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

Percentage Accurate: 44.3% → 56.7%

Time: 12.4s

Alternatives: 3

Speedup: 244.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.3% 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: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(x\_m \cdot 0.5\right)}^{-0.5}\\ t_1 := \frac{x\_m}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_1}{\sin t\_1} \leq 2:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{y}}{t\_0 \cdot t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (pow (* x_m 0.5) -0.5)) (t_1 (/ x_m (* y 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 2.0)
     (/ 1.0 (cos (/ (/ 1.0 y) (* t_0 t_0))))
     1.0)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = pow((x_m * 0.5), -0.5);
	double t_1 = x_m / (y * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 2.0) {
		tmp = 1.0 / cos(((1.0 / y) / (t_0 * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m * 0.5d0) ** (-0.5d0)
    t_1 = x_m / (y * 2.0d0)
    if ((tan(t_1) / sin(t_1)) <= 2.0d0) then
        tmp = 1.0d0 / cos(((1.0d0 / y) / (t_0 * t_0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = Math.pow((x_m * 0.5), -0.5);
	double t_1 = x_m / (y * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 2.0) {
		tmp = 1.0 / Math.cos(((1.0 / y) / (t_0 * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = math.pow((x_m * 0.5), -0.5)
	t_1 = x_m / (y * 2.0)
	tmp = 0
	if (math.tan(t_1) / math.sin(t_1)) <= 2.0:
		tmp = 1.0 / math.cos(((1.0 / y) / (t_0 * t_0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(x_m * 0.5) ^ -0.5
	t_1 = Float64(x_m / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 2.0)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / y) / Float64(t_0 * t_0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = (x_m * 0.5) ^ -0.5;
	t_1 = x_m / (y * 2.0);
	tmp = 0.0;
	if ((tan(t_1) / sin(t_1)) <= 2.0)
		tmp = 1.0 / cos(((1.0 / y) / (t_0 * t_0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[Power[N[(x$95$m * 0.5), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[Cos[N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.9%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   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 64.9

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

   4. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]
   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. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. exp-to-pow N/A

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

      11. lift-log.f64 N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. log-div N/A

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

      17. pow-sub N/A

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

      18. pow-exp N/A

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

      19. *-commutative N/A

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

      20. pow-to-exp N/A

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

      21. inv-pow N/A

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

      22. lift-/.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. pow-exp N/A

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

      25. neg-mul-1 N/A

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

      26. lower-exp.f64 N/A

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

   6. Applied rewrites 26.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. neg-log N/A

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

      5. rem-exp-log N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 28.0

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

   8. Applied rewrites 28.0%

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

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

   1. Initial program 2.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.4%

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

Alternative 2: 56.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 1.48:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{2}{\frac{x\_m}{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ x_m (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 1.48)
     (/ 1.0 (cos (/ 1.0 (/ 2.0 (/ x_m y)))))
     1.0)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = x_m / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.48) {
		tmp = 1.0 / cos((1.0 / (2.0 / (x_m / y))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m / (y * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 1.48d0) then
        tmp = 1.0d0 / cos((1.0d0 / (2.0d0 / (x_m / y))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = x_m / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.48) {
		tmp = 1.0 / Math.cos((1.0 / (2.0 / (x_m / y))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = x_m / (y * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 1.48:
		tmp = 1.0 / math.cos((1.0 / (2.0 / (x_m / y))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(x_m / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.48)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(2.0 / Float64(x_m / y)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = x_m / (y * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 1.48)
		tmp = 1.0 / cos((1.0 / (2.0 / (x_m / y))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(x$95$m / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.48], N[(1.0 / N[Cos[N[(1.0 / N[(2.0 / N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 1.48

   1. Initial program 67.6%

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

      1. 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 67.6

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

   4. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]
   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. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. exp-to-pow N/A

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

      11. lift-log.f64 N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. log-div N/A

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

      17. pow-sub N/A

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

      18. clear-num N/A

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

      19. lower-/.f64 N/A

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

      20. pow-exp N/A

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

      21. *-commutative N/A

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

      22. pow-to-exp N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

   6. Applied rewrites 28.5%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. neg-log N/A

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

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 67.9

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

   8. Applied rewrites 67.9%

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

   if 1.48 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.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 44.5%

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

Alternative 3: 55.5% accurate, 244.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 58.7%

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

Developer Target 1: 54.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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)))))