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

Percentage Accurate: 44.1% → 56.7%

Time: 24.2s

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: 44.1% 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: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-0.5 \cdot x\_m}{\sqrt{y\_m}}}{\sqrt{y\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 5e+68)
   (/ 1.0 (cos (/ (/ (* -0.5 x_m) (sqrt y_m)) (sqrt y_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)) <= 5e+68) {
		tmp = 1.0 / cos((((-0.5 * x_m) / sqrt(y_m)) / sqrt(y_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)) <= 5d+68) then
        tmp = 1.0d0 / cos(((((-0.5d0) * x_m) / sqrt(y_m)) / sqrt(y_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)) <= 5e+68) {
		tmp = 1.0 / Math.cos((((-0.5 * x_m) / Math.sqrt(y_m)) / Math.sqrt(y_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)) <= 5e+68:
		tmp = 1.0 / math.cos((((-0.5 * x_m) / math.sqrt(y_m)) / math.sqrt(y_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)) <= 5e+68)
		tmp = Float64(1.0 / cos(Float64(Float64(Float64(-0.5 * x_m) / sqrt(y_m)) / sqrt(y_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)) <= 5e+68)
		tmp = 1.0 / cos((((-0.5 * x_m) / sqrt(y_m)) / sqrt(y_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], 5e+68], N[(1.0 / N[Cos[N[(N[(N[(-0.5 * x$95$m), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.0000000000000004e68

   1. Initial program 46.0%

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

      1. remove-double-neg 46.0%

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

      2. distribute-frac-neg 46.0%

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

      3. tan-neg 46.0%

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

      4. distribute-frac-neg2 46.0%

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

      5. distribute-lft-neg-out 46.0%

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

      6. distribute-frac-neg2 46.0%

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

      7. distribute-lft-neg-out 46.0%

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

      8. distribute-frac-neg2 46.0%

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

      9. distribute-frac-neg 46.0%

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

      10. neg-mul-1 46.0%

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

      11. *-commutative 46.0%

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

      12. associate-/l* 45.6%

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

      13. *-commutative 45.6%

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

      14. associate-/r* 45.6%

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

      15. metadata-eval 45.6%

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

      16. sin-neg 45.6%

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

      17. distribute-frac-neg 45.6%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in x around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*r/ 65.2%

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

   7. Simplified 65.2%

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

      1. associate-*r/ 65.0%

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

      2. add-sqr-sqrt 36.4%

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

      3. associate-/r* 36.3%

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

   9. Applied egg-rr 36.3%

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

   if 5.0000000000000004e68 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.5%

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

      1. remove-double-neg 5.5%

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

      2. distribute-frac-neg 5.5%

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

      3. tan-neg 5.5%

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

      4. distribute-frac-neg2 5.5%

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

      5. distribute-lft-neg-out 5.5%

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

      6. distribute-frac-neg2 5.5%

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

      7. distribute-lft-neg-out 5.5%

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

      8. distribute-frac-neg2 5.5%

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

      9. distribute-frac-neg 5.5%

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

      10. neg-mul-1 5.5%

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

      11. *-commutative 5.5%

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

      12. associate-/l* 6.3%

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

      13. *-commutative 6.3%

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

      14. associate-/r* 6.3%

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

      15. metadata-eval 6.3%

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

      16. sin-neg 6.3%

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

      17. distribute-frac-neg 6.3%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in x around 0 11.6%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{-0.5 \cdot x}{\sqrt{y}}}{\sqrt{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+68}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+68)
   (/ 1.0 (cos (/ (* x_m (pow (cbrt -0.5) 3.0)) y_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+68) {
		tmp = 1.0 / cos(((x_m * pow(cbrt(-0.5), 3.0)) / y_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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+68) {
		tmp = 1.0 / Math.cos(((x_m * Math.pow(Math.cbrt(-0.5), 3.0)) / y_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+68)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * (cbrt(-0.5) ^ 3.0)) / y_m)));
	else
		tmp = 1.0;
	end
	return 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+68], N[(1.0 / N[Cos[N[(N[(x$95$m * N[Power[N[Power[-0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999953e67

