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

Percentage Accurate: 43.4% → 56.5%

Time: 9.1s

Alternatives: 5

Speedup: 244.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+285}:\\ \;\;\;\;{\cos \left(\frac{\sqrt{x\_m}}{-2} \cdot {\left(y\_m \cdot {x\_m}^{-0.5}\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 5e+285)
   (pow (cos (* (/ (sqrt x_m) -2.0) (pow (* y_m (pow x_m -0.5)) -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)) <= 5e+285) {
		tmp = pow(cos(((sqrt(x_m) / -2.0) * pow((y_m * pow(x_m, -0.5)), -1.0))), -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)) <= 5d+285) then
        tmp = cos(((sqrt(x_m) / (-2.0d0)) * ((y_m * (x_m ** (-0.5d0))) ** (-1.0d0)))) ** (-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)) <= 5e+285) {
		tmp = Math.pow(Math.cos(((Math.sqrt(x_m) / -2.0) * Math.pow((y_m * Math.pow(x_m, -0.5)), -1.0))), -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)) <= 5e+285:
		tmp = math.pow(math.cos(((math.sqrt(x_m) / -2.0) * math.pow((y_m * math.pow(x_m, -0.5)), -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)) <= 5e+285)
		tmp = cos(Float64(Float64(sqrt(x_m) / -2.0) * (Float64(y_m * (x_m ^ -0.5)) ^ -1.0))) ^ -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)) <= 5e+285)
		tmp = cos(((sqrt(x_m) / -2.0) * ((y_m * (x_m ^ -0.5)) ^ -1.0))) ^ -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], 5e+285], N[Power[N[Cos[N[(N[(N[Sqrt[x$95$m], $MachinePrecision] / -2.0), $MachinePrecision] * N[Power[N[(y$95$m * N[Power[x$95$m, -0.5], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000016e285

   1. Initial program 45.3%

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

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. rgt-mult-inverse N/A

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

      11. remove-double-div N/A

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

      12. cos-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/r* N/A

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

   4. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-frac-neg N/A

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

      11. distribute-frac-neg2 N/A

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

      12. div-inv N/A

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

      13. remove-double-div N/A

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

      14. frac-times N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. metadata-eval N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 57.4

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

   6. Applied rewrites 57.4%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. lift-pow.f64 N/A

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

      12. sqr-pow N/A

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

      13. associate-*r* N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   if 5.00000000000000016e285 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.7%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+285}:\\ \;\;\;\;{\cos \left(\frac{\sqrt{x}}{-2} \cdot {\left(y \cdot {x}^{-0.5}\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+268}:\\ \;\;\;\;{\cos \left({\left(\frac{-2}{\frac{x\_m}{y\_m}}\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+268)
   (pow (cos (pow (/ -2.0 (/ 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)) <= 1e+268) {
		tmp = pow(cos(pow((-2.0 / (x_m / y_m)), -1.0)), -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)) <= 1d+268) then
        tmp = cos((((-2.0d0) / (x_m / y_m)) ** (-1.0d0))) ** (-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)) <= 1e+268) {
		tmp = Math.pow(Math.cos(Math.pow((-2.0 / (x_m / y_m)), -1.0)), -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)) <= 1e+268:
		tmp = math.pow(math.cos(math.pow((-2.0 / (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)) <= 1e+268)
		tmp = cos((Float64(-2.0 / Float64(x_m / y_m)) ^ -1.0)) ^ -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)) <= 1e+268)
		tmp = cos(((-2.0 / (x_m / y_m)) ^ -1.0)) ^ -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], 1e+268], N[Power[N[Cos[N[Power[N[(-2.0 / N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.9999999999999997e267

   1. Initial program 45.9%

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

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. rgt-mult-inverse N/A

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

      11. remove-double-div N/A

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

      12. cos-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/r* N/A

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

   4. Applied rewrites 58.4%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-frac-neg N/A

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

      11. distribute-frac-neg2 N/A

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

      12. div-inv N/A

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

      13. remove-double-div N/A

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

      14. frac-times N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. metadata-eval N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 58.1

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

   6. Applied rewrites 58.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. remove-double-div N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if 9.9999999999999997e267 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+268}:\\ \;\;\;\;{\cos \left({\left(\frac{-2}{\frac{x}{y}}\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+280}:\\ \;\;\;\;{\cos \left(-0.5 \cdot \frac{x\_m}{y\_m}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+280)
   (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)) <= 1e+280) {
		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)) <= 1d+280) 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)) <= 1e+280) {
		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)) <= 1e+280:
		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)) <= 1e+280)
		tmp = cos(Float64(-0.5 * Float64(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)) <= 1e+280)
		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], 1e+280], N[Power[N[Cos[N[(-0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e280

   1. Initial program 45.8%

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

      1. lift-/.f64 N/A

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

      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. div-inv N/A

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. rgt-mult-inverse N/A

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

      11. remove-double-div N/A

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

      12. cos-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/r* N/A

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

   4. Applied rewrites 58.2%

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

   if 1e280 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 1.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+280}:\\ \;\;\;\;{\cos \left(-0.5 \cdot \frac{x}{y}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+244}:\\ \;\;\;\;{\cos \left(\frac{0.5}{y\_m} \cdot x\_m\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+244) (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)) <= 1e+244) {
		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)) <= 1d+244) 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)) <= 1e+244) {
		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)) <= 1e+244:
		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)) <= 1e+244)
		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)) <= 1e+244)
		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], 1e+244], 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))) < 1.00000000000000007e244

   1. Initial program 46.1%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

   if 1.00000000000000007e244 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+244}:\\ \;\;\;\;{\cos \left(\frac{0.5}{y} \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.0% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.0%

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

Developer Target 1: 55.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t\_1}{t\_1 \cdot \log \left(e^{\cos t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2024313 
(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)))))