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

Percentage Accurate: 44.0% → 56.9%

Time: 30.2s

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: 44.0% 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: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+253}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{x\_m}\right)}^{2}}{-2} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{x\_m}}\right)}^{3}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\sqrt[3]{0.015625}}\\ \end{array} \end{array} \]

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+253)
   (/
    1.0
    (cos
     (* (/ (pow (cbrt x_m) 2.0) -2.0) (/ (pow (cbrt (cbrt x_m)) 3.0) y_m))))
   (* -2.0 (sqrt (cbrt 0.015625)))))

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+253) {
		tmp = 1.0 / cos(((pow(cbrt(x_m), 2.0) / -2.0) * (pow(cbrt(cbrt(x_m)), 3.0) / y_m)));
	} else {
		tmp = -2.0 * sqrt(cbrt(0.015625));
	}
	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+253) {
		tmp = 1.0 / Math.cos(((Math.pow(Math.cbrt(x_m), 2.0) / -2.0) * (Math.pow(Math.cbrt(Math.cbrt(x_m)), 3.0) / y_m)));
	} else {
		tmp = -2.0 * Math.sqrt(Math.cbrt(0.015625));
	}
	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+253)
		tmp = Float64(1.0 / cos(Float64(Float64((cbrt(x_m) ^ 2.0) / -2.0) * Float64((cbrt(cbrt(x_m)) ^ 3.0) / y_m))));
	else
		tmp = Float64(-2.0 * sqrt(cbrt(0.015625)));
	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+253], N[(1.0 / N[Cos[N[(N[(N[Power[N[Power[x$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / -2.0), $MachinePrecision] * N[(N[Power[N[Power[N[Power[x$95$m, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.9999999999999994e252

   1. Initial program 48.5%

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

      1. remove-double-neg 48.5%

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

      2. distribute-frac-neg 48.5%

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

      3. tan-neg 48.5%

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

      4. distribute-frac-neg2 48.5%

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

      5. distribute-lft-neg-out 48.5%

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

      6. distribute-frac-neg2 48.5%

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

      7. distribute-lft-neg-out 48.5%

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

      8. distribute-frac-neg2 48.5%

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

      9. distribute-frac-neg 48.5%

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

      10. neg-mul-1 48.5%

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

      11. *-commutative 48.5%

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

      12. associate-/l* 48.5%

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

      13. *-commutative 48.5%

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

      14. associate-/r* 48.5%

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

      15. metadata-eval 48.5%

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

      16. sin-neg 48.5%

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

      17. distribute-frac-neg 48.5%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in x around inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. *-commutative 63.3%

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

      3. associate-*r/ 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in x around inf 63.3%

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

      1. metadata-eval 63.3%

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

      2. times-frac 63.3%

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

      3. neg-mul-1 63.3%

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

      4. *-commutative 63.3%

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

      5. cos-neg 63.3%

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

      6. neg-mul-1 63.3%

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

      7. *-commutative 63.3%

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

      8. times-frac 63.3%

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

      9. metadata-eval 63.3%

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

      10. distribute-rgt-neg-in 63.3%

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

      11. metadata-eval 63.3%

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

      12. associate-*l/ 63.3%

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

   10. Simplified 63.3%

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

       1. *-un-lft-identity 63.3%

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

       2. times-frac 63.5%

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

       3. metadata-eval 63.5%

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

       4. pow1 63.5%

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

       5. metadata-eval 63.5%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \frac{\sqrt{0.25}}{{y}^{\color{blue}{\left(\frac{2}{2}\right)}}}\right)} \]

       6. sqrt-pow1 59.4%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \frac{\sqrt{0.25}}{\color{blue}{\sqrt{{y}^{2}}}}\right)} \]

       7. sqrt-div 59.3%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \color{blue}{\sqrt{\frac{0.25}{{y}^{2}}}}\right)} \]

