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

Percentage Accurate: 43.9% → 56.5%

Time: 14.5s

Alternatives: 4

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: 43.9% 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: 56.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{y\_m \cdot 2}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+228}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x\_m}{t\_0}}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (* y_m 2.0))))
   (if (<= (/ x_m (* y_m 2.0)) 1e+228) (/ 1.0 (cos (/ (/ x_m 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 * 2.0));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+228) {
		tmp = 1.0 / cos(((x_m / 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 * 2.0d0))
    if ((x_m / (y_m * 2.0d0)) <= 1d+228) then
        tmp = 1.0d0 / cos(((x_m / 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 * 2.0));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+228) {
		tmp = 1.0 / Math.cos(((x_m / 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 * 2.0))
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+228:
		tmp = 1.0 / math.cos(((x_m / 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 * 2.0))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+228)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / 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 * 2.0));
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+228)
		tmp = 1.0 / cos(((x_m / 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 * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+228], N[(1.0 / N[Cos[N[(N[(x$95$m / 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))) < 9.9999999999999992e227

   1. Initial program 48.0%

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

      1. remove-double-neg 48.0%

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

      2. distribute-frac-neg 48.0%

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

      3. tan-neg 48.0%

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

      4. distribute-frac-neg2 48.0%

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

      5. distribute-lft-neg-out 48.0%

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

      6. distribute-frac-neg2 48.0%

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

      7. distribute-lft-neg-out 48.0%

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

      8. distribute-frac-neg2 48.0%

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

      9. distribute-frac-neg 48.0%

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

      10. neg-mul-1 48.0%

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

      11. *-commutative 48.0%

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

      12. associate-/l* 47.8%

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

      13. *-commutative 47.8%

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

      14. associate-/r* 47.8%

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

      15. metadata-eval 47.8%

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

      16. sin-neg 47.8%

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

      17. distribute-frac-neg 47.8%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in x around inf 61.7%

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

      1. associate-*r/ 61.7%

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

      2. *-commutative 61.7%

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

      3. associate-*r/ 61.4%

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

   7. Simplified 61.4%

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

      1. associate-*r/ 61.7%

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

      2. clear-num 61.6%

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

   9. Applied egg-rr 61.6%

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

       1. clear-num 61.7%

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

       2. associate-*r/ 61.4%

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

       3. add-sqr-sqrt 39.2%

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

       4. sqrt-unprod 60.2%

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

       5. associate-*r/ 60.5%

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

       6. *-commutative 60.5%

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

       7. associate-*r/ 60.5%

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

       8. associate-*r/ 60.6%

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

       9. *-commutative 60.6%

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

       10. associate-*r/ 60.6%

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

       11. swap-sqr 60.6%

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

       12. metadata-eval 60.6%

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

       13. metadata-eval 60.6%

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

       14. swap-sqr 60.6%

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

       15. sqrt-unprod 36.6%

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

       16. add-sqr-sqrt 61.7%

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

       17. metadata-eval 61.7%

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

       18. times-frac 61.7%

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

       19. *-commutative 61.7%

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

       20. add-sqr-sqrt 28.6%

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

       21. times-frac 28.9%

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

   11. Applied egg-rr 28.9%

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

       1. associate-*l/ 28.9%

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

       2. *-lft-identity 28.9%

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

       3. *-commutative 28.9%

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

       4. *-commutative 28.9%

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

   13. Simplified 28.9%

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

   if 9.9999999999999992e227 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.7%

