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

Percentage Accurate: 44.0% → 56.8%

Time: 12.5s

Alternatives: 7

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: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+106}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{y\_m}{x\_m \cdot 0.5}\right)}^{-0.5} \cdot \frac{\sqrt{\frac{x\_m}{y\_m}}}{\sqrt{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+106)
   (/
    1.0
    (cos (* (pow (/ y_m (* x_m 0.5)) -0.5) (/ (sqrt (/ x_m y_m)) (sqrt 2.0)))))
   -1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+106) {
		tmp = 1.0 / cos((pow((y_m / (x_m * 0.5)), -0.5) * (sqrt((x_m / y_m)) / sqrt(2.0))));
	} 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 / (y_m * 2.0d0)) <= 1d+106) then
        tmp = 1.0d0 / cos((((y_m / (x_m * 0.5d0)) ** (-0.5d0)) * (sqrt((x_m / y_m)) / sqrt(2.0d0))))
    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 / (y_m * 2.0)) <= 1e+106) {
		tmp = 1.0 / Math.cos((Math.pow((y_m / (x_m * 0.5)), -0.5) * (Math.sqrt((x_m / y_m)) / Math.sqrt(2.0))));
	} 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 / (y_m * 2.0)) <= 1e+106:
		tmp = 1.0 / math.cos((math.pow((y_m / (x_m * 0.5)), -0.5) * (math.sqrt((x_m / y_m)) / math.sqrt(2.0))))
	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(y_m * 2.0)) <= 1e+106)
		tmp = Float64(1.0 / cos(Float64((Float64(y_m / Float64(x_m * 0.5)) ^ -0.5) * Float64(sqrt(Float64(x_m / y_m)) / sqrt(2.0)))));
	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 / (y_m * 2.0)) <= 1e+106)
		tmp = 1.0 / cos((((y_m / (x_m * 0.5)) ^ -0.5) * (sqrt((x_m / y_m)) / sqrt(2.0))));
	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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+106], N[(1.0 / N[Cos[N[(N[Power[N[(y$95$m / N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[Sqrt[N[(x$95$m / y$95$m), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.00000000000000009e106

   1. Initial program 53.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. metadata-eval N/A

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

      2. div-inv N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. clear-num N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. div-inv N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower-pow.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 44.5

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

   10. Applied rewrites 44.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. sqrt-unprod N/A

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

       8. metadata-eval N/A

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

       9. div-inv N/A

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

       10. sqrt-div N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-sqrt.f64 44.6

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

   12. Applied rewrites 44.6%

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

   if 1.00000000000000009e106 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.7%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 3.9%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. *-inverses N/A

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

      17. frac-2neg N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 2.0%

      \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

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

   9. Applied rewrites 14.5%

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

Alternative 2: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;t\_0 \leq 10^{+86}:\\ \;\;\;\;\frac{1}{\cos \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))) (t_1 (sqrt t_0)))
   (if (<= t_0 1e+86) (/ 1.0 (cos (* t_1 t_1))) -1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double t_1 = sqrt(t_0);
	double tmp;
	if (t_0 <= 1e+86) {
		tmp = 1.0 / cos((t_1 * t_1));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m / (y_m * 2.0d0)
    t_1 = sqrt(t_0)
    if (t_0 <= 1d+86) then
        tmp = 1.0d0 / cos((t_1 * t_1))
    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 t_0 = x_m / (y_m * 2.0);
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (t_0 <= 1e+86) {
		tmp = 1.0 / Math.cos((t_1 * t_1));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	t_0 = x_m / (y_m * 2.0)
	t_1 = math.sqrt(t_0)
	tmp = 0
	if t_0 <= 1e+86:
		tmp = 1.0 / math.cos((t_1 * t_1))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (t_0 <= 1e+86)
		tmp = Float64(1.0 / cos(Float64(t_1 * t_1)));
	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)
	t_0 = x_m / (y_m * 2.0);
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (t_0 <= 1e+86)
		tmp = 1.0 / cos((t_1 * t_1));
	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_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+86], N[(1.0 / N[Cos[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e86

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

      1. metadata-eval N/A

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

      2. div-inv N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. clear-num N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. div-inv N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower-pow.f64 N/A

