%

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

   7. distribute-lft-neg-out 50.1%

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

   8. distribute-frac-neg2 50.1%

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

   9. distribute-frac-neg 50.1%

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

   10. neg-mul-1 50.1%

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

   11. *-commutative 50.1%

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

   12. associate-/l* 49.3%

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

   13. *-commutative 49.3%

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

   14. associate-/r* 49.3%

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

   15. metadata-eval 49.3%

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

   16. sin-neg 49.3%

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

   17. distribute-frac-neg 49.3%

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

3. Simplified 49.5%

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

   1. add-sqr-sqrt 24.4%

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

   2. sqrt-unprod 18.8%

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

   3. pow2 18.8%

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

6. Applied egg-rr 18.8%

   \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \color{blue}{\left(\sqrt{{\left(x \cdot \frac{-0.5}{y}\right)}^{2}}\right)}} \]
7. Step-by-step derivation

   1. unpow2 18.8%

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

   2. rem-sqrt-square 26.9%

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

   3. associate-*r/ 27.0%

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

   4. *-commutative 27.0%

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

   5. associate-*r/ 27.0%

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

8. Simplified 27.0%

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

   1. *-commutative 27.0%

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

   2. associate-*l/ 27.0%

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

   3. associate-*r/ 26.9%

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

   4. rem-cube-cbrt 26.6%

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

   5. add-sqr-sqrt 24.2%

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

   6. fabs-sqr 24.2%

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

   7. add-sqr-sqrt 48.4%

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

   8. rem-cube-cbrt 49.5%

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

   9. log1p-expm1-u 49.5%

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

   10. log1p-undefine 7.4%

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

   11. add-sqr-sqrt 4.1%

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

   12. add-sqr-sqrt 7.4%

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

   13. rem-cube-cbrt 7.2%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left({\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}\right)}\right)\right)} \]

   14. add-sqr-sqrt 4.0%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}} \cdot \sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right)}\right)\right)} \]

   15. fabs-sqr 4.0%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \color{blue}{\left(\left|\sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}} \cdot \sqrt{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right|\right)}\right)\right)} \]

   16. add-sqr-sqrt 6.2%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\left|\color{blue}{{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}}\right)}^{3}}\right|\right)\right)\right)} \]

   17. unpow3 5.9%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\log \left(1 + \mathsf{expm1}\left(\sin \left(\left|\color{blue}{\left(\sqrt[3]{x \cdot \frac{-0.5}{y}} \cdot \sqrt[3]{x \cdot \frac{-0.5}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{-0.5}{y}}}\right|\right)\right)\right)} \]

10. Applied egg-rr 3.9%

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

11. Taylor expanded in x around 0 6.1%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 6.1%

    \[\leadsto -1 \]
13. Add Preprocessing

Alternative 7: 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 50.1%

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

   1. remove-double-neg 50.1%

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

   2. distribute-frac-neg 50.1%

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

   3. tan-neg 50.1%

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

   4. distribute-frac-neg2 50.1%

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

   5. distribute-lft-neg-out 50.1%

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

   6. distribute-frac-neg2 50.1%

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

   7. distribute-lft-neg-out 50.1%

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

   8. distribute-frac-neg2 50.1%

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

   9. distribute-frac-neg 50.1%

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

   10. neg-mul-1 50.1%

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

   11. *-commutative 50.1%

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

   12. associate-/l* 49.3%

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

   13. *-commutative 49.3%

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

   14. associate-/r* 49.3%

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

   15. metadata-eval 49.3%

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

   16. sin-neg 49.3%

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

   17. distribute-frac-neg 49.3%

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

3. Simplified 49.5%

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

5. Taylor expanded in x around 0 60.2%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 60.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 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 2024115 
(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)))))