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

Percentage Accurate: 43.5% → 54.9%

Time: 14.4s

Alternatives: 10

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: 43.5% 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: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{y}}\\ \frac{1}{\cos \left(\frac{0.5 \cdot x}{{t\_0}^{2}} \cdot \frac{{\left(\sqrt[3]{y}\right)}^{-2}}{t\_0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt y))))
   (/ 1.0 (cos (* (/ (* 0.5 x) (pow t_0 2.0)) (/ (pow (cbrt y) -2.0) t_0))))))

C

double code(double x, double y) {
	double t_0 = cbrt(cbrt(y));
	return 1.0 / cos((((0.5 * x) / pow(t_0, 2.0)) * (pow(cbrt(y), -2.0) / t_0)));
}

Java

public static double code(double x, double y) {
	double t_0 = Math.cbrt(Math.cbrt(y));
	return 1.0 / Math.cos((((0.5 * x) / Math.pow(t_0, 2.0)) * (Math.pow(Math.cbrt(y), -2.0) / t_0)));
}

Julia

function code(x, y)
	t_0 = cbrt(cbrt(y))
	return Float64(1.0 / cos(Float64(Float64(Float64(0.5 * x) / (t_0 ^ 2.0)) * Float64((cbrt(y) ^ -2.0) / t_0))))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Power[y, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[Cos[N[(N[(N[(0.5 * x), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[y, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. associate-*r/ 54.6%

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

   2. add-sqr-sqrt 27.8%

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

   3. associate-/r* 27.6%

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

5. Applied egg-rr 27.6%

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

7. Applied egg-rr 55.1%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5 \cdot x}{{\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{2}} \cdot \frac{{\left(\sqrt[3]{y}\right)}^{-2}}{\sqrt[3]{\sqrt[3]{y}}}\right)}} \]
8. Add Preprocessing

Alternative 2: 28.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left(\frac{\left(0.5 \cdot x\right) \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  1.0
  (cos
   (/ (* (* 0.5 x) (/ (pow y -0.5) (pow y 0.16666666666666666))) (cbrt y)))))

C

double code(double x, double y) {
	return 1.0 / cos((((0.5 * x) * (pow(y, -0.5) / pow(y, 0.16666666666666666))) / cbrt(y)));
}

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos((((0.5 * x) * (Math.pow(y, -0.5) / Math.pow(y, 0.16666666666666666))) / Math.cbrt(y)));
}

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(Float64(Float64(0.5 * x) * Float64((y ^ -0.5) / (y ^ 0.16666666666666666))) / cbrt(y))))
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(N[(0.5 * x), $MachinePrecision] * N[(N[Power[y, -0.5], $MachinePrecision] / N[Power[y, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. associate-*r/ 54.6%

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

   2. add-sqr-sqrt 27.8%

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

   3. associate-/r* 27.6%

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

5. Applied egg-rr 27.6%

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

   1. div-inv 27.3%

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

   2. add-cbrt-cube 27.1%

      \[\leadsto \frac{1}{\cos \left(\frac{\left(0.5 \cdot x\right) \cdot \frac{1}{\sqrt{y}}}{\color{blue}{\sqrt[3]{\left(\sqrt{y} \cdot \sqrt{y}\right) \cdot \sqrt{y}}}}\right)} \]

   3. add-sqr-sqrt 27.0%

      \[\leadsto \frac{1}{\cos \left(\frac{\left(0.5 \cdot x\right) \cdot \frac{1}{\sqrt{y}}}{\sqrt[3]{\color{blue}{y} \cdot \sqrt{y}}}\right)} \]

   4. cbrt-prod 27.5%

      \[\leadsto \frac{1}{\cos \left(\frac{\left(0.5 \cdot x\right) \cdot \frac{1}{\sqrt{y}}}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{\sqrt{y}}}}\right)} \]

   5. times-frac 27.6%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{\frac{1}{\sqrt{y}}}{\sqrt[3]{\sqrt{y}}}\right)}} \]

   6. pow1/2 27.6%

      \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{\frac{1}{\color{blue}{{y}^{0.5}}}}{\sqrt[3]{\sqrt{y}}}\right)} \]

   7. pow-flip 27.2%

      \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{\color{blue}{{y}^{\left(-0.5\right)}}}{\sqrt[3]{\sqrt{y}}}\right)} \]

   8. metadata-eval 27.2%

      \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{\color{blue}{-0.5}}}{\sqrt[3]{\sqrt{y}}}\right)} \]