   1. Initial program 46.4%

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

      1. remove-double-neg 46.4%

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

      2. distribute-frac-neg 46.4%

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

      3. tan-neg 46.4%

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

      4. distribute-frac-neg2 46.4%

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

      5. distribute-lft-neg-out 46.4%

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

      6. distribute-frac-neg2 46.4%

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

      7. distribute-lft-neg-out 46.4%

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

      8. distribute-frac-neg2 46.4%

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

      9. distribute-frac-neg 46.4%

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

      10. neg-mul-1 46.4%

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

      11. *-commutative 46.4%

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

      12. associate-/l* 46.0%

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

      13. *-commutative 46.0%

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

      14. associate-/r* 46.0%

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

      15. metadata-eval 46.0%

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

      16. sin-neg 46.0%

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

      17. distribute-frac-neg 46.0%

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

   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 65.5%

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

      1. associate-*r/ 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r/ 65.7%

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

   7. Simplified 65.7%

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

      1. add-cube-cbrt 66.0%

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

      2. pow3 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in x around inf 65.9%

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

   if 9.99999999999999953e67 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.3%

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

      1. remove-double-neg 5.3%

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

      2. distribute-frac-neg 5.3%

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

      3. tan-neg 5.3%

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

      4. distribute-frac-neg2 5.3%

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

      5. distribute-lft-neg-out 5.3%

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

      6. distribute-frac-neg2 5.3%

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

      7. distribute-lft-neg-out 5.3%

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

      8. distribute-frac-neg2 5.3%

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

      9. distribute-frac-neg 5.3%

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

      10. neg-mul-1 5.3%

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

      11. *-commutative 5.3%

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

      12. associate-/l* 6.4%

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

      13. *-commutative 6.4%

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

      14. associate-/r* 6.4%

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

      15. metadata-eval 6.4%

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

      16. sin-neg 6.4%

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

      17. distribute-frac-neg 6.4%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in x around 0 11.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+68}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.8% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (pow (/ 1.0 (/ 1.0 (cbrt (/ -0.5 (/ y_m x_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((1.0 / (1.0 / cbrt((-0.5 / (y_m / x_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((1.0 / (1.0 / Math.cbrt((-0.5 / (y_m / x_m))))), 3.0));
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos((Float64(1.0 / Float64(1.0 / cbrt(Float64(-0.5 / Float64(y_m / x_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[(1.0 / N[(1.0 / N[Power[N[(-0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

