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

Percentage Accurate: 44.2% → 57.3%

Time: 24.5s

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: 44.2% 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: 57.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+123}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+123)
   (/ 1.0 (cos (/ 1.0 (* y_m (/ 2.0 x_m)))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+123) {
		tmp = 1.0 / cos((1.0 / (y_m * (2.0 / x_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+123) then
        tmp = 1.0d0 / cos((1.0d0 / (y_m * (2.0d0 / x_m))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+123) {
		tmp = 1.0 / Math.cos((1.0 / (y_m * (2.0 / x_m))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+123:
		tmp = 1.0 / math.cos((1.0 / (y_m * (2.0 / x_m))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+123)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(y_m * Float64(2.0 / x_m)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+123)
		tmp = 1.0 / cos((1.0 / (y_m * (2.0 / x_m))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+123], N[(1.0 / N[Cos[N[(1.0 / N[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.99999999999999978e122

   1. Initial program 51.0%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. *-commutative 65.4%

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

   5. Simplified 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. *-commutative 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-*r/ 65.4%

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

      5. clear-num 65.6%

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

      6. un-div-inv 65.6%

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

   7. Applied egg-rr 65.6%

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

      1. *-rgt-identity 65.6%

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

      2. associate-/r/ 65.5%

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

   9. Simplified 65.5%

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

       1. metadata-eval 65.5%

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

       2. associate-/r* 65.5%

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

       3. *-commutative 65.5%

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

       4. associate-/r/ 65.6%

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

       5. associate-/l* 65.6%

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

   11. Applied egg-rr 65.6%

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

   if 9.99999999999999978e122 < (/.f64 x (*.f64 y 2))

   1. Initial program 9.1%

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

      1. remove-double-neg 9.1%

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

      2. distribute-frac-neg 9.1%

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

      3. tan-neg 9.1%

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

      4. distribute-frac-neg2 9.1%

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

      5. distribute-lft-neg-out 9.1%

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

      6. distribute-frac-neg2 9.1%

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

      7. distribute-lft-neg-out 9.1%

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

      8. distribute-frac-neg2 9.1%

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

      9. distribute-frac-neg 9.1%

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

      10. neg-mul-1 9.1%

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

      11. *-commutative 9.1%

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

      12. associate-/l* 8.7%

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

      13. *-commutative 8.7%

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

      14. associate-/r* 8.7%

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

      15. metadata-eval 8.7%

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

      16. sin-neg 8.7%

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

      17. distribute-frac-neg 8.7%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 10.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+123}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y \cdot \frac{2}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \frac{1}{\cos \left(\frac{\frac{0.5}{{\left({\left({\left(\frac{y\_m}{x\_m}\right)}^{0.16666666666666666}\right)}^{2}\right)}^{2}}}{\sqrt[3]{\frac{y\_m}{x\_m}}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/
  1.0
  (cos
   (/
    (/ 0.5 (pow (pow (pow (/ y_m x_m) 0.16666666666666666) 2.0) 2.0))
    (cbrt (/ y_m x_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(((0.5 / pow(pow(pow((y_m / x_m), 0.16666666666666666), 2.0), 2.0)) / cbrt((y_m / x_m))));
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	return 1.0 / Math.cos(((0.5 / Math.pow(Math.pow(Math.pow((y_m / x_m), 0.16666666666666666), 2.0), 2.0)) / Math.cbrt((y_m / x_m))));
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(Float64(0.5 / (((Float64(y_m / x_m) ^ 0.16666666666666666) ^ 2.0) ^ 2.0)) / cbrt(Float64(y_m / x_m)))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(N[(0.5 / N[Power[N[Power[N[Power[N[(y$95$m / x$95$m), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(y$95$m / x$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

5. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. *-commutative 59.5%

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

   3. *-commutative 59.5%

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

   4. associate-*r/ 59.5%

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

   5. clear-num 59.7%

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

   6. un-div-inv 59.7%

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

7. Applied egg-rr 59.7%

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

   1. *-rgt-identity 59.7%

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

   2. associate-/r/ 59.5%

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

9. Simplified 59.5%

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

    1. associate-/r/ 59.7%

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

    2. add-cube-cbrt 60.1%

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

    3. associate-/r* 60.3%

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

    4. pow2 60.3%

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

11. Applied egg-rr 60.3%

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

    1. add-sqr-sqrt 26.5%

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

    2. pow2 26.5%

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

    3. pow1/3 35.6%

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

    4. sqrt-pow1 35.8%

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

    5. metadata-eval 35.8%

       \[\leadsto \frac{1}{\cos \left(\frac{\frac{0.5}{{\left({\left({\left(\frac{y}{x}\right)}^{\color{blue}{0.16666666666666666}}\right)}^{2}\right)}^{2}}}{\sqrt[3]{\frac{y}{x}}}\right)} \]

13. Applied egg-rr 35.8%

    \[\leadsto \frac{1}{\cos \left(\frac{\frac{0.5}{{\color{blue}{\left({\left({\left(\frac{y}{x}\right)}^{0.16666666666666666}\right)}^{2}\right)}}^{2}}}{\sqrt[3]{\frac{y}{x}}}\right)} \]