   9. add-cube-cbrt 27.5%

      \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}}\right)} \]

   10. unpow2 27.5%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt[3]{\sqrt{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}}}}\right)} \]

   11. sqrt-prod 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt{\sqrt[3]{y}}}}}\right)} \]

   12. unpow2 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt[3]{\sqrt{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \sqrt{\sqrt[3]{y}}}}\right)} \]

   13. sqrt-prod 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt[3]{\color{blue}{\left(\sqrt{\sqrt[3]{y}} \cdot \sqrt{\sqrt[3]{y}}\right)} \cdot \sqrt{\sqrt[3]{y}}}}\right)} \]

   14. add-cbrt-cube 27.3%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\color{blue}{\sqrt{\sqrt[3]{y}}}}\right)} \]

   15. pow1/3 27.9%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\sqrt{\color{blue}{{y}^{0.3333333333333333}}}}\right)} \]

   16. sqrt-pow1 27.9%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{\color{blue}{{y}^{\left(\frac{0.3333333333333333}{2}\right)}}}\right)} \]

   17. metadata-eval 27.9%

       \[\leadsto \frac{1}{\cos \left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{{y}^{\color{blue}{0.16666666666666666}}}\right)} \]

7. Applied egg-rr 27.9%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5 \cdot x}{\sqrt[3]{y}} \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}\right)}} \]
8. Step-by-step derivation

   1. associate-*l/ 28.1%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\left(0.5 \cdot x\right) \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)}} \]

   2. *-commutative 28.1%

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{\left(x \cdot 0.5\right)} \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)} \]

9. Simplified 28.1%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\left(x \cdot 0.5\right) \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)}} \]

10. Final simplification 28.1%

    \[\leadsto \frac{1}{\cos \left(\frac{\left(0.5 \cdot x\right) \cdot \frac{{y}^{-0.5}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)} \]
11. Add Preprocessing

Alternative 3: 28.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left(\frac{0.5}{\sqrt[3]{y}} \cdot \frac{x}{{y}^{0.16666666666666666} \cdot \sqrt{y}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  1.0
  (cos (* (/ 0.5 (cbrt y)) (/ x (* (pow y 0.16666666666666666) (sqrt y)))))))

C

double code(double x, double y) {
	return 1.0 / cos(((0.5 / cbrt(y)) * (x / (pow(y, 0.16666666666666666) * sqrt(y)))));
}

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos(((0.5 / Math.cbrt(y)) * (x / (Math.pow(y, 0.16666666666666666) * Math.sqrt(y)))));
}

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(Float64(0.5 / cbrt(y)) * Float64(x / Float64((y ^ 0.16666666666666666) * sqrt(y))))))
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(0.5 / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[Power[y, 0.16666666666666666], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. associate-*r/ 54.6%

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

   2. add-sqr-sqrt 27.8%

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

   3. associate-/r* 27.6%

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

5. Applied egg-rr 27.6%

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

   1. *-un-lft-identity 27.6%

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

   2. add-cbrt-cube 27.1%

      \[\leadsto \frac{1}{\cos \left(\frac{1 \cdot \frac{0.5 \cdot x}{\sqrt{y}}}{\color{blue}{\sqrt[3]{\left(\sqrt{y} \cdot \sqrt{y}\right) \cdot \sqrt{y}}}}\right)} \]

   3. add-sqr-sqrt 27.1%

      \[\leadsto \frac{1}{\cos \left(\frac{1 \cdot \frac{0.5 \cdot x}{\sqrt{y}}}{\sqrt[3]{\color{blue}{y} \cdot \sqrt{y}}}\right)} \]

   4. cbrt-prod 27.3%

      \[\leadsto \frac{1}{\cos \left(\frac{1 \cdot \frac{0.5 \cdot x}{\sqrt{y}}}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{\sqrt{y}}}}\right)} \]

   5. times-frac 26.8%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1}{\sqrt[3]{y}} \cdot \frac{\frac{0.5 \cdot x}{\sqrt{y}}}{\sqrt[3]{\sqrt{y}}}\right)}} \]

   6. associate-/l* 26.8%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{\color{blue}{0.5 \cdot \frac{x}{\sqrt{y}}}}{\sqrt[3]{\sqrt{y}}}\right)} \]

   7. add-cube-cbrt 26.8%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}}\right)} \]

   8. unpow2 26.8%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt[3]{\sqrt{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}}}}\right)} \]

   9. sqrt-prod 26.8%

      \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt{\sqrt[3]{y}}}}}\right)} \]

