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

Percentage Accurate: 43.8% → 56.1%

Time: 13.1s

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: 43.8% 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: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{1}{\cos \left({y\_m}^{-0.5} \cdot \left({y\_m}^{-0.5} \cdot \left(x\_m \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + \cos \left(\frac{y\_m}{x\_m}\right)\right)}^{-0.5} \cdot {0.5}^{-0.5}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+220)
   (/ 1.0 (cos (* (pow y_m -0.5) (* (pow y_m -0.5) (* x_m 0.5)))))
   (* (pow (+ 1.0 (cos (/ y_m x_m))) -0.5) (pow 0.5 -0.5))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+220) {
		tmp = 1.0 / cos((pow(y_m, -0.5) * (pow(y_m, -0.5) * (x_m * 0.5))));
	} else {
		tmp = pow((1.0 + cos((y_m / x_m))), -0.5) * pow(0.5, -0.5);
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 2d+220) then
        tmp = 1.0d0 / cos(((y_m ** (-0.5d0)) * ((y_m ** (-0.5d0)) * (x_m * 0.5d0))))
    else
        tmp = ((1.0d0 + cos((y_m / x_m))) ** (-0.5d0)) * (0.5d0 ** (-0.5d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+220) {
		tmp = 1.0 / Math.cos((Math.pow(y_m, -0.5) * (Math.pow(y_m, -0.5) * (x_m * 0.5))));
	} else {
		tmp = Math.pow((1.0 + Math.cos((y_m / x_m))), -0.5) * Math.pow(0.5, -0.5);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+220:
		tmp = 1.0 / math.cos((math.pow(y_m, -0.5) * (math.pow(y_m, -0.5) * (x_m * 0.5))))
	else:
		tmp = math.pow((1.0 + math.cos((y_m / x_m))), -0.5) * math.pow(0.5, -0.5)
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+220)
		tmp = Float64(1.0 / cos(Float64((y_m ^ -0.5) * Float64((y_m ^ -0.5) * Float64(x_m * 0.5)))));
	else
		tmp = Float64((Float64(1.0 + cos(Float64(y_m / x_m))) ^ -0.5) * (0.5 ^ -0.5));
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+220)
		tmp = 1.0 / cos(((y_m ^ -0.5) * ((y_m ^ -0.5) * (x_m * 0.5))));
	else
		tmp = ((1.0 + cos((y_m / x_m))) ^ -0.5) * (0.5 ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+220], N[(1.0 / N[Cos[N[(N[Power[y$95$m, -0.5], $MachinePrecision] * N[(N[Power[y$95$m, -0.5], $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + N[Cos[N[(y$95$m / x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2e220

   1. Initial program 48.7%

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

      1. lift-/.f64 N/A

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

      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. lower-cos.f64 61.8

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

   4. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. inv-pow N/A

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

      10. sqr-pow N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 33.8

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

   6. Applied rewrites 33.8%

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

   if 2e220 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.9%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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. 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. lower-cos.f64 2.9

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

   4. Applied rewrites 2.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. inv-pow N/A

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

      10. sqr-pow N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 0.3

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

   6. Applied rewrites 0.3%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 1.0

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

   8. Applied rewrites 1.0%

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

   10. Applied rewrites 11.5%

       \[\leadsto \color{blue}{{\left(1 + \cos \left(\frac{y}{x}\right)\right)}^{-0.5} \cdot {0.5}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 54.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \frac{1}{\cos \left({y\_m}^{-0.5} \cdot \left({\left({y\_m}^{-0.25}\right)}^{2} \cdot \left(x\_m \cdot 0.5\right)\right)\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (* (pow y_m -0.5) (* (pow (pow y_m -0.25) 2.0) (* x_m 0.5))))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos((pow(y_m, -0.5) * (pow(pow(y_m, -0.25), 2.0) * (x_m * 0.5))));
}

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 / cos(((y_m ** (-0.5d0)) * (((y_m ** (-0.25d0)) ** 2.0d0) * (x_m * 0.5d0))))
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 / Math.cos((Math.pow(y_m, -0.5) * (Math.pow(Math.pow(y_m, -0.25), 2.0) * (x_m * 0.5))));
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 1.0 / math.cos((math.pow(y_m, -0.5) * (math.pow(math.pow(y_m, -0.25), 2.0) * (x_m * 0.5))))

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64((y_m ^ -0.5) * Float64(((y_m ^ -0.25) ^ 2.0) * Float64(x_m * 0.5)))))
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((y_m ^ -0.5) * (((y_m ^ -0.25) ^ 2.0) * (x_m * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 44.1%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
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. lower-cos.f64 55.8

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

4. Applied rewrites 55.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. inv-pow N/A

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

   10. sqr-pow N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-pow.f64 30.4

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

6. Applied rewrites 30.4%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 30.3

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

8. Applied rewrites 30.3%

   \[\leadsto \frac{1}{\cos \left({y}^{-0.5} \cdot \left(\color{blue}{{\left({y}^{-0.25}\right)}^{2}} \cdot \left(x \cdot 0.5\right)\right)\right)} \]
9. Add Preprocessing

Alternative 3: 56.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{y\_m}}{\frac{2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+222)
   (/ 1.0 (cos (/ (/ 1.0 y_m) (/ 2.0 x_m))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+222) {
		tmp = 1.0 / cos(((1.0 / y_m) / (2.0 / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+222) then
        tmp = 1.0d0 / cos(((1.0d0 / y_m) / (2.0d0 / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+222) {
		tmp = 1.0 / Math.cos(((1.0 / y_m) / (2.0 / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+222:
		tmp = 1.0 / math.cos(((1.0 / y_m) / (2.0 / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+222)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / y_m) / Float64(2.0 / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+222)
		tmp = 1.0 / cos(((1.0 / y_m) / (2.0 / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+222], N[(1.0 / N[Cos[N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000023e222

   1. Initial program 48.7%

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

      1. lift-/.f64 N/A

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

      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. lower-cos.f64 61.8

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

   4. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 61.9

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

   6. Applied rewrites 61.9%

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

   if 5.00000000000000023e222 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.5%

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

Alternative 4: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* x_m (/ 0.5 y_m)))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos((x_m * (0.5 / y_m)));
}

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 / cos((x_m * (0.5d0 / y_m)))
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 / Math.cos((x_m * (0.5 / y_m)));
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 1.0 / math.cos((x_m * (0.5 / y_m)))

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_m))))
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((x_m * (0.5 / y_m)));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
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. lower-cos.f64 55.8

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

4. Applied rewrites 55.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 56.0

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

6. Applied rewrites 56.0%

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

7. Final simplification 56.0%

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

Alternative 5: 54.6% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.4%

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

Developer Target 1: 54.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(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)))))