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

Percentage Accurate: 44.1% → 56.6%

Time: 16.9s

Alternatives: 11

Speedup: 211.0×

• Could not converge on a ground truth

  1. reg0 = (bf "8.701136086126978290340077487065713999283e-232")
  2. reg1 = (bf "3.106012047232445361347965104329176525025e270")

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: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{\frac{-0.5}{y\_m}} \cdot {\left(\sqrt[3]{\sqrt[3]{x\_m}}\right)}^{3}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{-2 \cdot \frac{y\_m}{y\_m}}\right)}^{3}\\ \end{array} \end{array} \]

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)) 2e+291)
   (/ 1.0 (cos (pow (* (cbrt (/ -0.5 y_m)) (pow (cbrt (cbrt x_m)) 3.0)) 3.0)))
   (* 0.5 (pow (cbrt (* -2.0 (/ y_m y_m))) 3.0))))

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)) <= 2e+291) {
		tmp = 1.0 / cos(pow((cbrt((-0.5 / y_m)) * pow(cbrt(cbrt(x_m)), 3.0)), 3.0));
	} else {
		tmp = 0.5 * pow(cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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)) <= 2e+291) {
		tmp = 1.0 / Math.cos(Math.pow((Math.cbrt((-0.5 / y_m)) * Math.pow(Math.cbrt(Math.cbrt(x_m)), 3.0)), 3.0));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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)) <= 2e+291)
		tmp = Float64(1.0 / cos((Float64(cbrt(Float64(-0.5 / y_m)) * (cbrt(cbrt(x_m)) ^ 3.0)) ^ 3.0)));
	else
		tmp = Float64(0.5 * (cbrt(Float64(-2.0 * Float64(y_m / y_m))) ^ 3.0));
	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], 2e+291], N[(1.0 / N[Cos[N[Power[N[(N[Power[N[(-0.5 / y$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[x$95$m, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[N[(-2.0 * N[(y$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e291

   1. Initial program 49.3%

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

      1. remove-double-neg 49.3%

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

      2. distribute-frac-neg 49.3%

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

      3. tan-neg 49.3%

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

      4. distribute-frac-neg2 49.3%

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

      5. distribute-lft-neg-out 49.3%

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

      6. distribute-frac-neg2 49.3%

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

      7. distribute-lft-neg-out 49.3%

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

      8. distribute-frac-neg2 49.3%

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

      9. distribute-frac-neg 49.3%

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

      10. neg-mul-1 49.3%

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

      11. *-commutative 49.3%

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

      12. associate-/l* 49.3%

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

      13. *-commutative 49.3%

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

      14. associate-/r* 49.3%

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

      15. metadata-eval 49.3%

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

      16. sin-neg 49.3%

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

      17. distribute-frac-neg 49.3%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*r/ 59.8%

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

   7. Simplified 59.8%

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

      1. add-cube-cbrt 59.9%

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

      2. pow3 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. *-commutative 59.7%

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

       2. cbrt-prod 59.9%

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

   11. Applied egg-rr 59.9%

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

       1. add-cube-cbrt 60.0%

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

       2. pow3 60.4%

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

   13. Applied egg-rr 60.4%

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

   if 1.9999999999999999e291 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 1.4%

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

      1. add-exp-log 0.1%

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

   4. Applied egg-rr 0.1%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{e^{\log \left(y \cdot 2\right)}}}\right)} \]
   5. Step-by-step derivation

      1. rem-exp-log 1.4%

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

      2. metadata-eval 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

      3. div-inv 1.4%

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

      4. clear-num 1.4%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{\frac{1}{\frac{0.5}{y}}}}\right)} \]

      5. add-sqr-sqrt 0.1%

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

      6. sqrt-unprod 1.5%

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

      7. frac-times 1.5%

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

      8. metadata-eval 1.5%

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

      9. metadata-eval 1.5%

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

      10. frac-times 1.5%

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

      11. sqrt-unprod 1.5%

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

      12. add-sqr-sqrt 1.5%

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

      13. clear-num 1.5%

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

      14. div-inv 1.5%

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

      15. metadata-eval 1.5%

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

      16. metadata-eval 1.5%

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

      17. distribute-rgt-neg-in 1.5%

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

      18. rem-exp-log 1.1%

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

      19. add-log-exp 0.0%

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

      20. neg-log 0.0%

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

      21. rem-exp-log 0.0%

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

      22. *-commutative 0.0%

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

      23. exp-prod 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 3.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\log \left(\frac{1}{e^{2 \cdot y}}\right)}{y}} \]
   8. Step-by-step derivation

