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

Percentage Accurate: 44.3% → 55.5%

Time: 15.3s

Alternatives: 8

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: 44.3% 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: 55.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cbrt-cube 56.7%

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

   2. pow3 56.7%

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

   3. associate-/l* 56.6%

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

9. Applied egg-rr 56.6%

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

    1. metadata-eval 56.6%

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

    2. associate-/r* 56.6%

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

    3. *-commutative 56.6%

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

    4. div-inv 56.7%

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

    5. add-cube-cbrt 57.4%

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

    6. pow3 57.5%

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

    7. *-un-lft-identity 57.5%

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

    8. *-commutative 57.5%

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

    9. times-frac 57.5%

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

    10. metadata-eval 57.5%

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

    11. clear-num 57.6%

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

    12. un-div-inv 57.6%

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

11. Applied egg-rr 57.6%

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

Alternative 2: 55.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cube-cbrt 57.5%

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

   2. pow3 57.5%

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

   3. associate-/l* 57.4%

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

9. Applied egg-rr 57.4%

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

    1. associate-*r/ 57.5%

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

    2. cbrt-div 57.5%

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

11. Applied egg-rr 57.5%

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

12. Final simplification 57.5%

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

Alternative 3: 55.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cube-cbrt 57.5%

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

   2. pow3 57.5%

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

   3. associate-/l* 57.4%

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

9. Applied egg-rr 57.4%

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

Alternative 4: 55.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cube-cbrt 57.5%

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

   2. pow3 57.5%

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

   3. associate-/l* 57.4%

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

9. Applied egg-rr 57.4%

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

    1. rem-cube-cbrt 56.6%

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

    2. associate-*r/ 56.7%

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

    3. associate-*l/ 56.7%

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

    4. clear-num 56.8%

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

    5. associate-*l/ 56.8%

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

    6. metadata-eval 56.8%

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

11. Applied egg-rr 56.8%

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

    1. add-cube-cbrt 57.0%

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

    2. pow3 57.0%

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

13. Applied egg-rr 57.0%

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

Alternative 5: 55.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 / log(exp(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 / log(exp(cos((0.5d0 / (y / x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cube-cbrt 57.5%

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

   2. pow3 57.5%

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

   3. associate-/l* 57.4%

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

9. Applied egg-rr 57.4%

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

    1. rem-cube-cbrt 56.6%

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

    2. metadata-eval 56.6%

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

    3. associate-/r* 56.6%

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

    4. *-commutative 56.6%

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

    5. div-inv 56.7%

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

    6. add-log-exp 56.7%

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

    7. *-un-lft-identity 56.7%

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

    8. *-commutative 56.7%

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

    9. times-frac 56.7%

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

    10. metadata-eval 56.7%

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

    11. clear-num 56.8%

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

    12. un-div-inv 56.8%

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

11. Applied egg-rr 56.8%

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

Alternative 6: 55.5% 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 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*r/ 56.7%

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

   6. distribute-frac-neg 56.7%

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

   7. distribute-rgt-neg-in 56.7%

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

   8. metadata-eval 56.7%

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

7. Simplified 56.7%

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

   1. add-cube-cbrt 57.5%

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

   2. pow3 57.5%

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

   3. associate-/l* 57.4%

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

9. Applied egg-rr 57.4%

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

    1. rem-cube-cbrt 56.6%

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

    2. associate-*r/ 56.7%

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

    3. associate-*l/ 56.7%

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

    4. clear-num 56.8%

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

    5. associate-*l/ 56.8%

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

    6. metadata-eval 56.8%

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

11. Applied egg-rr 56.8%

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

Alternative 7: 55.5% 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 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around inf 56.7%

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

   1. associate-*r/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-*r/ 56.6%

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

7. Simplified 56.6%

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

Alternative 8: 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 44.2%

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

   1. remove-double-neg 44.2%

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

   2. distribute-frac-neg 44.2%

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

   3. tan-neg 44.2%

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

   4. distribute-frac-neg2 44.2%

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

   5. distribute-lft-neg-out 44.2%

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

   6. distribute-frac-neg2 44.2%

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

   7. distribute-lft-neg-out 44.2%

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

   8. distribute-frac-neg2 44.2%

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

   9. distribute-frac-neg 44.2%

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

   10. neg-mul-1 44.2%

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

   11. *-commutative 44.2%

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

   12. associate-/l* 44.2%

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

   13. *-commutative 44.2%

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

   14. associate-/r* 44.2%

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

   15. metadata-eval 44.2%

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

   16. sin-neg 44.2%

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

   17. distribute-frac-neg 44.2%

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

3. Simplified 44.5%

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

5. Taylor expanded in x around 0 55.7%

   \[\leadsto \color{blue}{1} \]
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 2024118 
(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)))))