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

Percentage Accurate: 43.9% → 55.2%

Time: 17.4s

Alternatives: 5

Speedup: 211.0×

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: 43.9% 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: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{1}{\cos \left({2}^{-0.125} \cdot \left({2}^{-0.125} \cdot \left({\left(\frac{y\_m}{0.5}\right)}^{-0.75} \cdot \left(x \cdot {y\_m}^{-0.25}\right)\right)\right)\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/
  1.0
  (cos
   (*
    (pow 2.0 -0.125)
    (* (pow 2.0 -0.125) (* (pow (/ y_m 0.5) -0.75) (* x (pow y_m -0.25))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return 1.0 / cos((pow(2.0, -0.125) * (pow(2.0, -0.125) * (pow((y_m / 0.5), -0.75) * (x * pow(y_m, -0.25))))));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = 1.0d0 / cos(((2.0d0 ** (-0.125d0)) * ((2.0d0 ** (-0.125d0)) * (((y_m / 0.5d0) ** (-0.75d0)) * (x * (y_m ** (-0.25d0)))))))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return 1.0 / Math.cos((Math.pow(2.0, -0.125) * (Math.pow(2.0, -0.125) * (Math.pow((y_m / 0.5), -0.75) * (x * Math.pow(y_m, -0.25))))));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return 1.0 / math.cos((math.pow(2.0, -0.125) * (math.pow(2.0, -0.125) * (math.pow((y_m / 0.5), -0.75) * (x * math.pow(y_m, -0.25))))))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(1.0 / cos(Float64((2.0 ^ -0.125) * Float64((2.0 ^ -0.125) * Float64((Float64(y_m / 0.5) ^ -0.75) * Float64(x * (y_m ^ -0.25)))))))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = 1.0 / cos(((2.0 ^ -0.125) * ((2.0 ^ -0.125) * (((y_m / 0.5) ^ -0.75) * (x * (y_m ^ -0.25))))));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Cos[N[(N[Power[2.0, -0.125], $MachinePrecision] * N[(N[Power[2.0, -0.125], $MachinePrecision] * N[(N[Power[N[(y$95$m / 0.5), $MachinePrecision], -0.75], $MachinePrecision] * N[(x * N[Power[y$95$m, -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}\right) \]

   3. tan-quot N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \]

   4. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x}{y \cdot 2}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right) \]

   13. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{x}{2}}{y}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{2}\right), y\right)\right)\right) \]

   15. /-lowering-/.f64 48.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), y\right)\right)\right) \]

4. Applied egg-rr 48.8%

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

   1. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{y \cdot 2} \cdot x\right)\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(y \cdot 2\right)}^{-1} \cdot x\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(y \cdot 2\right)}^{\left(\frac{-1}{2} \cdot 2\right)} \cdot x\right)\right)\right) \]

   6. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left({\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)}^{2} \cdot x\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left({\left(y \cdot \frac{1}{\frac{1}{2}}\right)}^{\frac{-1}{2}}\right)}^{2} \cdot x\right)\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}}\right)}^{2} \cdot x\right)\right)\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot {\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}}\right) \cdot x\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right) \]

   11. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(y \cdot \frac{1}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(y \cdot 2\right)}^{\frac{-1}{2}} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right) \]

   13. sqr-pow N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right) \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right) \]

   15. unpow-prod-down N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\left({y}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right) \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({y}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({2}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left({y}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right), \left({2}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

   18. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(y, \left(\frac{\frac{-1}{2}}{2}\right)\right), \left({2}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(y, \frac{-1}{4}\right), \left({2}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 24.4%

   \[\leadsto \frac{1}{\cos \color{blue}{\left({y}^{-0.25} \cdot \left({2}^{-0.25} \cdot \left({\left(\frac{y}{0.5}\right)}^{-0.75} \cdot x\right)\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\left({2}^{\frac{-1}{4}} \cdot \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right)\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({2}^{\frac{-1}{4}} \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {2}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right) \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right), \left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{-1}{4}}{2}\right)\right), \left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({2}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right), \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{-1}{4}}{2}\right)\right), \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot x\right) \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}} \cdot \left(x \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({\left(\frac{y}{\frac{1}{2}}\right)}^{\frac{-3}{4}}\right), \left(x \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right)\right) \]

   13. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{\frac{1}{2}}\right), \frac{-3}{4}\right), \left(x \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{2}\right), \frac{-3}{4}\right), \left(x \cdot {y}^{\frac{-1}{4}}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{2}\right), \frac{-3}{4}\right), \mathsf{*.f64}\left(x, \left({y}^{\frac{-1}{4}}\right)\right)\right)\right)\right)\right)\right) \]

   16. pow-lowering-pow.f64 24.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{2}\right), \frac{-3}{4}\right), \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 24.8%

   \[\leadsto \frac{1}{\cos \color{blue}{\left({2}^{-0.125} \cdot \left({2}^{-0.125} \cdot \left({\left(\frac{y}{0.5}\right)}^{-0.75} \cdot \left(x \cdot {y}^{-0.25}\right)\right)\right)\right)}} \]
9. Add Preprocessing

Alternative 2: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{\frac{x}{2}}{y\_m}\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (/ 1.0 (cos (/ (/ x 2.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return 1.0 / cos(((x / 2.0) / y_m));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = 1.0d0 / cos(((x / 2.0d0) / y_m))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return 1.0 / Math.cos(((x / 2.0) / y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return 1.0 / math.cos(((x / 2.0) / y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(1.0 / cos(Float64(Float64(x / 2.0) / y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = 1.0 / cos(((x / 2.0) / y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Cos[N[(N[(x / 2.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}\right) \]

   3. tan-quot N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \]

   4. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x}{y \cdot 2}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right) \]

   13. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{x}{2}}{y}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{2}\right), y\right)\right)\right) \]

   15. /-lowering-/.f64 48.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), y\right)\right)\right) \]

4. Applied egg-rr 48.8%

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

Alternative 3: 55.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = 1.0d0 / cos((0.5d0 / (y_m / x)))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return 1.0 / Math.cos((0.5 / (y_m / x)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}\right) \]

   3. tan-quot N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \]

   4. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x}{y \cdot 2}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right) \]

   13. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{x}{2}}{y}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{2}\right), y\right)\right)\right) \]

   15. /-lowering-/.f64 48.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), y\right)\right)\right) \]

4. Applied egg-rr 48.8%

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

   1. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y}{x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{\frac{y}{x}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{\frac{y}{x}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{y}{x}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 48.7%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

6. Applied egg-rr 48.7%

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

Alternative 4: 55.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = 1.0d0 / cos((x * (0.5d0 / y_m)))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return 1.0 / Math.cos((x * (0.5 / y_m)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\tan \left(\frac{x}{y \cdot 2}\right)}\right)}\right) \]

   3. tan-quot N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right)\right) \]

   4. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(1 \cdot \left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x}{y \cdot 2}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y \cdot 2}\right)\right)\right) \]

   13. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{x}{2}}{y}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{2}\right), y\right)\right)\right) \]

   15. /-lowering-/.f64 48.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), y\right)\right)\right) \]

4. Applied egg-rr 48.8%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{\frac{2}{x}}}{y}\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{y} \cdot x\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{y}\right), x\right)\right)\right) \]

   6. /-lowering-/.f64 48.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, y\right), x\right)\right)\right) \]

6. Applied egg-rr 48.5%

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

7. Final simplification 48.5%

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

Alternative 5: 55.3% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 47.4%

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

Developer Target 1: 55.3% 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 2024192 
(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)))))