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

Percentage Accurate: 44.8% → 56.1%

Time: 18.9s

Alternatives: 7

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: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\cos \left({\left({\left(\sqrt[3]{\frac{-0.5 \cdot x}{y}}\right)}^{2}\right)}^{1.5}\right)}\right)}^{-3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (pow (cbrt (cos (pow (pow (cbrt (/ (* -0.5 x) y)) 2.0) 1.5))) -3.0))

C

double code(double x, double y) {
	return pow(cbrt(cos(pow(pow(cbrt(((-0.5 * x) / y)), 2.0), 1.5))), -3.0);
}

Java

public static double code(double x, double y) {
	return Math.pow(Math.cbrt(Math.cos(Math.pow(Math.pow(Math.cbrt(((-0.5 * x) / y)), 2.0), 1.5))), -3.0);
}

Julia

function code(x, y)
	return cbrt(cos(((cbrt(Float64(Float64(-0.5 * x) / y)) ^ 2.0) ^ 1.5))) ^ -3.0
end

Wolfram

code[x_, y_] := N[Power[N[Power[N[Cos[N[Power[N[Power[N[Power[N[(N[(-0.5 * x), $MachinePrecision] / y), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], -3.0], $MachinePrecision]

Derivation

1. Initial program 46.5%

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

   1. remove-double-neg 46.5%

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

   2. distribute-frac-neg 46.5%

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

   3. tan-neg 46.5%

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

   4. distribute-frac-neg2 46.5%

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

   5. distribute-lft-neg-out 46.5%

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

   6. distribute-frac-neg2 46.5%

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

   7. distribute-lft-neg-out 46.5%

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

   8. distribute-frac-neg2 46.5%

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

   9. distribute-frac-neg 46.5%

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

   10. neg-mul-1 46.5%

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

   11. *-commutative 46.5%

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

   12. associate-/l* 46.7%

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

   13. *-commutative 46.7%

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

   14. associate-/r* 46.7%

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

   15. metadata-eval 46.7%

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

   16. sin-neg 46.7%

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

   17. distribute-frac-neg 46.7%

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

3. Simplified 46.4%

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

5. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

7. Simplified 57.8%

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

9. Applied egg-rr 57.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(-0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. log1p-expm1-u 57.8%

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

    2. rem-cube-cbrt 57.8%

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

    3. cbrt-div 57.8%

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

    4. metadata-eval 57.8%

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

    5. inv-pow 57.8%

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

    6. pow-pow 57.8%

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

    7. metadata-eval 57.8%

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

11. Applied egg-rr 57.8%

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

    1. associate-*r/ 57.8%

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

    2. associate-*l/ 57.7%

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

    3. *-commutative 57.7%

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

    4. add-cube-cbrt 58.4%

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

    5. unpow3 57.9%

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

    6. sqr-pow 35.2%

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

    7. pow-prod-down 58.6%

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

13. Applied egg-rr 58.2%

    \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left({\left({\left(\sqrt[3]{\frac{-0.5 \cdot x}{y}}\right)}^{2}\right)}^{1.5}\right)}}\right)}^{-3} \]

14. Final simplification 58.2%

    \[\leadsto {\left(\sqrt[3]{\cos \left({\left({\left(\sqrt[3]{\frac{-0.5 \cdot x}{y}}\right)}^{2}\right)}^{1.5}\right)}\right)}^{-3} \]
15. Add Preprocessing

Alternative 2: 56.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Taylor expanded in x around inf 46.2%

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

   1. *-commutative 46.2%

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

   2. associate-*l/ 46.5%

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

   3. associate-*r/ 45.9%

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

   4. *-commutative 45.9%

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

   5. associate-*l/ 45.9%

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

   6. associate-*r/ 46.7%

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

5. Simplified 46.7%

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

6. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-*r/ 57.7%

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

8. Simplified 57.7%

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

   1. associate-*r/ 57.8%

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

   2. associate-*l/ 57.8%

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

   3. *-commutative 57.8%

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

   4. clear-num 57.5%

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

   5. un-div-inv 57.5%

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

10. Applied egg-rr 57.5%

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

    1. add-sqr-sqrt 34.2%

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

    2. sqrt-unprod 55.2%

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

    3. div-inv 55.2%

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

    4. clear-num 55.6%

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

    5. div-inv 55.6%

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

    6. clear-num 55.6%

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

    7. swap-sqr 55.6%

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

    8. metadata-eval 55.6%

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

    9. metadata-eval 55.6%

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

    10. swap-sqr 55.6%

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

    11. sqrt-unprod 35.0%

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

    12. add-sqr-sqrt 57.8%

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

    13. associate-*r/ 57.8%

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

    14. associate-*l/ 57.7%

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

    15. *-commutative 57.7%

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

    16. add-cube-cbrt 58.4%

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

12. Applied egg-rr 58.1%

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

13. Final simplification 58.1%

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

Alternative 3: 56.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. remove-double-neg 46.5%

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

   2. distribute-frac-neg 46.5%

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

   3. tan-neg 46.5%

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

   4. distribute-frac-neg2 46.5%

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

   5. distribute-lft-neg-out 46.5%

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

   6. distribute-frac-neg2 46.5%

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

   7. distribute-lft-neg-out 46.5%

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

   8. distribute-frac-neg2 46.5%

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

   9. distribute-frac-neg 46.5%

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

   10. neg-mul-1 46.5%

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

   11. *-commutative 46.5%

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

   12. associate-/l* 46.7%

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

   13. *-commutative 46.7%

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

   14. associate-/r* 46.7%

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

   15. metadata-eval 46.7%

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

   16. sin-neg 46.7%

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

   17. distribute-frac-neg 46.7%

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

3. Simplified 46.4%

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

5. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

7. Simplified 57.8%

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

9. Applied egg-rr 57.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(-0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. log1p-expm1-u 57.8%