       8. metadata-eval 59.3%

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

       9. unpow2 59.3%

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

       10. frac-times 59.4%

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

       11. sqrt-unprod 33.9%

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

       12. add-sqr-sqrt 63.5%

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

       13. times-frac 63.3%

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

       14. *-commutative 63.3%

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

       15. times-frac 63.3%

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

       16. metadata-eval 63.3%

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

       17. metadata-eval 63.3%

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

       18. times-frac 63.3%

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

       19. *-un-lft-identity 63.3%

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

       20. add-cube-cbrt 64.0%

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

       21. times-frac 64.2%

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

       22. pow2 64.2%

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

   12. Applied egg-rr 64.2%

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

       1. add-cube-cbrt 64.2%

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

       2. pow3 64.3%

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

   14. Applied egg-rr 64.3%

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

   if 9.9999999999999994e252 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.7%

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

      1. remove-double-neg 5.7%

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

      2. distribute-frac-neg 5.7%

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

      3. tan-neg 5.7%

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

      4. distribute-frac-neg2 5.7%

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

      5. distribute-lft-neg-out 5.7%

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

      6. distribute-frac-neg2 5.7%

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

      7. distribute-lft-neg-out 5.7%

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

      8. distribute-frac-neg2 5.7%

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

      9. distribute-frac-neg 5.7%

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

      10. neg-mul-1 5.7%

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

      11. *-commutative 5.7%

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

      12. associate-/l* 6.7%

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

      13. *-commutative 6.7%

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

      14. associate-/r* 6.7%

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

      15. metadata-eval 6.7%

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

      16. sin-neg 6.7%

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

      17. distribute-frac-neg 6.7%

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

   3. Simplified 5.9%

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

      1. add-cbrt-cube 2.4%

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

      2. pow3 2.4%

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

   6. Applied egg-rr 2.4%

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

   7. Taylor expanded in x around 0 11.2%

      \[\leadsto \color{blue}{-2 \cdot \sqrt[3]{-0.125}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}\right)} \]

      2. sqrt-unprod 11.4%

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}} \]

      3. cbrt-unprod 11.4%

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}} \]

      4. metadata-eval 11.4%

         \[\leadsto -2 \cdot \sqrt{\sqrt[3]{\color{blue}{0.015625}}} \]

   9. Applied egg-rr 11.4%

      \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{0.015625}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 57.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{x\_m}\right)}^{2}}{-2} \cdot \frac{\sqrt[3]{x\_m}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\sqrt[3]{0.015625}}\\ \end{array} \end{array} \]

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)) 4e+133)
   (/ 1.0 (cos (* (/ (pow (cbrt x_m) 2.0) -2.0) (/ (cbrt x_m) y_m))))
   (* -2.0 (sqrt (cbrt 0.015625)))))

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)) <= 4e+133) {
		tmp = 1.0 / cos(((pow(cbrt(x_m), 2.0) / -2.0) * (cbrt(x_m) / y_m)));
	} else {
		tmp = -2.0 * sqrt(cbrt(0.015625));
	}
	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)) <= 4e+133) {
		tmp = 1.0 / Math.cos(((Math.pow(Math.cbrt(x_m), 2.0) / -2.0) * (Math.cbrt(x_m) / y_m)));
	} else {
		tmp = -2.0 * Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 4e+133)
		tmp = Float64(1.0 / cos(Float64(Float64((cbrt(x_m) ^ 2.0) / -2.0) * Float64(cbrt(x_m) / y_m))));
	else
		tmp = Float64(-2.0 * sqrt(cbrt(0.015625)));
	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], 4e+133], N[(1.0 / N[Cos[N[(N[(N[Power[N[Power[x$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / -2.0), $MachinePrecision] * N[(N[Power[x$95$m, 1/3], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.0000000000000001e133

   1. Initial program 51.1%

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

      1. remove-double-neg 51.1%

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

      2. distribute-frac-neg 51.1%

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

      3. tan-neg 51.1%

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

      4. distribute-frac-neg2 51.1%

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

      5. distribute-lft-neg-out 51.1%

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

      6. distribute-frac-neg2 51.1%

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

      7. distribute-lft-neg-out 51.1%

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

      8. distribute-frac-neg2 51.1%

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

      9. distribute-frac-neg 51.1%

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

      10. neg-mul-1 51.1%

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

      11. *-commutative 51.1%

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

      12. associate-/l* 50.9%

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

      13. *-commutative 50.9%

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

      14. associate-/r* 50.9%

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

      15. metadata-eval 50.9%

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

      16. sin-neg 50.9%

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

      17. distribute-frac-neg 50.9%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in x around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r/ 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in x around inf 66.9%