      1. log-rec 3.1%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-\log \left(e^{2 \cdot y}\right)}}{y} \]

      2. *-commutative 3.1%

         \[\leadsto 0.5 \cdot \frac{-\log \left(e^{\color{blue}{y \cdot 2}}\right)}{y} \]

      3. rem-log-exp 10.6%

         \[\leadsto 0.5 \cdot \frac{-\color{blue}{y \cdot 2}}{y} \]

      4. distribute-rgt-neg-in 10.6%

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

      5. metadata-eval 10.6%

         \[\leadsto 0.5 \cdot \frac{y \cdot \color{blue}{-2}}{y} \]

   9. Simplified 10.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot -2}{y}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 10.6%

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

       2. pow3 10.6%

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

       3. *-commutative 10.6%

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

       4. *-un-lft-identity 10.6%

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

       5. times-frac 10.6%

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

       6. metadata-eval 10.6%

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

   11. Applied egg-rr 10.6%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{-2 \cdot \frac{y}{y}}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 56.6% accurate, 0.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.0)
     (/ 1.0 (cos (pow (* (cbrt (/ x_m y_m)) (cbrt -0.5)) 3.0)))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 2.0) {
		tmp = 1.0 / cos(pow((cbrt((x_m / y_m)) * cbrt(-0.5)), 3.0));
	} 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 t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 2.0) {
		tmp = 1.0 / Math.cos(Math.pow((Math.cbrt((x_m / y_m)) * Math.cbrt(-0.5)), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 2.0)
		tmp = Float64(1.0 / cos((Float64(cbrt(Float64(x_m / y_m)) * cbrt(-0.5)) ^ 3.0)));
	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_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[Cos[N[Power[N[(N[Power[N[(x$95$m / y$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2

   1. Initial program 63.8%

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

      1. remove-double-neg 63.8%

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

      2. distribute-frac-neg 63.8%

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

      3. tan-neg 63.8%

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

      4. distribute-frac-neg2 63.8%

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

      5. distribute-lft-neg-out 63.8%

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

      6. distribute-frac-neg2 63.8%

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

      7. distribute-lft-neg-out 63.8%

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

      8. distribute-frac-neg2 63.8%

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

      9. distribute-frac-neg 63.8%

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

      10. neg-mul-1 63.8%

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

      11. *-commutative 63.8%

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

      12. associate-/l* 63.8%

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

      13. *-commutative 63.8%

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

      14. associate-/r* 63.8%

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

      15. metadata-eval 63.8%

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

      16. sin-neg 63.8%

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

      17. distribute-frac-neg 63.8%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in x around inf 63.8%

