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

Percentage Accurate: 44.0% → 57.4%

Time: 8.3s

Alternatives: 5

Speedup: 244.0×

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: 44.0% 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: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-1}{\frac{0.5}{y\_m} \cdot x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-\log \left(\frac{x\_m}{y\_m}\right)} \cdot \frac{0.125}{y\_m}, x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+78)
   (/ 1.0 (cos (/ 1.0 (/ -1.0 (* (/ 0.5 y_m) x_m)))))
   (fma (* (exp (- (log (/ x_m y_m)))) (/ 0.125 y_m)) x_m 1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+78) {
		tmp = 1.0 / cos((1.0 / (-1.0 / ((0.5 / y_m) * x_m))));
	} else {
		tmp = fma((exp(-log((x_m / y_m))) * (0.125 / y_m)), x_m, 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+78)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(-1.0 / Float64(Float64(0.5 / y_m) * x_m)))));
	else
		tmp = fma(Float64(exp(Float64(-log(Float64(x_m / y_m)))) * Float64(0.125 / y_m)), x_m, 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+78], N[(1.0 / N[Cos[N[(1.0 / N[(-1.0 / N[(N[(0.5 / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[Log[N[(x$95$m / y$95$m), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[(0.125 / y$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000002e78

   1. Initial program 46.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.3

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

   4. Applied rewrites 46.3%

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

   6. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. remove-double-div N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. *-commutative N/A

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

      21. lift-/.f64 N/A

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

      22. associate-*r/ N/A

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

      23. metadata-eval N/A

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

      24. div-inv N/A

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

      25. associate-/l/ N/A

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

      26. lift-*.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. lower-/.f64 63.5

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

      29. lift-/.f64 N/A

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

      30. lift-*.f64 N/A

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

   8. Applied rewrites 63.7%

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

   if 2.00000000000000002e78 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{8} \cdot \frac{{x}^{2}}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{{x}^{2}}{{y}^{2}} + 1} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot {x}^{2}}{{y}^{2}}} + 1 \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot {x}^{2}}{\color{blue}{y \cdot y}} + 1 \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8}}{y} \cdot \frac{{x}^{2}}{y}} + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{8}}{y}, \frac{{x}^{2}}{y}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{y}}, \frac{{x}^{2}}{y}, 1\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{y}, \color{blue}{\frac{{x}^{2}}{y}}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{y}, \frac{\color{blue}{x \cdot x}}{y}, 1\right) \]

      9. lower-*.f64 2.0

         \[\leadsto \mathsf{fma}\left(\frac{0.125}{y}, \frac{\color{blue}{x \cdot x}}{y}, 1\right) \]

   5. Applied rewrites 2.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{y}, \frac{x \cdot x}{y}, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 11.8%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{y} \cdot \frac{y}{x}, \color{blue}{x}, 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 12.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{y} \cdot e^{\log \left(\frac{x}{y}\right) \cdot -1}, x, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-1}{\frac{0.5}{y} \cdot x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-\log \left(\frac{x}{y}\right)} \cdot \frac{0.125}{y}, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-1}{\frac{0.5}{y\_m} \cdot x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 2e+78)
   (/ 1.0 (cos (/ 1.0 (/ -1.0 (* (/ 0.5 y_m) x_m)))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+78) {
		tmp = 1.0 / cos((1.0 / (-1.0 / ((0.5 / y_m) * x_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 2d+78) then
        tmp = 1.0d0 / cos((1.0d0 / ((-1.0d0) / ((0.5d0 / y_m) * x_m))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 2e+78) {
		tmp = 1.0 / Math.cos((1.0 / (-1.0 / ((0.5 / y_m) * x_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 2e+78:
		tmp = 1.0 / math.cos((1.0 / (-1.0 / ((0.5 / y_m) * x_m))))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 2e+78)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(-1.0 / Float64(Float64(0.5 / y_m) * x_m)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 2e+78)
		tmp = 1.0 / cos((1.0 / (-1.0 / ((0.5 / y_m) * x_m))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+78], N[(1.0 / N[Cos[N[(1.0 / N[(-1.0 / N[(N[(0.5 / y$95$m), $MachinePrecision] * 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))) < 2.00000000000000002e78

   1. Initial program 46.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.3

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

   4. Applied rewrites 46.3%

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

   6. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. remove-double-div N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. *-commutative N/A

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

      21. lift-/.f64 N/A

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

      22. associate-*r/ N/A

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

      23. metadata-eval N/A

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

      24. div-inv N/A

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

      25. associate-/l/ N/A

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

      26. lift-*.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. lower-/.f64 63.5

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

      29. lift-/.f64 N/A

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

      30. lift-*.f64 N/A

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

   8. Applied rewrites 63.7%

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

   if 2.00000000000000002e78 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.8%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-1}{\frac{0.5}{y} \cdot x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-2}{x\_m} \cdot y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 5e+81)
   (/ 1.0 (cos (/ 1.0 (* (/ -2.0 x_m) y_m))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+81) {
		tmp = 1.0 / cos((1.0 / ((-2.0 / x_m) * y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 5d+81) then
        tmp = 1.0d0 / cos((1.0d0 / (((-2.0d0) / x_m) * y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+81) {
		tmp = 1.0 / Math.cos((1.0 / ((-2.0 / x_m) * y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 5e+81:
		tmp = 1.0 / math.cos((1.0 / ((-2.0 / x_m) * y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 5e+81)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(Float64(-2.0 / x_m) * y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 5e+81)
		tmp = 1.0 / cos((1.0 / ((-2.0 / x_m) * y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+81], N[(1.0 / N[Cos[N[(1.0 / N[(N[(-2.0 / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999998e81

   1. Initial program 46.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. remove-double-div N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lower-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. *-commutative N/A

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

      21. lift-/.f64 N/A

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

      22. associate-*r/ N/A

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

      23. metadata-eval N/A

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

      24. div-inv N/A

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

      25. associate-/l/ N/A

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

      26. lift-*.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. lower-/.f64 63.3

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

      29. lift-/.f64 N/A

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

      30. lift-*.f64 N/A

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lft-identity N/A

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

      12. times-frac N/A

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

      13. /-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 63.3

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

   10. Applied rewrites 63.3%

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

   if 4.9999999999999998e81 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{-2}{x} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 5e+81) (/ 1.0 (cos (/ -0.5 (/ y_m x_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+81) {
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
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 / (2.0d0 * y_m)) <= 5d+81) then
        tmp = 1.0d0 / cos(((-0.5d0) / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 5e+81) {
		tmp = 1.0 / Math.cos((-0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 5e+81:
		tmp = 1.0 / math.cos((-0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 5e+81)
		tmp = Float64(1.0 / cos(Float64(-0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 5e+81)
		tmp = 1.0 / cos((-0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+81], N[(1.0 / N[Cos[N[(-0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999998e81

   1. Initial program 46.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. associate-/r/ N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 4.9999999999999998e81 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{-0.5}{\frac{y}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.1%

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

Developer Target 1: 55.6% 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 2024288 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

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