14. Final simplification 35.8%

    \[\leadsto \frac{1}{\cos \left(\frac{\frac{0.5}{{\left({\left({\left(\frac{y}{x}\right)}^{0.16666666666666666}\right)}^{2}\right)}^{2}}}{\sqrt[3]{\frac{y}{x}}}\right)} \]
15. Add Preprocessing

Alternative 3: 55.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

5. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. *-commutative 59.5%

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

   3. *-commutative 59.5%

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

   4. associate-*r/ 59.5%

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

   5. clear-num 59.7%

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

   6. un-div-inv 59.7%

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

7. Applied egg-rr 59.7%

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

   1. *-rgt-identity 59.7%

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

   2. associate-/r/ 59.5%

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

9. Simplified 59.5%

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

    1. associate-/r/ 59.7%

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

    2. add-cube-cbrt 60.1%

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

    3. associate-/r* 60.3%

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

    4. pow2 60.3%

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

11. Applied egg-rr 60.3%

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

12. Final simplification 60.3%

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

Alternative 4: 55.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

5. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. *-commutative 59.5%

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

   3. *-commutative 59.5%

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

   4. associate-*r/ 59.5%

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

   5. clear-num 59.7%

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

   6. un-div-inv 59.7%

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

7. Applied egg-rr 59.7%

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

   1. *-rgt-identity 59.7%

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

   2. associate-/r/ 59.5%

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

9. Simplified 59.5%

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

    1. associate-/r/ 59.7%

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

11. Applied egg-rr 59.7%

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

    1. add-cube-cbrt 60.3%

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

    2. pow3 60.0%

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

    3. div-inv 60.0%

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

    4. clear-num 60.3%

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

13. Applied egg-rr 60.3%

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

14. Final simplification 60.3%

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

Alternative 5: 55.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

5. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. *-commutative 59.5%

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

   3. *-commutative 59.5%

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

   4. associate-*r/ 59.5%

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

   5. clear-num 59.7%

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

   6. un-div-inv 59.7%

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

7. Applied egg-rr 59.7%

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

   1. *-rgt-identity 59.7%

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

   2. associate-/r/ 59.5%

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

9. Simplified 59.5%

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

    1. associate-/r/ 59.7%

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

11. Applied egg-rr 59.7%

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

    1. div-inv 59.7%

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

    2. clear-num 59.5%

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

    3. add-exp-log 33.9%

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

13. Applied egg-rr 33.9%

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

14. Final simplification 33.9%

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

Alternative 6: 55.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. remove-double-neg 46.6%

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

   2. distribute-frac-neg 46.6%

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

   3. tan-neg 46.6%

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

   4. distribute-frac-neg2 46.6%

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

   5. distribute-lft-neg-out 46.6%

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

   6. distribute-frac-neg2 46.6%

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

   7. distribute-lft-neg-out 46.6%

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

   8. distribute-frac-neg2 46.6%

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

   9. distribute-frac-neg 46.6%

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

   10. neg-mul-1 46.6%

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

   11. *-commutative 46.6%

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

   12. associate-/l* 46.2%

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

   13. *-commutative 46.2%

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

   14. associate-/r* 46.2%

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

   15. metadata-eval 46.2%

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

   16. sin-neg 46.2%

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

   17. distribute-frac-neg 46.2%

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

3. Simplified 46.6%

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

5. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

   3. associate-*r/ 59.5%

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

7. Simplified 59.5%

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

8. Final simplification 59.5%

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

Alternative 7: 55.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around inf 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-commutative 59.5%

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

5. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. *-commutative 59.5%

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

   3. *-commutative 59.5%

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

   4. associate-*r/ 59.5%

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

   5. clear-num 59.7%

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

   6. un-div-inv 59.7%

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

7. Applied egg-rr 59.7%

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

   1. *-rgt-identity 59.7%

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

   2. associate-/r/ 59.5%

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

9. Simplified 59.5%

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

    1. associate-/r/ 59.7%

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

11. Applied egg-rr 59.7%

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

12. Final simplification 59.7%

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

Alternative 8: 55.8% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. remove-double-neg 46.6%

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

   2. distribute-frac-neg 46.6%

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

   3. tan-neg 46.6%

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

   4. distribute-frac-neg2 46.6%

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

   5. distribute-lft-neg-out 46.6%

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

   6. distribute-frac-neg2 46.6%

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

   7. distribute-lft-neg-out 46.6%

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

   8. distribute-frac-neg2 46.6%

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

   9. distribute-frac-neg 46.6%

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

   10. neg-mul-1 46.6%

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

   11. *-commutative 46.6%

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

   12. associate-/l* 46.2%

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

   13. *-commutative 46.2%

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

   14. associate-/r* 46.2%

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

   15. metadata-eval 46.2%

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

   16. sin-neg 46.2%

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

   17. distribute-frac-neg 46.2%

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

3. Simplified 46.6%

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

5. Taylor expanded in x around 0 59.2%

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

6. Final simplification 59.2%

   \[\leadsto 1 \]
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 2024055 
(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)))))