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 45.7

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

   10. Applied rewrites 45.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. lift-/.f64 45.7

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

   12. Applied rewrites 45.8%

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

   if 1e86 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.5%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 3.5%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. *-inverses N/A

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

      17. frac-2neg N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 1.9%

      \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

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

   9. Applied rewrites 13.8%

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

Alternative 3: 56.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{y\_m}}{\frac{2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+106)
   (/ 1.0 (cos (/ (/ 1.0 y_m) (/ 2.0 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 / (y_m * 2.0)) <= 5e+106) {
		tmp = 1.0 / cos(((1.0 / y_m) / (2.0 / 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 / (y_m * 2.0d0)) <= 5d+106) then
        tmp = 1.0d0 / cos(((1.0d0 / y_m) / (2.0d0 / 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 / (y_m * 2.0)) <= 5e+106) {
		tmp = 1.0 / Math.cos(((1.0 / y_m) / (2.0 / 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 / (y_m * 2.0)) <= 5e+106:
		tmp = 1.0 / math.cos(((1.0 / y_m) / (2.0 / 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(y_m * 2.0)) <= 5e+106)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / y_m) / Float64(2.0 / 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 / (y_m * 2.0)) <= 5e+106)
		tmp = 1.0 / cos(((1.0 / y_m) / (2.0 / 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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+106], N[(1.0 / N[Cos[N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999998e106

   1. Initial program 53.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

      1. metadata-eval N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if 4.9999999999999998e106 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.5%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 4.0%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. *-inverses N/A

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

      17. frac-2neg N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 2.0%

      \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

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

   9. Applied rewrites 14.5%

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

Alternative 4: 57.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+31) (/ 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 / (y_m * 2.0)) <= 2e+31) {
		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 / (y_m * 2.0d0)) <= 2d+31) 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 / (y_m * 2.0)) <= 2e+31) {
		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 / (y_m * 2.0)) <= 2e+31:
		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(y_m * 2.0)) <= 2e+31)
		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 / (y_m * 2.0)) <= 2e+31)
		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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+31], 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))) < 1.9999999999999999e31

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 76.9

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

   7. Applied rewrites 76.9%

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

   if 1.9999999999999999e31 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 3.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. *-inverses N/A

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

      17. frac-2neg N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 1.9%

      \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

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

   9. Applied rewrites 13.7%

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

Alternative 5: 57.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+14) (/ 1.0 (cos (* x_m (/ 0.5 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 / (y_m * 2.0)) <= 5e+14) {
		tmp = 1.0 / cos((x_m * (0.5 / 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 / (y_m * 2.0d0)) <= 5d+14) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / 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 / (y_m * 2.0)) <= 5e+14) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / 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 / (y_m * 2.0)) <= 5e+14:
		tmp = 1.0 / math.cos((x_m * (0.5 / 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(y_m * 2.0)) <= 5e+14)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / 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 / (y_m * 2.0)) <= 5e+14)
		tmp = 1.0 / cos((x_m * (0.5 / 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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+14], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5e14

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

   if 5e14 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.6%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 3.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. *-inverses N/A

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

      17. frac-2neg N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 1.9%

      \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

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

   9. Applied rewrites 13.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.0% 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 45.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 59.6%

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

Alternative 7: 6.8% 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 45.4%

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

   1. associate-/l/ N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. neg-sub0 N/A

      \[\leadsto \frac{\tan \left(\color{blue}{\left(0 - x\right)} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   10. flip-- N/A

       \[\leadsto \frac{\tan \left(\color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   11. frac-times N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\left(0 + x\right) \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   12. +-lft-identity N/A

       \[\leadsto \frac{\tan \left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\color{blue}{x} \cdot y}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{\left(0 \cdot 0 - x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

4. Applied rewrites 21.6%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. remove-double-neg N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. times-frac N/A

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

   16. *-inverses N/A

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

   17. frac-2neg N/A

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

   18. clear-num N/A

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

   19. lift-/.f64 N/A

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

   20. div-inv N/A

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

6. Applied rewrites 6.7%

   \[\leadsto \frac{\tan \color{blue}{\left({\left(\frac{y \cdot y}{\left(x \cdot x\right) \cdot 0.25}\right)}^{-0.5}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

7. Taylor expanded in y around -inf

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

9. Applied rewrites 6.6%

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

Developer Target 1: 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 2024214 
(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)))))