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

Percentage Accurate: 44.5% → 57.4%

Time: 11.1s

Alternatives: 4

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: 44.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\left(--0.5\right) \cdot \frac{{y\_m}^{-1}}{\frac{-1}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 1e+153)
   (/ 1.0 (cos (* (- -0.5) (/ (pow y_m -1.0) (/ -1.0 x_m)))))
   (* (pow 0.25 0.25) (* (sqrt 0.5) -2.0))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / cos((-(-0.5) * (pow(y_m, -1.0) / (-1.0 / x_m))));
	} else {
		tmp = pow(0.25, 0.25) * (sqrt(0.5) * -2.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 1d+153) then
        tmp = 1.0d0 / cos((-(-0.5d0) * ((y_m ** (-1.0d0)) / ((-1.0d0) / x_m))))
    else
        tmp = (0.25d0 ** 0.25d0) * (sqrt(0.5d0) * (-2.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / Math.cos((-(-0.5) * (Math.pow(y_m, -1.0) / (-1.0 / x_m))));
	} else {
		tmp = Math.pow(0.25, 0.25) * (Math.sqrt(0.5) * -2.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 1e+153:
		tmp = 1.0 / math.cos((-(-0.5) * (math.pow(y_m, -1.0) / (-1.0 / x_m))))
	else:
		tmp = math.pow(0.25, 0.25) * (math.sqrt(0.5) * -2.0)
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 1e+153)
		tmp = Float64(1.0 / cos(Float64(Float64(-(-0.5)) * Float64((y_m ^ -1.0) / Float64(-1.0 / x_m)))));
	else
		tmp = Float64((0.25 ^ 0.25) * Float64(sqrt(0.5) * -2.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 1e+153)
		tmp = 1.0 / cos((-(-0.5) * ((y_m ^ -1.0) / (-1.0 / x_m))));
	else
		tmp = (0.25 ^ 0.25) * (sqrt(0.5) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 1e+153], N[(1.0 / N[Cos[N[((--0.5) * N[(N[Power[y$95$m, -1.0], $MachinePrecision] / N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.25, 0.25], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e153

   1. Initial program 48.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 48.1

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

   4. Applied rewrites 48.1%

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

   6. Applied rewrites 62.0%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

         \[\leadsto \frac{1}{\cos \left(\frac{0 - \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \cdot \frac{x}{y}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{-1} \cdot \color{blue}{\frac{x}{y}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      11. inv-pow N/A

          \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{-1} \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{-1}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      12. pow-prod-up N/A

          \[\leadsto \frac{1}{\cos \left(\frac{0 - \color{blue}{{\left(\frac{y}{x}\right)}^{\left(-1 + -1\right)}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{\color{blue}{-2}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{\color{blue}{-2}}}{0 - \frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      18. sub0-neg N/A

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

      19. lower-neg.f64 46.9

          \[\leadsto \frac{1}{\cos \left(\frac{0 - {\left(\frac{y}{x}\right)}^{-2}}{\color{blue}{-\frac{x}{y}}} \cdot -0.5\right)} \]

   8. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{0 - {\left(\frac{y}{x}\right)}^{-2}}{-\frac{x}{y}}} \cdot -0.5\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{0 - {\left(\frac{y}{x}\right)}^{-2}}{-\frac{x}{y}}} \cdot \frac{-1}{2}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{0 - {\left(\frac{y}{x}\right)}^{-2}}}{-\frac{x}{y}} \cdot \frac{-1}{2}\right)} \]

      3. sub0-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-div N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. frac-2neg N/A

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

      18. mul-1-neg N/A

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

      19. div-inv N/A

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

      20. lift-/.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. inv-pow N/A

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

      24. lower-pow.f64 62.4

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

   10. Applied rewrites 62.4%

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

   if 1e153 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.8%

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

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-pow.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 0.2

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

   4. Applied rewrites 0.2%

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

   6. Applied rewrites 2.6%

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{{4}^{-0.25}}{-1} \cdot \frac{{\left(y \cdot \left(2 \cdot y\right)\right)}^{-0.5}}{\frac{-1}{x}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-2 \cdot \left({\frac{1}{4}}^{\frac{1}{4}} \cdot \sqrt{\frac{1}{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot {\frac{1}{4}}^{\frac{1}{4}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right)} \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      6. lower-pow.f64 12.0

         \[\leadsto \left(-2 \cdot \sqrt{0.5}\right) \cdot \color{blue}{{0.25}^{0.25}} \]