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

      1. remove-double-neg 4.7%

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

      2. distribute-frac-neg 4.7%

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

      3. tan-neg 4.7%

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

      4. distribute-frac-neg2 4.7%

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

      5. distribute-lft-neg-out 4.7%

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

      6. distribute-frac-neg2 4.7%

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

      7. distribute-lft-neg-out 4.7%

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

      8. distribute-frac-neg2 4.7%

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

      9. distribute-frac-neg 4.7%

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

      10. neg-mul-1 4.7%

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

      11. *-commutative 4.7%

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

      12. associate-/l* 5.4%

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

      13. *-commutative 5.4%

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

      14. associate-/r* 5.4%

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

      15. metadata-eval 5.4%

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

      16. sin-neg 5.4%

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

      17. distribute-frac-neg 5.4%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in x around 0 15.1%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+228}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{x}{\sqrt{y \cdot 2}}}{\sqrt{y \cdot 2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{1}{\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x\_m \cdot 0.5}{y\_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 4.5e+134)
   (/ 1.0 (cos (expm1 (log1p (/ (* 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)) <= 4.5e+134) {
		tmp = 1.0 / cos(expm1(log1p(((x_m * 0.5) / y_m))));
	} 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)) <= 4.5e+134) {
		tmp = 1.0 / Math.cos(Math.expm1(Math.log1p(((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)) <= 4.5e+134:
		tmp = 1.0 / math.cos(math.expm1(math.log1p(((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)) <= 4.5e+134)
		tmp = Float64(1.0 / cos(expm1(log1p(Float64(Float64(x_m * 0.5) / y_m)))));
	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], 4.5e+134], N[(1.0 / N[Cos[N[(Exp[N[Log[1 + N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.4999999999999997e134

   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* 50.0%

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

      13. *-commutative 50.0%

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

      14. associate-/r* 50.0%

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

      15. metadata-eval 50.0%

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

      16. sin-neg 50.0%

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

      17. distribute-frac-neg 50.0%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in x around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-*r/ 64.3%

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

   7. Simplified 64.3%

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

      1. associate-*r/ 64.6%

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

      2. add-sqr-sqrt 29.9%

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

      3. associate-/r* 29.9%

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

   9. Applied egg-rr 29.9%

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

       1. associate-/l/ 29.9%

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

       2. *-commutative 29.9%

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

       3. add-sqr-sqrt 64.6%

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

       4. associate-*l/ 64.3%

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

       5. add-sqr-sqrt 34.1%

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

       6. sqrt-unprod 59.9%

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

       7. frac-times 59.9%

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

       8. metadata-eval 59.9%

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

       9. metadata-eval 59.9%

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

       10. frac-times 59.9%

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

       11. sqrt-unprod 29.9%

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

       12. add-sqr-sqrt 64.3%

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

       13. associate-*l/ 64.6%

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

       14. expm1-log1p-u 62.5%

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

       15. associate-*r/ 62.5%

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

       16. expm1-undefine 62.4%

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

   11. Applied egg-rr 62.4%

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

       1. expm1-define 62.5%

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

       2. *-commutative 62.5%

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

       3. associate-*l/ 62.5%

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

   13. Simplified 62.5%

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

   if 4.4999999999999997e134 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.9%

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

      1. remove-double-neg 5.9%

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

      2. distribute-frac-neg 5.9%

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

      3. tan-neg 5.9%

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

      4. distribute-frac-neg2 5.9%

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

      5. distribute-lft-neg-out 5.9%

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

      6. distribute-frac-neg2 5.9%

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

      7. distribute-lft-neg-out 5.9%

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

      8. distribute-frac-neg2 5.9%

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

      9. distribute-frac-neg 5.9%

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

      10. neg-mul-1 5.9%

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

      11. *-commutative 5.9%

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

      12. associate-/l* 6.4%

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

      13. *-commutative 6.4%

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

      14. associate-/r* 6.4%

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

      15. metadata-eval 6.4%

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

      16. sin-neg 6.4%

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

      17. distribute-frac-neg 6.4%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in x around 0 12.2%

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

Alternative 3: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 5e+56)
   (/ 1.0 (cos (/ 1.0 (* y_m (/ 2.0 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)) <= 5e+56) {
		tmp = 1.0 / cos((1.0 / (y_m * (2.0 / 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)) <= 5d+56) then
        tmp = 1.0d0 / cos((1.0d0 / (y_m * (2.0d0 / 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)) <= 5e+56) {
		tmp = 1.0 / Math.cos((1.0 / (y_m * (2.0 / 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)) <= 5e+56:
		tmp = 1.0 / math.cos((1.0 / (y_m * (2.0 / 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)) <= 5e+56)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(y_m * Float64(2.0 / 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)) <= 5e+56)
		tmp = 1.0 / cos((1.0 / (y_m * (2.0 / 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], 5e+56], N[(1.0 / N[Cos[N[(1.0 / N[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000024e56