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

    2. rem-cube-cbrt 57.8%

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

    3. cbrt-div 57.8%

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

    4. metadata-eval 57.8%

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

    5. inv-pow 57.8%

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

    6. pow-pow 57.8%

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

    7. metadata-eval 57.8%

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

11. Applied egg-rr 57.8%

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

    1. add-sqr-sqrt 35.0%

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

    2. sqrt-unprod 55.6%

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

    3. swap-sqr 55.6%

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

    4. metadata-eval 55.6%

       \[\leadsto {\left(\sqrt[3]{\cos \left(\sqrt{\color{blue}{0.25} \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)}\right)}\right)}^{-3} \]

    5. metadata-eval 55.6%

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

    6. swap-sqr 55.6%

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

    7. clear-num 55.6%

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

    8. div-inv 55.6%

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

    9. clear-num 55.2%

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

    10. div-inv 55.2%

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

    11. sqrt-unprod 34.2%

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

    12. add-sqr-sqrt 57.6%

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

    13. div-inv 57.4%

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

    14. associate-/r* 57.9%

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

13. Applied egg-rr 57.9%

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

14. Final simplification 57.9%

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

Alternative 4: 56.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. remove-double-neg 46.5%

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

   2. distribute-frac-neg 46.5%

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

   3. tan-neg 46.5%

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

   4. distribute-frac-neg2 46.5%

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

   5. distribute-lft-neg-out 46.5%

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

   6. distribute-frac-neg2 46.5%

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

   7. distribute-lft-neg-out 46.5%

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

   8. distribute-frac-neg2 46.5%

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

   9. distribute-frac-neg 46.5%

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

   10. neg-mul-1 46.5%

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

   11. *-commutative 46.5%

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

   12. associate-/l* 46.7%

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

   13. *-commutative 46.7%

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

   14. associate-/r* 46.7%

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

   15. metadata-eval 46.7%

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

   16. sin-neg 46.7%

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

   17. distribute-frac-neg 46.7%

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

3. Simplified 46.4%

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

5. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

7. Simplified 57.8%

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

9. Applied egg-rr 57.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(-0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. log1p-expm1-u 57.8%

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

    2. rem-cube-cbrt 57.8%

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

    3. cbrt-div 57.8%

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

    4. metadata-eval 57.8%

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

    5. inv-pow 57.8%

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

    6. pow-pow 57.8%

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

    7. metadata-eval 57.8%

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

11. Applied egg-rr 57.8%

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

12. Final simplification 57.8%

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

Alternative 5: 56.2% 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 46.5%

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

3. Taylor expanded in x around inf 46.2%

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

   1. *-commutative 46.2%

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

   2. associate-*l/ 46.5%

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

   3. associate-*r/ 45.9%

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

   4. *-commutative 45.9%

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

   5. associate-*l/ 45.9%

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

   6. associate-*r/ 46.7%

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

5. Simplified 46.7%

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

6. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-*r/ 57.7%

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

8. Simplified 57.7%

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

9. Final simplification 57.7%

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

Alternative 6: 56.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. remove-double-neg 46.5%

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

   2. distribute-frac-neg 46.5%

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

   3. tan-neg 46.5%

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

   4. distribute-frac-neg2 46.5%

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

   5. distribute-lft-neg-out 46.5%

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

   6. distribute-frac-neg2 46.5%

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

   7. distribute-lft-neg-out 46.5%

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

   8. distribute-frac-neg2 46.5%

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

   9. distribute-frac-neg 46.5%

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

   10. neg-mul-1 46.5%

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

   11. *-commutative 46.5%

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

   12. associate-/l* 46.7%

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

   13. *-commutative 46.7%

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

   14. associate-/r* 46.7%

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

   15. metadata-eval 46.7%

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

   16. sin-neg 46.7%

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

   17. distribute-frac-neg 46.7%

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

3. Simplified 46.4%

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

5. Taylor expanded in x around inf 57.8%

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

   1. associate-*r/ 57.8%

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

7. Simplified 57.8%

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

8. Final simplification 57.8%

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

Alternative 7: 56.0% 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 46.5%

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

   1. remove-double-neg 46.5%

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

   2. distribute-frac-neg 46.5%

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

   3. tan-neg 46.5%

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

   4. distribute-frac-neg2 46.5%

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

   5. distribute-lft-neg-out 46.5%

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

   6. distribute-frac-neg2 46.5%

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

   7. distribute-lft-neg-out 46.5%

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

   8. distribute-frac-neg2 46.5%

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

   9. distribute-frac-neg 46.5%

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

   10. neg-mul-1 46.5%

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

   11. *-commutative 46.5%

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

   12. associate-/l* 46.7%

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

   13. *-commutative 46.7%

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

   14. associate-/r* 46.7%

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

   15. metadata-eval 46.7%

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

   16. sin-neg 46.7%

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

   17. distribute-frac-neg 46.7%

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

3. Simplified 46.4%

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

5. Taylor expanded in x around 0 57.5%

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

6. Final simplification 57.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 55.5% 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 2024072 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (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)))))