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

      1. metadata-eval 66.9%

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

      2. times-frac 66.9%

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

      3. neg-mul-1 66.9%

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

      4. *-commutative 66.9%

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

      5. cos-neg 66.9%

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

      6. neg-mul-1 66.9%

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

      7. *-commutative 66.9%

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

      8. times-frac 66.9%

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

      9. metadata-eval 66.9%

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

      10. distribute-rgt-neg-in 66.9%

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

      11. metadata-eval 66.9%

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

      12. associate-*l/ 66.9%

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

   10. Simplified 66.9%

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

       1. *-un-lft-identity 66.9%

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

       2. times-frac 66.9%

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

       3. metadata-eval 66.9%

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

       4. pow1 66.9%

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

       5. metadata-eval 66.9%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \frac{\sqrt{0.25}}{{y}^{\color{blue}{\left(\frac{2}{2}\right)}}}\right)} \]

       6. sqrt-pow1 62.9%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \frac{\sqrt{0.25}}{\color{blue}{\sqrt{{y}^{2}}}}\right)} \]

       7. sqrt-div 62.8%

          \[\leadsto \frac{1}{\cos \left(\frac{x}{1} \cdot \color{blue}{\sqrt{\frac{0.25}{{y}^{2}}}}\right)} \]

       8. metadata-eval 62.8%

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

       9. unpow2 62.8%

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

       10. frac-times 62.9%

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

       11. sqrt-unprod 35.8%

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

       12. add-sqr-sqrt 66.9%

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

       13. times-frac 66.9%

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

       14. *-commutative 66.9%

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

       15. times-frac 66.9%

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

       16. metadata-eval 66.9%

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

       17. metadata-eval 66.9%

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

       18. times-frac 66.9%

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

       19. *-un-lft-identity 66.9%

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

       20. add-cube-cbrt 67.6%

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

       21. times-frac 67.7%

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

       22. pow2 67.7%

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

   12. Applied egg-rr 67.7%

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

   if 4.0000000000000001e133 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.9%

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

      1. remove-double-neg 7.9%

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

      2. distribute-frac-neg 7.9%

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

      3. tan-neg 7.9%

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

      4. distribute-frac-neg2 7.9%

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

      5. distribute-lft-neg-out 7.9%

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

      6. distribute-frac-neg2 7.9%

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

      7. distribute-lft-neg-out 7.9%

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

      8. distribute-frac-neg2 7.9%

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

      9. distribute-frac-neg 7.9%

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

      10. neg-mul-1 7.9%

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

      11. *-commutative 7.9%

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

      12. associate-/l* 9.8%

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

      13. *-commutative 9.8%

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

      14. associate-/r* 9.8%

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

      15. metadata-eval 9.8%

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

      16. sin-neg 9.8%

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

      17. distribute-frac-neg 9.8%

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

   3. Simplified 9.1%

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

      1. add-cbrt-cube 2.1%

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

      2. pow3 2.1%

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

   6. Applied egg-rr 2.1%

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

   7. Taylor expanded in x around 0 11.2%

      \[\leadsto \color{blue}{-2 \cdot \sqrt[3]{-0.125}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}\right)} \]

      2. sqrt-unprod 12.5%

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}} \]

      3. cbrt-unprod 12.5%

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}} \]

      4. metadata-eval 12.5%

         \[\leadsto -2 \cdot \sqrt{\sqrt[3]{\color{blue}{0.015625}}} \]

   9. Applied egg-rr 12.5%

      \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{0.015625}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 57.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{{\left(\sqrt[3]{x\_m \cdot 0.5}\right)}^{3}}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\sqrt[3]{0.015625}}\\ \end{array} \end{array} \]

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)) 4e+133)
   (/ 1.0 (cos (/ (pow (cbrt (* x_m 0.5)) 3.0) y_m)))
   (* -2.0 (sqrt (cbrt 0.015625)))))

