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

Percentage Accurate: 44.1% → 56.2%

Time: 11.6s

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: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t\_0}{\sin t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 56.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\cos \left(\mathsf{fma}\left(-0.5, \frac{x\_m}{y\_m}, \mathsf{PI}\left(\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

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+138)
   (/ -1.0 (cos (fma -0.5 (/ x_m y_m) (PI))))
   -1.0))

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000016e138

   1. Initial program 54.1%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 68.2%

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

   if 5.00000000000000016e138 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. flip-+ N/A

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

      11. distribute-neg-frac2 N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. count-2-rev N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

      20. frac-2neg N/A

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

      21. lift-*.f64 N/A

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

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

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

      23. associate-/r* N/A

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

      24. lower-/.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. metadata-eval 4.7

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

   4. Applied rewrites 4.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 13.4%

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

Alternative 2: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;{\cos \left(\frac{0.5 \cdot x\_m}{y\_m}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

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)) 1.5e+16)
   (pow (cos (/ (* 0.5 x_m) y_m)) -1.0)
   -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)) <= 1.5e+16) {
		tmp = pow(cos(((0.5 * x_m) / y_m)), -1.0);
	} 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)) <= 1.5d+16) then
        tmp = cos(((0.5d0 * x_m) / y_m)) ** (-1.0d0)
    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)) <= 1.5e+16) {
		tmp = Math.pow(Math.cos(((0.5 * x_m) / y_m)), -1.0);
	} 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)) <= 1.5e+16:
		tmp = math.pow(math.cos(((0.5 * x_m) / y_m)), -1.0)
	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)) <= 1.5e+16)
		tmp = cos(Float64(Float64(0.5 * x_m) / y_m)) ^ -1.0;
	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)) <= 1.5e+16)
		tmp = cos(((0.5 * x_m) / y_m)) ^ -1.0;
	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], 1.5e+16], N[Power[N[Cos[N[(N[(0.5 * x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.5e16

   1. Initial program 58.0%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 71.5%

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

   if 1.5e16 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. flip-+ N/A

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

      11. distribute-neg-frac2 N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. count-2-rev N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

      20. frac-2neg N/A

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

      21. lift-*.f64 N/A

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

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

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

      23. associate-/r* N/A

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

      24. lower-/.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. metadata-eval 7.5

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

   4. Applied rewrites 7.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 13.0%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;{\cos \left(\frac{0.5 \cdot x}{y}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;{\cos \left(\frac{0.5}{y\_m} \cdot x\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

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)) 1.5e+16)
   (pow (cos (* (/ 0.5 y_m) x_m)) -1.0)
   -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)) <= 1.5e+16) {
		tmp = pow(cos(((0.5 / y_m) * x_m)), -1.0);
	} 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)) <= 1.5d+16) then
        tmp = cos(((0.5d0 / y_m) * x_m)) ** (-1.0d0)
    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)) <= 1.5e+16) {
		tmp = Math.pow(Math.cos(((0.5 / y_m) * x_m)), -1.0);
	} 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)) <= 1.5e+16:
		tmp = math.pow(math.cos(((0.5 / y_m) * x_m)), -1.0)
	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)) <= 1.5e+16)
		tmp = cos(Float64(Float64(0.5 / y_m) * x_m)) ^ -1.0;
	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)) <= 1.5e+16)
		tmp = cos(((0.5 / y_m) * x_m)) ^ -1.0;
	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], 1.5e+16], N[Power[N[Cos[N[(N[(0.5 / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.5e16

   1. Initial program 58.0%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 1.5e16 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. flip-+ N/A

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

      11. distribute-neg-frac2 N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. count-2-rev N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

      20. frac-2neg N/A

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

      21. lift-*.f64 N/A

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

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

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

      23. associate-/r* N/A

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

      24. lower-/.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. metadata-eval 7.5

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

   4. Applied rewrites 7.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 13.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;{\cos \left(\frac{0.5}{y} \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.0% accurate, 1.6× speedup

Math

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

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)) 0.5)
   (pow (fma (/ (* -0.125 x_m) y_m) (/ x_m y_m) 1.0) -1.0)
   -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)) <= 0.5) {
		tmp = pow(fma(((-0.125 * x_m) / y_m), (x_m / y_m), 1.0), -1.0);
	} 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)) <= 0.5)
		tmp = fma(Float64(Float64(-0.125 * x_m) / y_m), Float64(x_m / y_m), 1.0) ^ -1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 0.5

   1. Initial program 58.1%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.8%

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

   if 0.5 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 9.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. flip-+ N/A

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

      11. distribute-neg-frac2 N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. count-2-rev N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

      20. frac-2neg N/A

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

      21. lift-*.f64 N/A

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

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

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

      23. associate-/r* N/A

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

      24. lower-/.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. metadata-eval 7.1

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

   4. Applied rewrites 7.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 12.7%

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

4. Final simplification 55.9%

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

Alternative 5: 54.8% accurate, 10.6× speedup

Math

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

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)) 0.5) 1.0 -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)) <= 0.5) {
		tmp = 1.0;
	} 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)) <= 0.5d0) then
        tmp = 1.0d0
    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)) <= 0.5) {
		tmp = 1.0;
	} 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)) <= 0.5:
		tmp = 1.0
	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)) <= 0.5)
		tmp = 1.0;
	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)) <= 0.5)
		tmp = 1.0;
	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], 0.5], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 0.5

   1. Initial program 58.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 69.7%

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

   if 0.5 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 9.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. flip-+ N/A

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

      11. distribute-neg-frac2 N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. count-2-rev N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. remove-double-neg N/A

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

      20. frac-2neg N/A

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

      21. lift-*.f64 N/A

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

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

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

      23. associate-/r* N/A

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

      24. lower-/.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. metadata-eval 7.1

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

   4. Applied rewrites 7.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 12.7%

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

Alternative 6: 54.8% 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 47.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.0%

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

Alternative 7: 3.1% accurate, 244.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 0.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 0.0;
}

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 = 0.0d0
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return 0.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 0.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return 0.0
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 0.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := 0.0

Derivation

1. Initial program 47.0%

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

4. Applied rewrites 3.1%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Developer Target 1: 54.8% 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 2024339 
(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)))))