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

      1. associate-*r/ 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r/ 64.2%

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

   7. Simplified 64.2%

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

      1. add-cube-cbrt 64.5%

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

      2. pow3 64.4%

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

   9. Applied egg-rr 64.4%

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

   10. Taylor expanded in x around 0 64.9%

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

   if 2 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 2.8%

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

      1. remove-double-neg 2.8%

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

      2. distribute-frac-neg 2.8%

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

      3. tan-neg 2.8%

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

      4. distribute-frac-neg2 2.8%

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

      5. distribute-lft-neg-out 2.8%

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

      6. distribute-frac-neg2 2.8%

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

      7. distribute-lft-neg-out 2.8%

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

      8. distribute-frac-neg2 2.8%

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

      9. distribute-frac-neg 2.8%

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

      10. neg-mul-1 2.8%

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

      11. *-commutative 2.8%

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

      12. associate-/l* 2.9%

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

      13. *-commutative 2.9%

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

      14. associate-/r* 2.9%

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

      15. metadata-eval 2.9%

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

      16. sin-neg 2.9%

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

      17. distribute-frac-neg 2.9%

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

   3. Simplified 2.9%

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

   5. Taylor expanded in x around 0 40.7%

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

Alternative 3: 56.6% accurate, 0.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.0)
     (/ 1.0 (cos (pow (* (cbrt (/ -0.5 y_m)) (cbrt x_m)) 3.0)))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 2.0) {
		tmp = 1.0 / cos(pow((cbrt((-0.5 / y_m)) * cbrt(x_m)), 3.0));
	} 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 t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 2.0) {
		tmp = 1.0 / Math.cos(Math.pow((Math.cbrt((-0.5 / y_m)) * Math.cbrt(x_m)), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 2.0)
		tmp = Float64(1.0 / cos((Float64(cbrt(Float64(-0.5 / y_m)) * cbrt(x_m)) ^ 3.0)));
	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_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[Cos[N[Power[N[(N[Power[N[(-0.5 / y$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[x$95$m, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2

   1. Initial program 63.8%

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

      1. remove-double-neg 63.8%

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

      2. distribute-frac-neg 63.8%

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

      3. tan-neg 63.8%

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

      4. distribute-frac-neg2 63.8%

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

      5. distribute-lft-neg-out 63.8%

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

      6. distribute-frac-neg2 63.8%

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

      7. distribute-lft-neg-out 63.8%

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

      8. distribute-frac-neg2 63.8%

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

      9. distribute-frac-neg 63.8%

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

      10. neg-mul-1 63.8%

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

      11. *-commutative 63.8%

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

      12. associate-/l* 63.8%

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

      13. *-commutative 63.8%

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

      14. associate-/r* 63.8%

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

      15. metadata-eval 63.8%

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

      16. sin-neg 63.8%

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

      17. distribute-frac-neg 63.8%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in x around inf 63.8%

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

      1. associate-*r/ 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r/ 64.2%

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

   7. Simplified 64.2%

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

      1. add-cube-cbrt 64.5%

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

      2. pow3 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. *-commutative 64.4%

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

       2. cbrt-prod 64.9%

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

   11. Applied egg-rr 64.9%

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

   if 2 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 2.8%

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

      1. remove-double-neg 2.8%

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

      2. distribute-frac-neg 2.8%

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

      3. tan-neg 2.8%

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

      4. distribute-frac-neg2 2.8%

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

      5. distribute-lft-neg-out 2.8%

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

      6. distribute-frac-neg2 2.8%

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

      7. distribute-lft-neg-out 2.8%

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

      8. distribute-frac-neg2 2.8%

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

      9. distribute-frac-neg 2.8%

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

      10. neg-mul-1 2.8%

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

      11. *-commutative 2.8%

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

      12. associate-/l* 2.9%

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

      13. *-commutative 2.9%

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

      14. associate-/r* 2.9%

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

      15. metadata-eval 2.9%

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

      16. sin-neg 2.9%

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

      17. distribute-frac-neg 2.9%

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

   3. Simplified 2.9%

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

   5. Taylor expanded in x around 0 40.7%

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

Alternative 4: 56.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+80}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{\frac{1}{\frac{y\_m}{x\_m \cdot -0.5}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{-2 \cdot \frac{y\_m}{y\_m}}\right)}^{3}\\ \end{array} \end{array} \]

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+80)
   (/ 1.0 (cos (pow (cbrt (/ 1.0 (/ y_m (* x_m -0.5)))) 3.0)))
   (* 0.5 (pow (cbrt (* -2.0 (/ y_m y_m))) 3.0))))

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+80) {
		tmp = 1.0 / cos(pow(cbrt((1.0 / (y_m / (x_m * -0.5)))), 3.0));
	} else {
		tmp = 0.5 * pow(cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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+80) {
		tmp = 1.0 / Math.cos(Math.pow(Math.cbrt((1.0 / (y_m / (x_m * -0.5)))), 3.0));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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+80)
		tmp = Float64(1.0 / cos((cbrt(Float64(1.0 / Float64(y_m / Float64(x_m * -0.5)))) ^ 3.0)));
	else
		tmp = Float64(0.5 * (cbrt(Float64(-2.0 * Float64(y_m / y_m))) ^ 3.0));
	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+80], N[(1.0 / N[Cos[N[Power[N[Power[N[(1.0 / N[(y$95$m / N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[N[(-2.0 * N[(y$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e80