   1. remove-double-neg 39.6%

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

   2. distribute-frac-neg 39.6%

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

   3. tan-neg 39.6%

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

   4. distribute-frac-neg2 39.6%

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

   5. distribute-lft-neg-out 39.6%

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

   6. distribute-frac-neg2 39.6%

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

   7. distribute-lft-neg-out 39.6%

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

   8. distribute-frac-neg2 39.6%

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

   9. distribute-frac-neg 39.6%

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

   10. neg-mul-1 39.6%

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

   11. *-commutative 39.6%

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

   12. associate-/l* 39.5%

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

   13. *-commutative 39.5%

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

   14. associate-/r* 39.5%

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

   15. metadata-eval 39.5%

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

   16. sin-neg 39.5%

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

   17. distribute-frac-neg 39.5%

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

3. Simplified 40.1%

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

5. Taylor expanded in x around inf 55.7%

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

   1. associate-*r/ 55.7%

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

   2. *-commutative 55.7%

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

   3. associate-*r/ 56.1%

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

7. Simplified 56.1%

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

   1. add-cube-cbrt 56.6%

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

   2. pow3 56.4%

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

9. Applied egg-rr 56.4%

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

    1. associate-*r/ 56.3%

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

    2. metadata-eval 56.3%

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

    3. div-inv 56.3%

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

    4. clear-num 56.5%

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

    5. cbrt-div 56.3%

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

    6. metadata-eval 56.3%

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

    7. div-inv 56.5%

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

    8. clear-num 56.5%

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

11. Applied egg-rr 56.5%

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

    1. /-rgt-identity 56.5%

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

    2. clear-num 56.5%

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

    3. metadata-eval 56.5%

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

    4. clear-num 56.5%

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

    5. div-inv 56.3%

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

    6. div-inv 56.3%

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

    7. metadata-eval 56.3%

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

    8. associate-/r* 56.3%

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

    9. cbrt-div 56.9%

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

    10. clear-num 56.9%

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

    11. div-inv 56.9%

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

    12. clear-num 56.5%

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

13. Applied egg-rr 56.5%

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

    1. associate-*r/ 56.5%

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

    2. associate-*l/ 56.5%

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

    3. associate-/r/ 56.9%

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

15. Simplified 56.9%

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

16. Final simplification 56.9%

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

Alternative 4: 54.9% accurate, 0.7× speedup

Math

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

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 (/ y_m x_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 / (y_m / x_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 / (y_m / x_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(y_m / x_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[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

   1. remove-double-neg 39.6%

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

   2. distribute-frac-neg 39.6%

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

   3. tan-neg 39.6%

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

   4. distribute-frac-neg2 39.6%

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

   5. distribute-lft-neg-out 39.6%

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

   6. distribute-frac-neg2 39.6%

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

   7. distribute-lft-neg-out 39.6%

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

   8. distribute-frac-neg2 39.6%

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

   9. distribute-frac-neg 39.6%

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

   10. neg-mul-1 39.6%

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

   11. *-commutative 39.6%

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

   12. associate-/l* 39.5%

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

   13. *-commutative 39.5%

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

   14. associate-/r* 39.5%

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

   15. metadata-eval 39.5%

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

   16. sin-neg 39.5%

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

   17. distribute-frac-neg 39.5%

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

3. Simplified 40.1%

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

5. Taylor expanded in x around inf 55.7%

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

   1. associate-*r/ 55.7%

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

   2. *-commutative 55.7%

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

   3. associate-*r/ 56.1%

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

7. Simplified 56.1%

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

   1. add-cube-cbrt 56.6%

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

   2. pow3 56.4%

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

9. Applied egg-rr 56.4%

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

    1. associate-*r/ 56.3%

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

    2. metadata-eval 56.3%

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

    3. div-inv 56.3%

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

    4. clear-num 56.5%

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

    5. cbrt-div 56.3%

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

    6. metadata-eval 56.3%

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

    7. div-inv 56.5%

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

    8. clear-num 56.5%

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

11. Applied egg-rr 56.5%

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

    1. metadata-eval 56.5%

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

    2. cbrt-div 56.5%

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

    3. associate-*r/ 56.5%

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

    4. clear-num 56.3%

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

    5. associate-/l/ 56.3%

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

    6. *-un-lft-identity 56.3%

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

    7. cbrt-prod 56.3%

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

    8. metadata-eval 56.3%

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

    9. cbrt-div 56.3%

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

    10. div-inv 56.3%

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

    11. metadata-eval 56.3%

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

    12. cbrt-prod 55.6%

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

    13. associate-*r/ 55.9%

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

    14. cbrt-div 56.8%

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

    15. *-commutative 56.8%

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

    16. cbrt-unprod 56.4%

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

    17. associate-/r/ 56.5%

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

    18. div-inv 56.5%

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

    19. clear-num 56.3%

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

13. Applied egg-rr 56.3%

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

    1. *-lft-identity 56.3%

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

    2. associate-*r/ 56.3%

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

    3. associate-*l/ 56.4%

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

    4. associate-/r/ 56.5%

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

15. Simplified 56.5%

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

16. Final simplification 56.5%

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

Alternative 5: 55.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 / cos(((1.0 / y_m) / (-2.0 / 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(((1.0d0 / y_m) / ((-2.0d0) / 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(((1.0 / y_m) / (-2.0 / x_m)));
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 / cos(((1.0 / y_m) / (-2.0 / 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[(1.0 / y$95$m), $MachinePrecision] / N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