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)) <= 4e+133) {
		tmp = 1.0 / cos((pow(cbrt((x_m * 0.5)), 3.0) / y_m));
	} else {
		tmp = -2.0 * sqrt(cbrt(0.015625));
	}
	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)) <= 4e+133) {
		tmp = 1.0 / Math.cos((Math.pow(Math.cbrt((x_m * 0.5)), 3.0) / y_m));
	} else {
		tmp = -2.0 * Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 4e+133)
		tmp = Float64(1.0 / cos(Float64((cbrt(Float64(x_m * 0.5)) ^ 3.0) / y_m)));
	else
		tmp = Float64(-2.0 * sqrt(cbrt(0.015625)));
	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], 4e+133], N[(1.0 / N[Cos[N[(N[Power[N[Power[N[(x$95$m * 0.5), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.0000000000000001e133

   1. Initial program 51.1%

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

      1. remove-double-neg 51.1%

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

      2. distribute-frac-neg 51.1%

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

      3. tan-neg 51.1%

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

      4. distribute-frac-neg2 51.1%

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

      5. distribute-lft-neg-out 51.1%

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

      6. distribute-frac-neg2 51.1%

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

      7. distribute-lft-neg-out 51.1%

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

      8. distribute-frac-neg2 51.1%

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

      9. distribute-frac-neg 51.1%

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

      10. neg-mul-1 51.1%

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

      11. *-commutative 51.1%

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

      12. associate-/l* 50.9%

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

      13. *-commutative 50.9%

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

      14. associate-/r* 50.9%

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

      15. metadata-eval 50.9%

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

      16. sin-neg 50.9%

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

      17. distribute-frac-neg 50.9%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in x around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r/ 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in x around inf 66.9%

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

      1. metadata-eval 66.9%

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

      2. times-frac 66.9%

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

      3. neg-mul-1 66.9%

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

      4. *-commutative 66.9%

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

      5. cos-neg 66.9%

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

      6. neg-mul-1 66.9%

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

      7. *-commutative 66.9%

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

      8. times-frac 66.9%

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

      9. metadata-eval 66.9%

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

      10. distribute-rgt-neg-in 66.9%

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

      11. metadata-eval 66.9%

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

      12. associate-*l/ 66.9%

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

   10. Simplified 66.9%

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

       1. add-cube-cbrt 66.8%

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

       2. pow3 66.9%

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

   12. Applied egg-rr 66.9%

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

   if 4.0000000000000001e133 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.9%

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

      1. remove-double-neg 7.9%

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

      2. distribute-frac-neg 7.9%

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

      3. tan-neg 7.9%

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

      4. distribute-frac-neg2 7.9%

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

      5. distribute-lft-neg-out 7.9%

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

      6. distribute-frac-neg2 7.9%

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

      7. distribute-lft-neg-out 7.9%

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

      8. distribute-frac-neg2 7.9%

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

      9. distribute-frac-neg 7.9%

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

      10. neg-mul-1 7.9%

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

      11. *-commutative 7.9%

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

      12. associate-/l* 9.8%

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

      13. *-commutative 9.8%

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

      14. associate-/r* 9.8%

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

      15. metadata-eval 9.8%

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

      16. sin-neg 9.8%

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

      17. distribute-frac-neg 9.8%

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

   3. Simplified 9.1%

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

      1. add-cbrt-cube 2.1%

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

      2. pow3 2.1%

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

   6. Applied egg-rr 2.1%

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

   7. Taylor expanded in x around 0 11.2%

      \[\leadsto \color{blue}{-2 \cdot \sqrt[3]{-0.125}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}\right)} \]

      2. sqrt-unprod 12.5%

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}} \]

      3. cbrt-unprod 12.5%

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}} \]

      4. metadata-eval 12.5%

         \[\leadsto -2 \cdot \sqrt{\sqrt[3]{\color{blue}{0.015625}}} \]

   9. Applied egg-rr 12.5%

      \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{0.015625}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{-0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\sqrt[3]{0.015625}}\\ \end{array} \end{array} \]

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)) 4e+104)
   (/ 1.0 (cos (* x_m (/ -0.5 y_m))))
   (* -2.0 (sqrt (cbrt 0.015625)))))