   1. Initial program 55.6%

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

      1. remove-double-neg 55.6%

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

      2. distribute-frac-neg 55.6%

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

      3. tan-neg 55.6%

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

      4. distribute-frac-neg2 55.6%

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

      5. distribute-lft-neg-out 55.6%

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

      6. distribute-frac-neg2 55.6%

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

      7. distribute-lft-neg-out 55.6%

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

      8. distribute-frac-neg2 55.6%

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

      9. distribute-frac-neg 55.6%

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

      10. neg-mul-1 55.6%

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

      11. *-commutative 55.6%

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

      12. associate-/l* 55.6%

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

      13. *-commutative 55.6%

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

      14. associate-/r* 55.6%

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

      15. metadata-eval 55.6%

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

      16. sin-neg 55.6%

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

      17. distribute-frac-neg 55.6%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in x around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. *-commutative 67.3%

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

      3. associate-*r/ 67.6%

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

   7. Simplified 67.6%

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

      1. add-cube-cbrt 67.9%

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

      2. pow3 67.9%

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

   9. Applied egg-rr 67.9%

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

       1. associate-*r/ 67.7%

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

       2. clear-num 68.1%

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

   11. Applied egg-rr 68.1%

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

   if 1e80 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.1%

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

      1. add-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{e^{\log \left(y \cdot 2\right)}}}\right)} \]
   5. Step-by-step derivation

      1. rem-exp-log 7.1%

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

      2. metadata-eval 7.1%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

      3. div-inv 7.1%

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

      4. clear-num 7.8%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{\frac{1}{\frac{0.5}{y}}}}\right)} \]

      5. add-sqr-sqrt 3.5%

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

      6. sqrt-unprod 2.5%

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

      7. frac-times 3.4%

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

      8. metadata-eval 3.4%

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

      9. metadata-eval 3.4%

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

      10. frac-times 2.5%

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

      11. sqrt-unprod 2.0%

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

      12. add-sqr-sqrt 6.6%

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

      13. clear-num 7.8%

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

      14. div-inv 7.8%

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

      15. metadata-eval 7.8%

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

      16. metadata-eval 7.8%

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

      17. distribute-rgt-neg-in 7.8%

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

      18. rem-exp-log 2.7%

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

      19. add-log-exp 0.4%

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

      20. neg-log 0.4%

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

      21. rem-exp-log 0.9%

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

      22. *-commutative 0.9%

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

      23. exp-prod 0.9%

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

   6. Applied egg-rr 0.9%

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

   7. Taylor expanded in x around 0 3.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\log \left(\frac{1}{e^{2 \cdot y}}\right)}{y}} \]
   8. Step-by-step derivation

      1. log-rec 3.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-\log \left(e^{2 \cdot y}\right)}}{y} \]

      2. *-commutative 3.9%

         \[\leadsto 0.5 \cdot \frac{-\log \left(e^{\color{blue}{y \cdot 2}}\right)}{y} \]

      3. rem-log-exp 10.3%

         \[\leadsto 0.5 \cdot \frac{-\color{blue}{y \cdot 2}}{y} \]

      4. distribute-rgt-neg-in 10.3%

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

      5. metadata-eval 10.3%

         \[\leadsto 0.5 \cdot \frac{y \cdot \color{blue}{-2}}{y} \]

   9. Simplified 10.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot -2}{y}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 10.3%

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

       2. pow3 10.3%

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

       3. *-commutative 10.3%

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

       4. *-un-lft-identity 10.3%

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

       5. times-frac 10.3%

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

       6. metadata-eval 10.3%

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

   11. Applied egg-rr 10.3%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{-2 \cdot \frac{y}{y}}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\cos \left({\left(\sqrt[3]{x\_m \cdot \frac{-0.5}{y\_m}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{-2 \cdot \frac{y\_m}{y\_m}}\right)}^{3}\\ \end{array} \end{array} \]

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+88)
   (/ 1.0 (cos (pow (cbrt (* x_m (/ -0.5 y_m))) 3.0)))
   (* 0.5 (pow (cbrt (* -2.0 (/ y_m y_m))) 3.0))))