   1. remove-double-neg 39.6%

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

   2. distribute-frac-neg 39.6%

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

   3. tan-neg 39.6%

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

   4. distribute-frac-neg2 39.6%

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

   5. distribute-lft-neg-out 39.6%

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

   6. distribute-frac-neg2 39.6%

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

   7. distribute-lft-neg-out 39.6%

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

   8. distribute-frac-neg2 39.6%

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

   9. distribute-frac-neg 39.6%

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

   10. neg-mul-1 39.6%

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

   11. *-commutative 39.6%

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

   12. associate-/l* 39.5%

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

   13. *-commutative 39.5%

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

   14. associate-/r* 39.5%

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

   15. metadata-eval 39.5%

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

   16. sin-neg 39.5%

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

   17. distribute-frac-neg 39.5%

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

3. Simplified 40.1%

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

5. Taylor expanded in x around inf 55.7%

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

   1. associate-*r/ 55.7%

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

   2. *-commutative 55.7%

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

   3. associate-*r/ 56.1%

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

7. Simplified 56.1%

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

   1. add-cube-cbrt 56.6%

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

   2. pow3 56.4%

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

9. Applied egg-rr 56.4%

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

    1. rem-cube-cbrt 56.1%

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

    2. *-commutative 56.1%

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

    3. associate-*l/ 55.7%

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

    4. div-inv 56.1%

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

    5. metadata-eval 56.1%

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

    6. associate-/r/ 56.2%

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

    7. associate-*l/ 56.3%

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

    8. *-un-lft-identity 56.3%

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

11. Applied egg-rr 56.3%

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

12. Final simplification 56.3%

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

Alternative 6: 55.0% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* 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((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((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((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((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(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((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[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

3. Taylor expanded in x around inf 55.7%

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

4. Final simplification 55.7%

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

Alternative 7: 55.0% 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 39.6%

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

   1. remove-double-neg 39.6%

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

   2. distribute-frac-neg 39.6%

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

   3. tan-neg 39.6%

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

   4. distribute-frac-neg2 39.6%

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

   5. distribute-lft-neg-out 39.6%

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

   6. distribute-frac-neg2 39.6%

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

   7. distribute-lft-neg-out 39.6%

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

   8. distribute-frac-neg2 39.6%

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

   9. distribute-frac-neg 39.6%

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

   10. neg-mul-1 39.6%

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

   11. *-commutative 39.6%

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

   12. associate-/l* 39.5%

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

   13. *-commutative 39.5%

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

   14. associate-/r* 39.5%

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

   15. metadata-eval 39.5%

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

   16. sin-neg 39.5%

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

   17. distribute-frac-neg 39.5%

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

3. Simplified 40.1%

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

5. Taylor expanded in x around inf 55.7%

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

   1. associate-*r/ 55.7%

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

   2. *-commutative 55.7%

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

   3. associate-*r/ 56.1%

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

7. Simplified 56.1%

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

8. Final simplification 56.1%

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

Alternative 8: 54.9% 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 39.6%

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

   1. remove-double-neg 39.6%

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

   2. distribute-frac-neg 39.6%

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

   3. tan-neg 39.6%

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

   4. distribute-frac-neg2 39.6%

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

   5. distribute-lft-neg-out 39.6%

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

   6. distribute-frac-neg2 39.6%

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

   7. distribute-lft-neg-out 39.6%

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

   8. distribute-frac-neg2 39.6%

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

   9. distribute-frac-neg 39.6%

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

   10. neg-mul-1 39.6%

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

   11. *-commutative 39.6%

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

   12. associate-/l* 39.5%

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

   13. *-commutative 39.5%

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

   14. associate-/r* 39.5%

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

   15. metadata-eval 39.5%

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

   16. sin-neg 39.5%

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

   17. distribute-frac-neg 39.5%

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

3. Simplified 40.1%

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

5. Taylor expanded in x around 0 55.3%

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

6. Final simplification 55.3%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t\_1}{t\_1 \cdot \log \left(e^{\cos t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2024071 
(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)))))