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

Percentage Accurate: 44.2% → 55.5%

Time: 16.9s

Alternatives: 6

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: 44.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. add-log-exp 41.8%

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

   2. *-un-lft-identity 41.8%

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

   3. log-prod 41.8%

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

   4. metadata-eval 41.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

   5. add-log-exp 41.8%

      \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   6. div-inv 40.6%

      \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   7. tan-quot 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. associate-*l/ 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

   9. pow1 40.6%

      \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   10. inv-pow 40.6%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   11. pow-prod-up 54.3%

       \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   12. metadata-eval 54.3%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   14. div-inv 54.6%

       \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

   15. *-commutative 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

   16. associate-/r* 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

   17. metadata-eval 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

3. Applied egg-rr 54.6%

   \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]

   2. add-cbrt-cube 47.1%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}\right)} \]

   3. cbrt-prod 50.9%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y \cdot y} \cdot \sqrt[3]{y}}}\right)} \]

   4. associate-/r* 50.9%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{\sqrt[3]{y \cdot y}}}{\sqrt[3]{y}}\right)}} \]

   5. cbrt-prod 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\sqrt[3]{y}}\right)} \]

   6. pow2 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}}{\sqrt[3]{y}}\right)} \]

5. Applied egg-rr 54.7%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{y}}\right)}} \]
6. Step-by-step derivation

   1. div-inv 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]

   2. div-inv 54.5%

      \[\leadsto 0 + \frac{1}{\cos \left(\color{blue}{\left(\left(x \cdot 0.5\right) \cdot \frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right)} \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   3. pow-flip 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(-2\right)}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   4. metadata-eval 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{\color{blue}{-2}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

7. Applied egg-rr 54.6%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{-2}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]
8. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]

9. Simplified 55.4%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]
10. Step-by-step derivation

    1. un-div-inv 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{-2}}{\sqrt[3]{y}}}\right)} \]

11. Applied egg-rr 55.4%

    \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{-2}}{\sqrt[3]{y}}}\right)} \]

12. Final simplification 55.4%

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

Alternative 2: 55.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (* (* x 0.5) (pow (/ 1.0 (cbrt y)) 3.0)))))

C

double code(double x, double y) {
	return 1.0 / cos(((x * 0.5) * pow((1.0 / cbrt(y)), 3.0)));
}

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos(((x * 0.5) * Math.pow((1.0 / Math.cbrt(y)), 3.0)));
}

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(Float64(x * 0.5) * (Float64(1.0 / cbrt(y)) ^ 3.0))))
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(x * 0.5), $MachinePrecision] * N[Power[N[(1.0 / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

   1. add-log-exp 41.8%

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

   2. *-un-lft-identity 41.8%

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

   3. log-prod 41.8%

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

   4. metadata-eval 41.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

   5. add-log-exp 41.8%

      \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   6. div-inv 40.6%

      \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   7. tan-quot 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. associate-*l/ 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

   9. pow1 40.6%

      \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   10. inv-pow 40.6%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   11. pow-prod-up 54.3%

       \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   12. metadata-eval 54.3%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   14. div-inv 54.6%

       \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

   15. *-commutative 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

   16. associate-/r* 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

   17. metadata-eval 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

3. Applied egg-rr 54.6%

   \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]

   2. add-cbrt-cube 47.1%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}\right)} \]

   3. cbrt-prod 50.9%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y \cdot y} \cdot \sqrt[3]{y}}}\right)} \]

   4. associate-/r* 50.9%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{\sqrt[3]{y \cdot y}}}{\sqrt[3]{y}}\right)}} \]

   5. cbrt-prod 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\sqrt[3]{y}}\right)} \]

   6. pow2 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}}{\sqrt[3]{y}}\right)} \]

5. Applied egg-rr 54.7%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{y}}\right)}} \]
6. Step-by-step derivation

   1. div-inv 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]

   2. div-inv 54.5%

      \[\leadsto 0 + \frac{1}{\cos \left(\color{blue}{\left(\left(x \cdot 0.5\right) \cdot \frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right)} \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   3. pow-flip 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(-2\right)}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   4. metadata-eval 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{\color{blue}{-2}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

7. Applied egg-rr 54.6%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{-2}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]
8. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]

9. Simplified 55.4%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]
10. Step-by-step derivation

    1. metadata-eval 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{\color{blue}{\left(-1 + -1\right)}} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)} \]

    2. pow-prod-up 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(\color{blue}{\left({\left(\sqrt[3]{y}\right)}^{-1} \cdot {\left(\sqrt[3]{y}\right)}^{-1}\right)} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)} \]

    3. inv-pow 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(\left(\color{blue}{\frac{1}{\sqrt[3]{y}}} \cdot {\left(\sqrt[3]{y}\right)}^{-1}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)\right)} \]

    4. inv-pow 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(\left(\frac{1}{\sqrt[3]{y}} \cdot \color{blue}{\frac{1}{\sqrt[3]{y}}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)\right)} \]

    5. pow3 55.4%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\frac{1}{\sqrt[3]{y}}\right)}^{3}}\right)} \]

11. Applied egg-rr 55.4%

    \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\frac{1}{\sqrt[3]{y}}\right)}^{3}}\right)} \]

12. Final simplification 55.4%

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

Alternative 3: 55.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (* (* x 0.5) (pow (cbrt y) -3.0)))))

C

double code(double x, double y) {
	return 1.0 / cos(((x * 0.5) * pow(cbrt(y), -3.0)));
}