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)) <= 4e+104) {
		tmp = 1.0 / cos((x_m * (-0.5 / y_m)));
	} else {
		tmp = -2.0 * sqrt(cbrt(0.015625));
	}
	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)) <= 4e+104) {
		tmp = 1.0 / Math.cos((x_m * (-0.5 / y_m)));
	} else {
		tmp = -2.0 * Math.sqrt(Math.cbrt(0.015625));
	}
	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)) <= 4e+104)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(-0.5 / y_m))));
	else
		tmp = Float64(-2.0 * sqrt(cbrt(0.015625)));
	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], 4e+104], N[(1.0 / N[Cos[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Sqrt[N[Power[0.015625, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4e104

   1. Initial program 51.8%

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

      1. remove-double-neg 51.8%

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

      2. distribute-frac-neg 51.8%

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

      3. tan-neg 51.8%

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

      4. distribute-frac-neg2 51.8%

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

      5. distribute-lft-neg-out 51.8%

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

      6. distribute-frac-neg2 51.8%

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

      7. distribute-lft-neg-out 51.8%

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

      8. distribute-frac-neg2 51.8%

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

      9. distribute-frac-neg 51.8%

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

      10. neg-mul-1 51.8%

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

      11. *-commutative 51.8%

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

      12. associate-/l* 51.6%

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

      13. *-commutative 51.6%

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

      14. associate-/r* 51.6%

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

      15. metadata-eval 51.6%

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

      16. sin-neg 51.6%

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

      17. distribute-frac-neg 51.6%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in x around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r/ 67.9%

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

   7. Simplified 67.9%

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

   if 4e104 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.5%

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

      1. remove-double-neg 8.5%

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

      2. distribute-frac-neg 8.5%

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

      3. tan-neg 8.5%

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

      4. distribute-frac-neg2 8.5%

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

      5. distribute-lft-neg-out 8.5%

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

      6. distribute-frac-neg2 8.5%

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

      7. distribute-lft-neg-out 8.5%

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

      8. distribute-frac-neg2 8.5%

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

      9. distribute-frac-neg 8.5%

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

      10. neg-mul-1 8.5%

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

      11. *-commutative 8.5%

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

      12. associate-/l* 10.0%

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

      13. *-commutative 10.0%

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

      14. associate-/r* 10.0%

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

      15. metadata-eval 10.0%

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

      16. sin-neg 10.0%

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

      17. distribute-frac-neg 10.0%

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

   3. Simplified 9.4%

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

      1. add-cbrt-cube 2.8%

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

      2. pow3 2.8%

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

   6. Applied egg-rr 2.8%

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

   7. Taylor expanded in x around 0 11.7%

      \[\leadsto \color{blue}{-2 \cdot \sqrt[3]{-0.125}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{-0.125}} \cdot \sqrt{\sqrt[3]{-0.125}}\right)} \]

      2. sqrt-unprod 12.5%

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{-0.125} \cdot \sqrt[3]{-0.125}}} \]

      3. cbrt-unprod 12.5%

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\sqrt[3]{-0.125 \cdot -0.125}}} \]

      4. metadata-eval 12.5%

         \[\leadsto -2 \cdot \sqrt{\sqrt[3]{\color{blue}{0.015625}}} \]

   9. Applied egg-rr 12.5%

      \[\leadsto -2 \cdot \color{blue}{\sqrt{\sqrt[3]{0.015625}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 55.3% accurate, 1.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (/ 1.0 (cos (/ 1.0 (/ 1.0 (* 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((1.0 / (1.0 / (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((1.0d0 / (1.0d0 / (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((1.0 / (1.0 / (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((1.0 / (1.0 / (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(1.0 / Float64(1.0 / 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((1.0 / (1.0 / (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[(1.0 / N[(1.0 / N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.2%