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)) <= 4e+88) {
		tmp = 1.0 / cos(pow(cbrt((x_m * (-0.5 / y_m))), 3.0));
	} else {
		tmp = 0.5 * pow(cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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)) <= 4e+88) {
		tmp = 1.0 / Math.cos(Math.pow(Math.cbrt((x_m * (-0.5 / y_m))), 3.0));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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)) <= 4e+88)
		tmp = Float64(1.0 / cos((cbrt(Float64(x_m * Float64(-0.5 / y_m))) ^ 3.0)));
	else
		tmp = Float64(0.5 * (cbrt(Float64(-2.0 * Float64(y_m / y_m))) ^ 3.0));
	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], 4e+88], N[(1.0 / N[Cos[N[Power[N[Power[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[N[(-2.0 * N[(y$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 3.99999999999999984e88

   1. Initial program 55.4%

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

      1. remove-double-neg 55.4%

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

      2. distribute-frac-neg 55.4%

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

      3. tan-neg 55.4%

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

      4. distribute-frac-neg2 55.4%

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

      5. distribute-lft-neg-out 55.4%

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

      6. distribute-frac-neg2 55.4%

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

      7. distribute-lft-neg-out 55.4%

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

      8. distribute-frac-neg2 55.4%

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

      9. distribute-frac-neg 55.4%

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

      10. neg-mul-1 55.4%

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

      11. *-commutative 55.4%

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

      12. associate-/l* 55.3%

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

      13. *-commutative 55.3%

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

      14. associate-/r* 55.3%

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

      15. metadata-eval 55.3%

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

      16. sin-neg 55.3%

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

      17. distribute-frac-neg 55.3%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. associate-*r/ 67.4%

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

   7. Simplified 67.4%

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

      1. add-cube-cbrt 67.6%

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

      2. pow3 67.6%

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

   9. Applied egg-rr 67.6%

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

   if 3.99999999999999984e88 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.8%

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

      1. add-exp-log 4.0%

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

   4. Applied egg-rr 4.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{e^{\log \left(y \cdot 2\right)}}}\right)} \]
   5. Step-by-step derivation

      1. rem-exp-log 6.8%

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

      2. metadata-eval 6.8%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

      3. div-inv 6.8%

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

      4. clear-num 7.5%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{\frac{1}{\frac{0.5}{y}}}}\right)} \]

      5. add-sqr-sqrt 3.6%

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

      6. sqrt-unprod 2.5%

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

      7. frac-times 3.5%

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

      8. metadata-eval 3.5%

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

      9. metadata-eval 3.5%

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

      10. frac-times 2.5%

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

      11. sqrt-unprod 2.0%

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

      12. add-sqr-sqrt 6.7%

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

      13. clear-num 7.9%

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

      14. div-inv 7.9%

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

      15. metadata-eval 7.9%

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

      16. metadata-eval 7.9%

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

      17. distribute-rgt-neg-in 7.9%

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

      18. rem-exp-log 2.8%

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

      19. add-log-exp 0.4%

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

      20. neg-log 0.4%

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

      21. rem-exp-log 0.9%

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

      22. *-commutative 0.9%

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

      23. exp-prod 0.9%

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

   6. Applied egg-rr 0.9%

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

   7. Taylor expanded in x around 0 3.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\log \left(\frac{1}{e^{2 \cdot y}}\right)}{y}} \]
   8. Step-by-step derivation

      1. log-rec 3.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-\log \left(e^{2 \cdot y}\right)}}{y} \]

      2. *-commutative 3.7%

         \[\leadsto 0.5 \cdot \frac{-\log \left(e^{\color{blue}{y \cdot 2}}\right)}{y} \]

      3. rem-log-exp 10.2%

         \[\leadsto 0.5 \cdot \frac{-\color{blue}{y \cdot 2}}{y} \]

      4. distribute-rgt-neg-in 10.2%

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

      5. metadata-eval 10.2%

         \[\leadsto 0.5 \cdot \frac{y \cdot \color{blue}{-2}}{y} \]

   9. Simplified 10.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot -2}{y}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 10.2%

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

       2. pow3 10.2%

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

       3. *-commutative 10.2%

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

       4. *-un-lft-identity 10.2%

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

       5. times-frac 10.2%

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

       6. metadata-eval 10.2%

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

   11. Applied egg-rr 10.2%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{-2 \cdot \frac{y}{y}}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+75}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y\_m \cdot \frac{2}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{-2 \cdot \frac{y\_m}{y\_m}}\right)}^{3}\\ \end{array} \end{array} \]