Java

public static double code(double x, double y) {
	return 1.0 / Math.cos(((x * 0.5) * Math.pow(Math.cbrt(y), -3.0)));
}

Julia

function code(x, y)
	return Float64(1.0 / cos(Float64(Float64(x * 0.5) * (cbrt(y) ^ -3.0))))
end

Wolfram

code[x_, y_] := N[(1.0 / N[Cos[N[(N[(x * 0.5), $MachinePrecision] * N[Power[N[Power[y, 1/3], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

   1. add-log-exp 41.8%

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

   2. *-un-lft-identity 41.8%

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

   3. log-prod 41.8%

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

   4. metadata-eval 41.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

   5. add-log-exp 41.8%

      \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   6. div-inv 40.6%

      \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   7. tan-quot 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. associate-*l/ 40.6%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

   9. pow1 40.6%

      \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   10. inv-pow 40.6%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   11. pow-prod-up 54.3%

       \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   12. metadata-eval 54.3%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   14. div-inv 54.6%

       \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

   15. *-commutative 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

   16. associate-/r* 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

   17. metadata-eval 54.6%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

3. Applied egg-rr 54.6%

   \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]

   2. add-cbrt-cube 47.1%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}\right)} \]

   3. cbrt-prod 50.9%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y \cdot y} \cdot \sqrt[3]{y}}}\right)} \]

   4. associate-/r* 50.9%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{\sqrt[3]{y \cdot y}}}{\sqrt[3]{y}}\right)}} \]

   5. cbrt-prod 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\sqrt[3]{y}}\right)} \]

   6. pow2 54.7%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{\frac{x \cdot 0.5}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}}{\sqrt[3]{y}}\right)} \]

5. Applied egg-rr 54.7%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}}}{\sqrt[3]{y}}\right)}} \]
6. Step-by-step derivation

   1. div-inv 54.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]

   2. div-inv 54.5%

      \[\leadsto 0 + \frac{1}{\cos \left(\color{blue}{\left(\left(x \cdot 0.5\right) \cdot \frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right)} \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   3. pow-flip 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(-2\right)}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

   4. metadata-eval 54.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{\color{blue}{-2}}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)} \]

7. Applied egg-rr 54.6%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(\left(x \cdot 0.5\right) \cdot {\left(\sqrt[3]{y}\right)}^{-2}\right) \cdot \frac{1}{\sqrt[3]{y}}\right)}} \]
8. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]

9. Simplified 55.4%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\left(x \cdot 0.5\right) \cdot \left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 46.1%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)\right)}\right)} \]

    2. expm1-udef 45.0%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \frac{1}{\sqrt[3]{y}}\right)} - 1\right)}\right)} \]

    3. inv-pow 45.0%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{-2} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{-1}}\right)} - 1\right)\right)} \]

    4. pow-prod-up 45.0%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{\left(-2 + -1\right)}}\right)} - 1\right)\right)} \]

    5. metadata-eval 45.0%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{\color{blue}{-3}}\right)} - 1\right)\right)} \]

11. Applied egg-rr 45.0%

    \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{-3}\right)} - 1\right)}\right)} \]
12. Step-by-step derivation

    1. expm1-def 46.1%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{y}\right)}^{-3}\right)\right)}\right)} \]

    2. expm1-log1p 55.1%

       \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{-3}}\right)} \]

13. Simplified 55.1%

    \[\leadsto 0 + \frac{1}{\cos \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{-3}}\right)} \]

14. Final simplification 55.1%

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

Alternative 4: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return fabs((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 = abs((1.0d0 / cos((0.5d0 * (x / y)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. add-cube-cbrt 41.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\left(\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

   2. pow3 41.1%

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

   3. div-inv 41.4%

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

   4. *-commutative 41.4%

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

   5. associate-/r* 41.4%

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

   6. metadata-eval 41.4%

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

3. Applied egg-rr 41.4%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{{\left(\sqrt[3]{\sin \left(x \cdot \frac{0.5}{y}\right)}\right)}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 38.8%

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

   2. sqrt-unprod 41.2%

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

   3. pow2 41.2%

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

   4. metadata-eval 41.2%

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

   5. div-inv 41.2%

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

   6. associate-/l* 41.2%

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

   7. associate-*r/ 41.4%

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

   8. rem-cube-cbrt 42.1%

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

5. Applied egg-rr 42.1%

   \[\leadsto \color{blue}{\sqrt{{\left(\frac{\tan \left(x \cdot \frac{0.5}{y}\right)}{\sin \left(x \cdot \frac{0.5}{y}\right)}\right)}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 42.1%

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

   2. rem-sqrt-square 42.1%

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

7. Simplified 42.1%

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

8. Taylor expanded in x around inf 54.6%

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

9. Final simplification 54.6%

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

Alternative 5: 55.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. tan-quot 41.8%

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

   2. clear-num 40.6%

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

   3. div-inv 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-/r* 40.9%

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

   6. metadata-eval 40.9%

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

   7. div-inv 40.4%

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

   8. *-commutative 40.4%

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

   9. associate-/r* 40.4%

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

   10. metadata-eval 40.4%

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

3. Applied egg-rr 40.4%

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

4. Taylor expanded in x around inf 54.3%

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

   1. associate-*r/ 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*r/ 54.6%

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

6. Simplified 54.6%

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

7. Final simplification 54.6%

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

Alternative 6: 55.6% 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 41.8%

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

2. Taylor expanded in x around 0 54.5%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 54.5%

   \[\leadsto 1 \]

Developer target: 55.1% 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 2023195 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))