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

   1. remove-double-neg 45.2%

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

   2. distribute-frac-neg 45.2%

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

   3. tan-neg 45.2%

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

   4. distribute-frac-neg2 45.2%

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

   5. distribute-lft-neg-out 45.2%

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

   6. distribute-frac-neg2 45.2%

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

   7. distribute-lft-neg-out 45.2%

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

   8. distribute-frac-neg2 45.2%

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

   9. distribute-frac-neg 45.2%

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

   10. neg-mul-1 45.2%

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

   11. *-commutative 45.2%

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

   12. associate-/l* 45.3%

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

   13. *-commutative 45.3%

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

   14. associate-/r* 45.3%

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

   15. metadata-eval 45.3%

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

   16. sin-neg 45.3%

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

   17. distribute-frac-neg 45.3%

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

3. Simplified 45.4%

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

5. Taylor expanded in x around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

   3. associate-*r/ 59.0%

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

7. Simplified 59.0%

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

   1. associate-*r/ 58.8%

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

   2. clear-num 58.7%

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

9. Applied egg-rr 58.7%

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

    1. clear-num 58.9%

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

    2. associate-*r/ 59.0%

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

    3. inv-pow 59.0%

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

    4. rem-cbrt-cube 49.3%

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

    5. rem-cbrt-cube 59.0%

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

    6. add-sqr-sqrt 31.5%

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

    7. sqrt-unprod 54.8%

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

    8. frac-times 55.0%

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

    9. metadata-eval 55.0%

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

    10. unpow2 55.0%

        \[\leadsto \frac{1}{\cos \left(\frac{1}{{\left(x \cdot \sqrt{\frac{0.25}{\color{blue}{{y}^{2}}}}\right)}^{-1}}\right)} \]

    11. sqrt-div 54.8%

        \[\leadsto \frac{1}{\cos \left(\frac{1}{{\left(x \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{y}^{2}}}}\right)}^{-1}}\right)} \]

    12. metadata-eval 54.8%

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

    13. sqrt-pow1 59.0%

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

    14. metadata-eval 59.0%

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

    15. pow1 59.0%

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

    16. associate-/l* 58.9%

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

    17. *-un-lft-identity 58.9%

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

    18. *-commutative 58.9%

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

    19. times-frac 58.9%

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

    20. metadata-eval 58.9%

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

11. Applied egg-rr 58.9%

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

    1. unpow-1 58.9%

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

    2. associate-*r/ 58.9%

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

    3. associate-*l/ 59.0%

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

    4. *-commutative 59.0%

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

13. Simplified 59.0%

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

Alternative 6: 55.3% 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 45.2%

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

   1. remove-double-neg 45.2%

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

   2. distribute-frac-neg 45.2%

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

   3. tan-neg 45.2%

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

   4. distribute-frac-neg2 45.2%

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

   5. distribute-lft-neg-out 45.2%

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

   6. distribute-frac-neg2 45.2%

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

   7. distribute-lft-neg-out 45.2%

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

   8. distribute-frac-neg2 45.2%

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

   9. distribute-frac-neg 45.2%

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

   10. neg-mul-1 45.2%

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

   11. *-commutative 45.2%

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

   12. associate-/l* 45.3%

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

   13. *-commutative 45.3%

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

   14. associate-/r* 45.3%

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

   15. metadata-eval 45.3%

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

   16. sin-neg 45.3%

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

   17. distribute-frac-neg 45.3%

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

3. Simplified 45.4%

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

5. Taylor expanded in x around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

   3. associate-*r/ 59.0%

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

7. Simplified 59.0%

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

Alternative 7: 55.4% 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 45.2%

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

3. Taylor expanded in x around inf 58.8%

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

Alternative 8: 55.3% 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 45.2%

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

   1. remove-double-neg 45.2%

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

   2. distribute-frac-neg 45.2%

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

   3. tan-neg 45.2%

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

   4. distribute-frac-neg2 45.2%

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

   5. distribute-lft-neg-out 45.2%

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

   6. distribute-frac-neg2 45.2%

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

   7. distribute-lft-neg-out 45.2%

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

   8. distribute-frac-neg2 45.2%

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

   9. distribute-frac-neg 45.2%

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

   10. neg-mul-1 45.2%

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

   11. *-commutative 45.2%

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

   12. associate-/l* 45.3%

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

   13. *-commutative 45.3%

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

   14. associate-/r* 45.3%

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

   15. metadata-eval 45.3%

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

   16. sin-neg 45.3%

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

   17. distribute-frac-neg 45.3%

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

3. Simplified 45.4%

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

5. Taylor expanded in x around 0 58.4%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 55.3% 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 2024143 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))