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+75)
   (/ 1.0 (cos (/ 1.0 (* y_m (/ 2.0 x_m)))))
   (* 0.5 (pow (cbrt (* -2.0 (/ y_m y_m))) 3.0))))

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+75) {
		tmp = 1.0 / cos((1.0 / (y_m * (2.0 / x_m))));
	} else {
		tmp = 0.5 * pow(cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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+75) {
		tmp = 1.0 / Math.cos((1.0 / (y_m * (2.0 / x_m))));
	} else {
		tmp = 0.5 * Math.pow(Math.cbrt((-2.0 * (y_m / y_m))), 3.0);
	}
	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+75)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(y_m * Float64(2.0 / x_m)))));
	else
		tmp = Float64(0.5 * (cbrt(Float64(-2.0 * Float64(y_m / y_m))) ^ 3.0));
	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+75], N[(1.0 / N[Cos[N[(1.0 / N[(y$95$m * N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[N[(-2.0 * N[(y$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 9.99999999999999927e74

   1. Initial program 55.9%

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

      1. remove-double-neg 55.9%

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

      2. distribute-frac-neg 55.9%

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

      3. tan-neg 55.9%

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

      4. distribute-frac-neg2 55.9%

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

      5. distribute-lft-neg-out 55.9%

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

      6. distribute-frac-neg2 55.9%

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

      7. distribute-lft-neg-out 55.9%

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

      8. distribute-frac-neg2 55.9%

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

      9. distribute-frac-neg 55.9%

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

      10. neg-mul-1 55.9%

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

      11. *-commutative 55.9%

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

      12. associate-/l* 55.9%

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

      13. *-commutative 55.9%

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

      14. associate-/r* 55.9%

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

      15. metadata-eval 55.9%

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

      16. sin-neg 55.9%

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

      17. distribute-frac-neg 55.9%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in x around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*r/ 67.9%

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

   7. Simplified 67.9%

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

      1. add-cube-cbrt 68.1%

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

      2. pow3 68.1%

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

   9. Applied egg-rr 68.1%

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

       1. rem-cube-cbrt 67.9%

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

       2. clear-num 67.9%

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

       3. un-div-inv 67.6%

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

       4. div-inv 67.6%

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

       5. metadata-eval 67.6%

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

       6. metadata-eval 67.6%

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

       7. distribute-rgt-neg-in 67.6%

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

       8. rem-log-exp 67.6%

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

       9. log-pow 46.8%

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

       10. log-rec 46.8%

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

       11. add-sqr-sqrt 29.1%

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

       12. sqrt-unprod 46.8%

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

       13. log-rec 46.9%

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

       14. log-pow 46.8%

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

       15. rem-log-exp 46.8%

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

       16. log-rec 46.7%

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

       17. log-pow 61.5%

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

       18. rem-log-exp 61.5%

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

       19. sqr-neg 61.5%

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

       20. sqrt-unprod 27.0%

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

       21. add-sqr-sqrt 67.6%

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

   11. Applied egg-rr 67.9%

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

   if 9.99999999999999927e74 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.0%

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

      1. add-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{e^{\log \left(y \cdot 2\right)}}}\right)} \]
   5. Step-by-step derivation

      1. rem-exp-log 7.0%

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

      2. metadata-eval 7.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

      3. div-inv 7.0%

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

      4. clear-num 7.7%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{\frac{1}{\frac{0.5}{y}}}}\right)} \]

      5. add-sqr-sqrt 3.5%

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

      6. sqrt-unprod 2.5%

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

      7. frac-times 3.4%

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

      8. metadata-eval 3.4%

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

      9. metadata-eval 3.4%

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

      10. frac-times 2.5%

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

      11. sqrt-unprod 1.9%

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

      12. add-sqr-sqrt 7.0%

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

      13. clear-num 8.1%

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

      14. div-inv 8.1%

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

      15. metadata-eval 8.1%

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

      16. metadata-eval 8.1%

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

      17. distribute-rgt-neg-in 8.1%

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

      18. rem-exp-log 2.9%

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

      19. add-log-exp 0.4%

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

      20. neg-log 0.4%

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

      21. rem-exp-log 0.9%

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

      22. *-commutative 0.9%

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

      23. exp-prod 0.9%

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

   6. Applied egg-rr 0.9%

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

   7. Taylor expanded in x around 0 3.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\log \left(\frac{1}{e^{2 \cdot y}}\right)}{y}} \]
   8. Step-by-step derivation