   9. Applied rewrites 12.0%

      \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{0.5}\right) \cdot {0.25}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\left(--0.5\right) \cdot \frac{{y}^{-1}}{\frac{-1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m}{y\_m} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 1e+153)
   (/ 1.0 (cos (* (/ x_m y_m) -0.5)))
   (* (pow 0.25 0.25) (* (sqrt 0.5) -2.0))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	} else {
		tmp = pow(0.25, 0.25) * (sqrt(0.5) * -2.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 1d+153) then
        tmp = 1.0d0 / cos(((x_m / y_m) * (-0.5d0)))
    else
        tmp = (0.25d0 ** 0.25d0) * (sqrt(0.5d0) * (-2.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / Math.cos(((x_m / y_m) * -0.5));
	} else {
		tmp = Math.pow(0.25, 0.25) * (Math.sqrt(0.5) * -2.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 1e+153:
		tmp = 1.0 / math.cos(((x_m / y_m) * -0.5))
	else:
		tmp = math.pow(0.25, 0.25) * (math.sqrt(0.5) * -2.0)
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 1e+153)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m / y_m) * -0.5)));
	else
		tmp = Float64((0.25 ^ 0.25) * Float64(sqrt(0.5) * -2.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 1e+153)
		tmp = 1.0 / cos(((x_m / y_m) * -0.5));
	else
		tmp = (0.25 ^ 0.25) * (sqrt(0.5) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 1e+153], N[(1.0 / N[Cos[N[(N[(x$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.25, 0.25], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e153

   1. Initial program 48.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. cos-neg N/A

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

      16. lift-/.f64 N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lower-cos.f64 N/A

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

      19. distribute-frac-neg2 N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 62.0%

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

   if 1e153 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.8%

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

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-pow.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 0.2

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

   4. Applied rewrites 0.2%

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

   6. Applied rewrites 2.6%

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{{4}^{-0.25}}{-1} \cdot \frac{{\left(y \cdot \left(2 \cdot y\right)\right)}^{-0.5}}{\frac{-1}{x}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-2 \cdot \left({\frac{1}{4}}^{\frac{1}{4}} \cdot \sqrt{\frac{1}{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot {\frac{1}{4}}^{\frac{1}{4}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right)} \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      6. lower-pow.f64 12.0

         \[\leadsto \left(-2 \cdot \sqrt{0.5}\right) \cdot \color{blue}{{0.25}^{0.25}} \]

   9. Applied rewrites 12.0%

      \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{0.5}\right) \cdot {0.25}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y\_m} \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* 2.0 y_m)) 1e+153)
   (/ 1.0 (cos (* (/ 0.5 y_m) x_m)))
   (* (pow 0.25 0.25) (* (sqrt 0.5) -2.0))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / cos(((0.5 / y_m) * x_m));
	} else {
		tmp = pow(0.25, 0.25) * (sqrt(0.5) * -2.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (2.0d0 * y_m)) <= 1d+153) then
        tmp = 1.0d0 / cos(((0.5d0 / y_m) * x_m))
    else
        tmp = (0.25d0 ** 0.25d0) * (sqrt(0.5d0) * (-2.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (2.0 * y_m)) <= 1e+153) {
		tmp = 1.0 / Math.cos(((0.5 / y_m) * x_m));
	} else {
		tmp = Math.pow(0.25, 0.25) * (Math.sqrt(0.5) * -2.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (2.0 * y_m)) <= 1e+153:
		tmp = 1.0 / math.cos(((0.5 / y_m) * x_m))
	else:
		tmp = math.pow(0.25, 0.25) * (math.sqrt(0.5) * -2.0)
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(2.0 * y_m)) <= 1e+153)
		tmp = Float64(1.0 / cos(Float64(Float64(0.5 / y_m) * x_m)));
	else
		tmp = Float64((0.25 ^ 0.25) * Float64(sqrt(0.5) * -2.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (2.0 * y_m)) <= 1e+153)
		tmp = 1.0 / cos(((0.5 / y_m) * x_m));
	else
		tmp = (0.25 ^ 0.25) * (sqrt(0.5) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 1e+153], N[(1.0 / N[Cos[N[(N[(0.5 / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.25, 0.25], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e153

   1. Initial program 48.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if 1e153 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.8%

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

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-pow.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 0.2

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

   4. Applied rewrites 0.2%

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

   6. Applied rewrites 2.6%

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{{4}^{-0.25}}{-1} \cdot \frac{{\left(y \cdot \left(2 \cdot y\right)\right)}^{-0.5}}{\frac{-1}{x}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-2 \cdot \left({\frac{1}{4}}^{\frac{1}{4}} \cdot \sqrt{\frac{1}{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot {\frac{1}{4}}^{\frac{1}{4}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{\frac{1}{2}}\right)} \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot {\frac{1}{4}}^{\frac{1}{4}} \]

      6. lower-pow.f64 12.0

         \[\leadsto \left(-2 \cdot \sqrt{0.5}\right) \cdot \color{blue}{{0.25}^{0.25}} \]

   9. Applied rewrites 12.0%

      \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{0.5}\right) \cdot {0.25}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2 \cdot y} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\sqrt{0.5} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.9% 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 42.2%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 54.1%

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

Developer Target 1: 55.9% 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 2024255 
(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)))))