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

Percentage Accurate: 43.7% → 56.8%

Time: 14.2s

Alternatives: 7

Speedup: 244.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.7% 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: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+78) (/ 1.0 (cos (* x_m (/ 0.5 y_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)) <= 2e+78) {
		tmp = 1.0 / cos((x_m * (0.5 / y_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)) <= 2d+78) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / y_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)) <= 2e+78) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / y_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)) <= 2e+78:
		tmp = 1.0 / math.cos((x_m * (0.5 / y_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)) <= 2e+78)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_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)) <= 2e+78)
		tmp = 1.0 / cos((x_m * (0.5 / y_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], 2e+78], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.00000000000000002e78

   1. Initial program 49.3%

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

      1. clear-num N/A

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

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

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

      3. tan-quot N/A

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

      4. associate-/r/ N/A

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

      5. *-inverses N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. associate-/l/ N/A

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

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

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

      15. div-inv N/A

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

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

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

      17. metadata-eval 62.8

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

   4. Applied egg-rr 62.8%

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 62.8

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

   6. Applied egg-rr 62.8%

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

   if 2.00000000000000002e78 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 12.7%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\log y\_m, 2, \log y\_m \cdot -3\right)}}{\frac{2}{x\_m}}\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ (exp (fma (log y_m) 2.0 (* (log y_m) -3.0))) (/ 2.0 x_m)))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos((exp(fma(log(y_m), 2.0, (log(y_m) * -3.0))) / (2.0 / x_m)));
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(exp(fma(log(y_m), 2.0, Float64(log(y_m) * -3.0))) / Float64(2.0 / x_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[(N[Exp[N[(N[Log[y$95$m], $MachinePrecision] * 2.0 + N[(N[Log[y$95$m], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.7%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 52.6

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

4. Applied egg-rr 52.6%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. inv-pow N/A

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

   10. un-div-inv N/A

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

   11. /-lowering-/.f64 N/A

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

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

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

   13. /-lowering-/.f64 53.1

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

6. Applied egg-rr 53.1%

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

   1. inv-pow N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\cos \left(\frac{{y}^{\color{blue}{\left(2 - 3\right)}}}{\frac{2}{x}}\right)} \]

   3. pow-div N/A

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{\frac{{y}^{2}}{{y}^{3}}}}{\frac{2}{x}}\right)} \]

   4. pow-to-exp N/A

      \[\leadsto \frac{1}{\cos \left(\frac{\frac{\color{blue}{e^{\log y \cdot 2}}}{{y}^{3}}}{\frac{2}{x}}\right)} \]

   5. pow-to-exp N/A

      \[\leadsto \frac{1}{\cos \left(\frac{\frac{e^{\log y \cdot 2}}{\color{blue}{e^{\log y \cdot 3}}}}{\frac{2}{x}}\right)} \]

   6. div-exp N/A

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{e^{\log y \cdot 2 - \log y \cdot 3}}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{e^{\log y \cdot 2 - \log y \cdot 3}}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\log y \cdot 2 - \log y \cdot 3}}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\log y \cdot 2} - \log y \cdot 3}}{\frac{2}{x}}\right)} \]

   10. log-lowering-log.f64 N/A

       \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\log y} \cdot 2 - \log y \cdot 3}}{\frac{2}{x}}\right)} \]

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

       \[\leadsto \frac{1}{\cos \left(\frac{e^{\log y \cdot 2 - \color{blue}{\log y \cdot 3}}}{\frac{2}{x}}\right)} \]

   12. log-lowering-log.f64 26.2

       \[\leadsto \frac{1}{\cos \left(\frac{e^{\log y \cdot 2 - \color{blue}{\log y} \cdot 3}}{\frac{2}{x}}\right)} \]

8. Applied egg-rr 26.2%

   \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{e^{\log y \cdot 2 - \log y \cdot 3}}}{\frac{2}{x}}\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\log y \cdot 2 + \left(\mathsf{neg}\left(\log y \cdot 3\right)\right)}}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\mathsf{fma}\left(\log y, 2, \mathsf{neg}\left(\log y \cdot 3\right)\right)}}}{\frac{2}{x}}\right)} \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\color{blue}{\log y}, 2, \mathsf{neg}\left(\log y \cdot 3\right)\right)}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\log y, 2, \color{blue}{\log y \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\log y, 2, \color{blue}{\log y \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)}}{\frac{2}{x}}\right)} \]

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

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\log y, 2, \color{blue}{\log y} \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}}{\frac{2}{x}}\right)} \]

   7. metadata-eval 26.2

      \[\leadsto \frac{1}{\cos \left(\frac{e^{\mathsf{fma}\left(\log y, 2, \log y \cdot \color{blue}{-3}\right)}}{\frac{2}{x}}\right)} \]

10. Applied egg-rr 26.2%

    \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\mathsf{fma}\left(\log y, 2, \log y \cdot -3\right)}}}{\frac{2}{x}}\right)} \]
11. Add Preprocessing

Alternative 3: 55.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos((pow(pow(y_m, -0.5), 2.0) / (2.0 / x_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((((y_m ** (-0.5d0)) ** 2.0d0) / (2.0d0 / x_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((Math.pow(Math.pow(y_m, -0.5), 2.0) / (2.0 / x_m)));
}

Python

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

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) ^ 2.0) / Float64(2.0 / x_m))))
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) ^ 2.0) / (2.0 / x_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[(N[Power[N[Power[y$95$m, -0.5], $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.7%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 52.6

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

4. Applied egg-rr 52.6%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. inv-pow N/A

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

   10. un-div-inv N/A

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

   11. /-lowering-/.f64 N/A

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

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

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

   13. /-lowering-/.f64 53.1

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

6. Applied egg-rr 53.1%

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

   1. inv-pow N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. metadata-eval N/A

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

   6. pow-lowering-pow.f64 26.0

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

8. Applied egg-rr 26.0%

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

Alternative 4: 55.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos(((1.0 / y_m) / (2.0 / x_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(((1.0d0 / y_m) / (2.0d0 / x_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(((1.0 / y_m) / (2.0 / x_m)));
}

Python

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

Julia

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

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((1.0 / y_m) / (2.0 / x_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[(N[(1.0 / y$95$m), $MachinePrecision] / N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.7%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 52.6

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

4. Applied egg-rr 52.6%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. inv-pow N/A

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

   10. un-div-inv N/A

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

   11. /-lowering-/.f64 N/A

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

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

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

   13. /-lowering-/.f64 53.1

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

6. Applied egg-rr 53.1%

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

Alternative 5: 55.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0 / cos((0.5 / (y_m / x_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((0.5d0 / (y_m / x_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((0.5 / (y_m / x_m)));
}

Python

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

Julia

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

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos((0.5 / (y_m / x_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[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.7%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 52.6

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

4. Applied egg-rr 52.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

   6. /-lowering-/.f64 52.7

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

6. Applied egg-rr 52.7%

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

Alternative 6: 55.1% accurate, 1.9× speedup

Math

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

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(Float64(x_m * 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[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.7%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/l/ N/A

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

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

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 52.6

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

4. Applied egg-rr 52.6%

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

Alternative 7: 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 41.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 51.5%

   \[\leadsto \color{blue}{1} \]
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 2024204 
(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)))))