      1. log-rec 3.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-\log \left(e^{2 \cdot y}\right)}}{y} \]

      2. *-commutative 3.9%

         \[\leadsto 0.5 \cdot \frac{-\log \left(e^{\color{blue}{y \cdot 2}}\right)}{y} \]

      3. rem-log-exp 10.6%

         \[\leadsto 0.5 \cdot \frac{-\color{blue}{y \cdot 2}}{y} \]

      4. distribute-rgt-neg-in 10.6%

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

      5. metadata-eval 10.6%

         \[\leadsto 0.5 \cdot \frac{y \cdot \color{blue}{-2}}{y} \]

   9. Simplified 10.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot -2}{y}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 10.6%

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

       2. pow3 10.6%

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

       3. *-commutative 10.6%

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

       4. *-un-lft-identity 10.6%

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

       5. times-frac 10.6%

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

       6. metadata-eval 10.6%

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

   11. Applied egg-rr 10.6%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{-2 \cdot \frac{y}{y}}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return 1.0 + ((1.0 / cos((x_m * (-0.5 / y_m)))) + -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 + ((1.0d0 / cos((x_m * ((-0.5d0) / y_m)))) + (-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 + ((1.0 / Math.cos((x_m * (-0.5 / y_m)))) + -1.0);
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = 1.0 + ((1.0 / cos((x_m * (-0.5 / y_m)))) + -1.0);
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[(N[(1.0 / N[Cos[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. remove-double-neg 45.9%

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

   2. distribute-frac-neg 45.9%

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

   3. tan-neg 45.9%

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

   4. distribute-frac-neg2 45.9%

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

   5. distribute-lft-neg-out 45.9%

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

   6. distribute-frac-neg2 45.9%

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

   7. distribute-lft-neg-out 45.9%

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

   8. distribute-frac-neg2 45.9%

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

   9. distribute-frac-neg 45.9%

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

   10. neg-mul-1 45.9%

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

   11. *-commutative 45.9%

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

   12. associate-/l* 45.9%

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

   13. *-commutative 45.9%

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

   14. associate-/r* 45.9%

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

   15. metadata-eval 45.9%

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

   16. sin-neg 45.9%

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

   17. distribute-frac-neg 45.9%

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

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 55.3%

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

   1. associate-*r/ 55.3%

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

   2. *-commutative 55.3%

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

   3. associate-*r/ 55.6%

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

7. Simplified 55.6%

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

   1. add-cube-cbrt 55.8%

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

   2. pow3 55.7%

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

9. Applied egg-rr 55.7%

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

    1. *-commutative 55.7%

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

    2. cbrt-prod 55.8%

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

11. Applied egg-rr 55.8%

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

    1. cbrt-unprod 55.7%

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

    2. *-commutative 55.7%

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

    3. rem-cube-cbrt 55.6%

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

    4. associate-*r/ 55.3%

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

    5. *-commutative 55.3%

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

    6. associate-*r/ 55.3%

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

    7. associate-*r/ 55.3%

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

    8. *-commutative 55.3%

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

    9. expm1-log1p-u 52.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{x \cdot -0.5}{y}\right)}\right)\right)} \]

    10. expm1-define 52.5%

        \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{x \cdot -0.5}{y}\right)}\right)} - 1} \]

    11. sub-neg 52.5%

        \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{x \cdot -0.5}{y}\right)}\right)} + \left(-1\right)} \]

    12. metadata-eval 52.5%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\cos \left(\frac{x \cdot -0.5}{y}\right)}\right)} + \color{blue}{-1} \]