   10. unpow2 26.8%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt[3]{\sqrt{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \sqrt{\sqrt[3]{y}}}}\right)} \]

   11. sqrt-prod 26.9%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt[3]{\color{blue}{\left(\sqrt{\sqrt[3]{y}} \cdot \sqrt{\sqrt[3]{y}}\right)} \cdot \sqrt{\sqrt[3]{y}}}}\right)} \]

   12. add-cbrt-cube 27.1%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\color{blue}{\sqrt{\sqrt[3]{y}}}}\right)} \]

   13. pow1/3 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\sqrt{\color{blue}{{y}^{0.3333333333333333}}}}\right)} \]

   14. sqrt-pow1 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{\color{blue}{{y}^{\left(\frac{0.3333333333333333}{2}\right)}}}\right)} \]

   15. metadata-eval 27.6%

       \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{{y}^{\color{blue}{0.16666666666666666}}}\right)} \]

7. Applied egg-rr 27.6%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1}{\sqrt[3]{y}} \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{{y}^{0.16666666666666666}}\right)}} \]
8. Step-by-step derivation

   1. associate-*l/ 27.8%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1 \cdot \frac{0.5 \cdot \frac{x}{\sqrt{y}}}{{y}^{0.16666666666666666}}}{\sqrt[3]{y}}\right)}} \]

   2. *-lft-identity 27.8%

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{\frac{0.5 \cdot \frac{x}{\sqrt{y}}}{{y}^{0.16666666666666666}}}}{\sqrt[3]{y}}\right)} \]

   3. associate-/l* 27.8%

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{0.5 \cdot \frac{\frac{x}{\sqrt{y}}}{{y}^{0.16666666666666666}}}}{\sqrt[3]{y}}\right)} \]

   4. associate-*l/ 27.6%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\sqrt[3]{y}} \cdot \frac{\frac{x}{\sqrt{y}}}{{y}^{0.16666666666666666}}\right)}} \]

   5. associate-/l/ 27.9%

      \[\leadsto \frac{1}{\cos \left(\frac{0.5}{\sqrt[3]{y}} \cdot \color{blue}{\frac{x}{{y}^{0.16666666666666666} \cdot \sqrt{y}}}\right)} \]

9. Simplified 27.9%

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\sqrt[3]{y}} \cdot \frac{x}{{y}^{0.16666666666666666} \cdot \sqrt{y}}\right)}} \]
10. Add Preprocessing

Alternative 4: 33.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left({\left({\left({\left(\frac{x}{y \cdot 2}\right)}^{0.25}\right)}^{2}\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (pow (pow (pow (/ x (* y 2.0)) 0.25) 2.0) 2.0))))

C

double code(double x, double y) {
	return 1.0 / cos(pow(pow(pow((x / (y * 2.0)), 0.25), 2.0), 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos(((((x / (y * 2.0d0)) ** 0.25d0) ** 2.0d0) ** 2.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos(Math.pow(Math.pow(Math.pow((x / (y * 2.0)), 0.25), 2.0), 2.0));
}

Python

def code(x, y):
	return 1.0 / math.cos(math.pow(math.pow(math.pow((x / (y * 2.0)), 0.25), 2.0), 2.0))

Julia

function code(x, y)
	return Float64(1.0 / cos((((Float64(x / Float64(y * 2.0)) ^ 0.25) ^ 2.0) ^ 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / cos(((((x / (y * 2.0)) ^ 0.25) ^ 2.0) ^ 2.0));
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[Power[N[Power[N[Power[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. metadata-eval 54.6%

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

   2. times-frac 54.6%

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

   3. *-un-lft-identity 54.6%

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

   4. *-commutative 54.6%

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

   5. add-sqr-sqrt 25.2%

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

   6. frac-times 25.3%

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

   7. add-sqr-sqrt 17.5%

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

   8. pow2 17.5%

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

   9. frac-times 17.7%

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

   10. add-sqr-sqrt 37.8%

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

5. Applied egg-rr 37.8%

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

   1. add-sqr-sqrt 37.6%

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

   2. pow2 37.6%

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

   3. pow1/2 37.6%

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

   4. sqrt-pow1 37.8%

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

   5. metadata-eval 37.8%

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

7. Applied egg-rr 37.8%

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

Alternative 5: 33.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left({\left(e^{0.5 \cdot \log \left(\frac{x}{y \cdot 2}\right)}\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (pow (exp (* 0.5 (log (/ x (* y 2.0))))) 2.0))))

