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

Percentage Accurate: 44.2% → 55.8%

Time: 19.9s

Alternatives: 6

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.8% accurate, 0.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ 1.0 (cbrt (pow (cos (/ (pow y_m -0.5) (* (sqrt y_m) (/ -2.0 x)))) 3.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return 1.0 / cbrt(pow(cos((pow(y_m, -0.5) / (sqrt(y_m) * (-2.0 / x)))), 3.0));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return 1.0 / Math.cbrt(Math.pow(Math.cos((Math.pow(y_m, -0.5) / (Math.sqrt(y_m) * (-2.0 / x)))), 3.0));
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(1.0 / cbrt((cos(Float64((y_m ^ -0.5) / Float64(sqrt(y_m) * Float64(-2.0 / x)))) ^ 3.0)))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Power[N[Power[N[Cos[N[(N[Power[y$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] * N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-/l* 54.7%

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

10. Simplified 54.7%

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

    1. add-cbrt-cube 54.7%

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

    2. pow3 54.7%

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

12. Applied egg-rr 54.7%

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

    1. add-sqr-sqrt 27.9%

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

    2. sqrt-unprod 50.8%

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

    3. frac-times 51.0%

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

    4. metadata-eval 51.0%

       \[\leadsto \frac{1}{\sqrt[3]{{\cos \left(x \cdot \sqrt{\frac{\color{blue}{0.25}}{y \cdot y}}\right)}^{3}}} \]

    5. metadata-eval 51.0%

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

    6. frac-times 50.8%

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

    7. sqrt-unprod 26.8%

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

    8. add-sqr-sqrt 54.7%

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

    9. associate-*r/ 54.5%

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

    10. add-sqr-sqrt 28.3%

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

    11. associate-/l/ 28.4%

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

    12. div-inv 28.3%

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

    13. clear-num 28.3%

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

    14. associate-*l/ 28.3%

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

    15. *-un-lft-identity 28.3%

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

    16. pow1/2 28.3%

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

    17. pow-flip 28.6%

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

    18. metadata-eval 28.6%

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

    19. div-inv 28.6%

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

    20. *-commutative 28.6%

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

    21. associate-/r* 28.6%

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

    22. metadata-eval 28.6%

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

14. Applied egg-rr 28.6%

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

15. Final simplification 28.6%

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

Alternative 2: 55.8% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (cbrt (/ x y_m)))) (/ 1.0 (cos (* (pow t_0 2.0) (* -0.5 t_0))))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-/l* 54.7%

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

10. Simplified 54.7%

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

    1. associate-*r/ 54.5%

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

    2. add-sqr-sqrt 28.3%

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

    3. associate-/r* 28.4%

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

12. Applied egg-rr 28.4%

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

    1. associate-/l/ 28.3%

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

    2. add-sqr-sqrt 54.5%

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

    3. associate-*l/ 54.5%

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

    4. add-cube-cbrt 55.3%

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

    5. associate-*l* 55.3%

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

    6. pow2 55.3%

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

14. Applied egg-rr 55.3%

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

15. Final simplification 55.3%

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

Alternative 3: 55.8% accurate, 0.7× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ 1.0 (cos (/ (/ (* -0.5 x) (sqrt y_m)) (sqrt y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Cos[N[(N[(N[(-0.5 * x), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-/l* 54.7%

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

10. Simplified 54.7%

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

    1. associate-*r/ 54.5%

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

    2. add-sqr-sqrt 28.3%

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

    3. associate-/r* 28.4%

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

12. Applied egg-rr 28.4%

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

13. Final simplification 28.4%

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

Alternative 4: 55.9% accurate, 1.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ 1.0 (cos (* (/ 1.0 y_m) (/ 1.0 (/ -2.0 x))))))

C

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

Fortran

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

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(1.0 / cos(Float64(Float64(1.0 / y_m) * Float64(1.0 / Float64(-2.0 / x)))))
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Cos[N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(1.0 / N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-/l* 54.7%

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

10. Simplified 54.7%

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

    1. associate-*r/ 54.5%

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

    2. add-sqr-sqrt 28.3%

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

    3. associate-/r* 28.4%

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

12. Applied egg-rr 28.4%

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

    1. associate-/l/ 28.3%

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

    2. add-sqr-sqrt 54.5%

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

    3. clear-num 54.4%

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

    4. inv-pow 54.4%

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

    5. div-inv 54.6%

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

    6. unpow-prod-down 55.0%

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

    7. inv-pow 55.0%

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

    8. *-commutative 55.0%

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

    9. associate-/r* 55.0%

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

    10. metadata-eval 55.0%

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

14. Applied egg-rr 55.0%

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

    1. unpow-1 55.0%

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

16. Simplified 55.0%

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

17. Final simplification 55.0%

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

Alternative 5: 55.9% accurate, 2.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (/ 1.0 (cos (* x (/ -0.5 y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(1.0 / N[Cos[N[(x * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in x around inf 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-/l* 54.7%

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

10. Simplified 54.7%

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

11. Final simplification 54.7%

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

Alternative 6: 55.8% accurate, 211.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := 1.0

Derivation

1. Initial program 40.4%

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

   1. remove-double-neg 40.4%

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

   2. distribute-frac-neg 40.4%

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

   3. tan-neg 40.4%

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

   4. distribute-frac-neg2 40.4%

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

   5. distribute-lft-neg-out 40.4%

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

   6. distribute-frac-neg2 40.4%

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

   7. distribute-lft-neg-out 40.4%

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

   8. distribute-frac-neg2 40.4%

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

   9. distribute-frac-neg 40.4%

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

   10. neg-mul-1 40.4%

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

   11. *-commutative 40.4%

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

   12. associate-/l* 40.5%

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

   13. *-commutative 40.5%

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

   14. associate-/r* 40.5%

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

   15. metadata-eval 40.5%

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

   16. sin-neg 40.5%

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

   17. distribute-frac-neg 40.5%

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

3. Simplified 40.6%

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

5. Taylor expanded in x around 0 54.6%

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

6. Final simplification 54.6%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 55.8% 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 2024066 
(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)))))