   1. Initial program 53.2%

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

      1. remove-double-neg 53.2%

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

      2. distribute-frac-neg 53.2%

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

      3. tan-neg 53.2%

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

      4. distribute-frac-neg2 53.2%

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

      5. distribute-lft-neg-out 53.2%

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

      6. distribute-frac-neg2 53.2%

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

      7. distribute-lft-neg-out 53.2%

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

      8. distribute-frac-neg2 53.2%

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

      9. distribute-frac-neg 53.2%

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

      10. neg-mul-1 53.2%

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

      11. *-commutative 53.2%

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

      12. associate-/l* 53.0%

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

      13. *-commutative 53.0%

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

      14. associate-/r* 53.0%

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

      15. metadata-eval 53.0%

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

      16. sin-neg 53.0%

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

      17. distribute-frac-neg 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in x around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*r/ 68.5%

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

   7. Simplified 68.5%

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

      1. associate-*r/ 68.8%

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

      2. add-sqr-sqrt 31.9%

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

      3. associate-/r* 32.0%

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

   9. Applied egg-rr 32.0%

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

       1. associate-/l/ 31.9%

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

       2. add-sqr-sqrt 68.8%

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

       3. associate-*r/ 68.5%

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

       4. add-sqr-sqrt 36.2%

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

       5. sqrt-unprod 63.7%

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

       6. frac-times 63.7%

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

       7. metadata-eval 63.7%

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

       8. metadata-eval 63.7%

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

       9. frac-times 63.7%

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

       10. sqrt-unprod 32.0%

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

       11. add-sqr-sqrt 68.5%

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

       12. metadata-eval 68.5%

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

       13. associate-/r* 68.5%

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

       14. *-commutative 68.5%

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

       15. div-inv 68.8%

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

       16. clear-num 68.8%

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

       17. inv-pow 68.8%

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

       18. associate-/l* 68.9%

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

   11. Applied egg-rr 68.9%

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

       1. unpow-1 68.9%

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

   13. Simplified 68.9%

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

   if 5.00000000000000024e56 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.5%

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

      1. remove-double-neg 6.5%

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

      2. distribute-frac-neg 6.5%

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

      3. tan-neg 6.5%

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

      4. distribute-frac-neg2 6.5%

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

      5. distribute-lft-neg-out 6.5%

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

      6. distribute-frac-neg2 6.5%

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

      7. distribute-lft-neg-out 6.5%

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

      8. distribute-frac-neg2 6.5%

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

      9. distribute-frac-neg 6.5%

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

      10. neg-mul-1 6.5%

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

      11. *-commutative 6.5%

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

      12. associate-/l* 7.0%

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

      13. *-commutative 7.0%

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

      14. associate-/r* 7.0%

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

      15. metadata-eval 7.0%

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

      16. sin-neg 7.0%

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

      17. distribute-frac-neg 7.0%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in x around 0 12.0%

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

Alternative 4: 55.0% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

   1. remove-double-neg 44.1%

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

   2. distribute-frac-neg 44.1%

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

   3. tan-neg 44.1%

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

   4. distribute-frac-neg2 44.1%

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

   5. distribute-lft-neg-out 44.1%

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

   6. distribute-frac-neg2 44.1%

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

   7. distribute-lft-neg-out 44.1%

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

   8. distribute-frac-neg2 44.1%

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

   9. distribute-frac-neg 44.1%

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

   10. neg-mul-1 44.1%

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

   11. *-commutative 44.1%

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

   12. associate-/l* 44.0%

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

   13. *-commutative 44.0%

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

   14. associate-/r* 44.0%

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

   15. metadata-eval 44.0%

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

   16. sin-neg 44.0%

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

   17. distribute-frac-neg 44.0%

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

3. Simplified 43.9%

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

5. Taylor expanded in x around 0 57.8%

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

Developer target: 55.0% 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 2024101 
(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)))))