C

double code(double x, double y) {
	return 1.0 / cos(pow(exp((0.5 * log((x / (y * 2.0))))), 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos((exp((0.5d0 * log((x / (y * 2.0d0))))) ** 2.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos(Math.pow(Math.exp((0.5 * Math.log((x / (y * 2.0))))), 2.0));
}

Python

def code(x, y):
	return 1.0 / math.cos(math.pow(math.exp((0.5 * math.log((x / (y * 2.0))))), 2.0))

Julia

function code(x, y)
	return Float64(1.0 / cos((exp(Float64(0.5 * log(Float64(x / Float64(y * 2.0))))) ^ 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / cos((exp((0.5 * log((x / (y * 2.0))))) ^ 2.0));
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[Power[N[Exp[N[(0.5 * N[Log[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. metadata-eval 54.6%

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

   2. times-frac 54.6%

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

   3. *-un-lft-identity 54.6%

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

   4. *-commutative 54.6%

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

   5. add-sqr-sqrt 25.2%

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

   6. frac-times 25.3%

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

   7. add-sqr-sqrt 17.5%

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

   8. pow2 17.5%

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

   9. frac-times 17.7%

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

   10. add-sqr-sqrt 37.8%

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

5. Applied egg-rr 37.8%

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

   1. pow1/2 37.8%

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

   2. pow-to-exp 37.5%

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

7. Applied egg-rr 37.5%

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

8. Final simplification 37.5%

   \[\leadsto \frac{1}{\cos \left({\left(e^{0.5 \cdot \log \left(\frac{x}{y \cdot 2}\right)}\right)}^{2}\right)} \]
9. Add Preprocessing

Alternative 6: 28.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (/ (/ (* 0.5 x) (sqrt y)) (sqrt y)))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(Float64(Float64(0.5 * x) / sqrt(y)) / sqrt(y))))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(N[(0.5 * x), $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. associate-*r/ 54.6%

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

   2. add-sqr-sqrt 27.8%

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

   3. associate-/r* 27.6%

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

5. Applied egg-rr 27.6%

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

Alternative 7: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cos \left(\frac{1}{y \cdot \frac{2}{x}}\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (cos (/ 1.0 (* y (/ 2.0 x))))))

C

double code(double x, double y) {
	return 1.0 / cos((1.0 / (y * (2.0 / x))));
}

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 / math.cos((1.0 / (y * (2.0 / x))))

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(1.0 / Float64(y * Float64(2.0 / x)))))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(1.0 / N[(y * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. metadata-eval 54.6%

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

   2. times-frac 54.6%

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

   3. *-un-lft-identity 54.6%

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

   4. *-commutative 54.6%

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

   5. clear-num 54.6%

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

   6. associate-/l* 54.6%

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

5. Applied egg-rr 54.6%

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

Alternative 8: 55.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (cos (/ 0.5 (/ y x)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

   1. clear-num 54.6%

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

   2. un-div-inv 54.6%

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

5. Applied egg-rr 54.6%

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

Alternative 9: 55.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (cos (* 0.5 (/ x y)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around inf 54.6%

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

Alternative 10: 55.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 40.1%

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

   1. remove-double-neg 40.1%

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

   2. distribute-frac-neg 40.1%

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

   3. tan-neg 40.1%

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

   4. distribute-frac-neg2 40.1%

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

   5. distribute-lft-neg-out 40.1%

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

   6. distribute-frac-neg2 40.1%

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

   7. distribute-lft-neg-out 40.1%

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

   8. distribute-frac-neg2 40.1%

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

   9. distribute-frac-neg 40.1%

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

   10. neg-mul-1 40.1%

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

   11. *-commutative 40.1%

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

   12. associate-/l* 39.8%

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

   13. *-commutative 39.8%

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

   14. associate-/r* 39.8%

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

   15. metadata-eval 39.8%

       \[\leadsto \frac{\tan \left(x \cdot \frac{\color{blue}{-0.5}}{y}\right)}{-\sin \left(\frac{x}{y \cdot 2}\right)} \]

   16. sin-neg 39.8%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\color{blue}{\sin \left(-\frac{x}{y \cdot 2}\right)}} \]

   17. distribute-frac-neg 39.8%

       \[\leadsto \frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \color{blue}{\left(\frac{-x}{y \cdot 2}\right)}} \]

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{-0.5}{y}\right)}{\sin \left(x \cdot \frac{-0.5}{y}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 53.8%

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

Developer Target 1: 54.4% 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 2024163 
(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)))))