13. Applied egg-rr 55.3%

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

    1. associate-+l+ 55.3%

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

15. Simplified 55.6%

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

Alternative 8: 55.4% 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 45.9%

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

   1. remove-double-neg 45.9%

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

   2. distribute-frac-neg 45.9%

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

   3. tan-neg 45.9%

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

   4. distribute-frac-neg2 45.9%

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

   5. distribute-lft-neg-out 45.9%

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

   6. distribute-frac-neg2 45.9%

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

   7. distribute-lft-neg-out 45.9%

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

   8. distribute-frac-neg2 45.9%

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

   9. distribute-frac-neg 45.9%

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

   10. neg-mul-1 45.9%

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

   11. *-commutative 45.9%

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

   12. associate-/l* 45.9%

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

   13. *-commutative 45.9%

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

   14. associate-/r* 45.9%

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

   15. metadata-eval 45.9%

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

   16. sin-neg 45.9%

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

   17. distribute-frac-neg 45.9%

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

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 55.3%

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

   1. associate-*r/ 55.3%

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

   2. *-commutative 55.3%

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

   3. associate-*r/ 55.6%

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

7. Simplified 55.6%

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

Alternative 9: 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 45.9%

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

3. Taylor expanded in x around inf 55.3%

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

Alternative 10: 55.0% 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 45.9%

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

   1. remove-double-neg 45.9%

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

   2. distribute-frac-neg 45.9%

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

   3. tan-neg 45.9%

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

   4. distribute-frac-neg2 45.9%

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

   5. distribute-lft-neg-out 45.9%

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

   6. distribute-frac-neg2 45.9%

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

   7. distribute-lft-neg-out 45.9%

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

   8. distribute-frac-neg2 45.9%

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

   9. distribute-frac-neg 45.9%

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

   10. neg-mul-1 45.9%

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

   11. *-commutative 45.9%

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

   12. associate-/l* 45.9%

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

   13. *-commutative 45.9%

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

   14. associate-/r* 45.9%

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

   15. metadata-eval 45.9%

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

   16. sin-neg 45.9%

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

   17. distribute-frac-neg 45.9%

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

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 0 54.9%

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

Alternative 11: 6.6% 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 45.9%

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

   1. add-exp-log 16.8%

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

4. Applied egg-rr 16.8%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{e^{\log \left(y \cdot 2\right)}}}\right)} \]
5. Step-by-step derivation

   1. rem-exp-log 45.9%

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

   2. metadata-eval 45.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

   3. div-inv 45.9%

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

   4. clear-num 46.0%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{\color{blue}{\frac{1}{\frac{0.5}{y}}}}\right)} \]

   5. add-sqr-sqrt 18.0%

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

   6. sqrt-unprod 11.3%

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

   7. frac-times 11.2%

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

   8. metadata-eval 11.2%

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

   9. metadata-eval 11.2%

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

   10. frac-times 11.3%

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

   11. sqrt-unprod 1.8%

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

   12. add-sqr-sqrt 4.4%

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

   13. clear-num 4.7%

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

   14. div-inv 4.7%

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

   15. metadata-eval 4.7%

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

   16. metadata-eval 4.7%

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

   17. distribute-rgt-neg-in 4.7%

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

   18. rem-exp-log 2.0%

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

   19. add-log-exp 0.3%

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

   20. neg-log 0.3%

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

   21. rem-exp-log 0.7%

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

   22. *-commutative 0.7%

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

   23. exp-prod 0.7%

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

6. Applied egg-rr 0.7%

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

7. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\log \left(\frac{1}{e^{2 \cdot y}}\right)}{y}} \]
8. Step-by-step derivation

   1. log-rec 2.3%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{-\log \left(e^{2 \cdot y}\right)}}{y} \]

   2. *-commutative 2.3%

      \[\leadsto 0.5 \cdot \frac{-\log \left(e^{\color{blue}{y \cdot 2}}\right)}{y} \]

   3. rem-log-exp 6.6%

      \[\leadsto 0.5 \cdot \frac{-\color{blue}{y \cdot 2}}{y} \]

   4. distribute-rgt-neg-in 6.6%

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

   5. metadata-eval 6.6%

      \[\leadsto 0.5 \cdot \frac{y \cdot \color{blue}{-2}}{y} \]

9. Simplified 6.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot -2}{y}} \]

10. Taylor expanded in y around 0 6.6%

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

Developer Target 1: 55.0% 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 2024139 
(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)))))