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

Percentage Accurate: 43.9% → 56.7%

Time: 18.2s

Alternatives: 6

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt{x\_m}}{y\_m} \cdot \left(\sqrt[3]{x\_m \cdot 0.25} \cdot \sqrt[3]{\sqrt{x\_m} \cdot -0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 4e+192)
   (/
    1.0
    (cos
     (*
      (/ (sqrt x_m) y_m)
      (* (cbrt (* x_m 0.25)) (cbrt (* (sqrt x_m) -0.5))))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+192) {
		tmp = 1.0 / cos(((sqrt(x_m) / y_m) * (cbrt((x_m * 0.25)) * cbrt((sqrt(x_m) * -0.5)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+192) {
		tmp = 1.0 / Math.cos(((Math.sqrt(x_m) / y_m) * (Math.cbrt((x_m * 0.25)) * Math.cbrt((Math.sqrt(x_m) * -0.5)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 4e+192)
		tmp = Float64(1.0 / cos(Float64(Float64(sqrt(x_m) / y_m) * Float64(cbrt(Float64(x_m * 0.25)) * cbrt(Float64(sqrt(x_m) * -0.5))))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 4e+192], N[(1.0 / N[Cos[N[(N[(N[Sqrt[x$95$m], $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[Power[N[(x$95$m * 0.25), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[x$95$m], $MachinePrecision] * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.00000000000000016e192

   1. Initial program 51.4%

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

      1. remove-double-neg 51.4%

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

      2. distribute-frac-neg 51.4%

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

      3. tan-neg 51.4%

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

      4. distribute-frac-neg2 51.4%

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

      5. distribute-lft-neg-out 51.4%

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

      6. distribute-frac-neg2 51.4%

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

      7. distribute-lft-neg-out 51.4%

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

      8. distribute-frac-neg2 51.4%

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

      9. distribute-frac-neg 51.4%

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

      10. neg-mul-1 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-/l* 50.7%

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

      13. *-commutative 50.7%

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

      14. associate-/r* 50.7%

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

      15. metadata-eval 50.7%

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

      16. sin-neg 50.7%

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

      17. distribute-frac-neg 50.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 63.2%

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

      1. associate-*r/ 63.2%

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

      2. *-commutative 63.2%

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

      3. associate-*r/ 63.3%

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

   7. Simplified 63.3%

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

      1. associate-*r/ 63.2%

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

      2. clear-num 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. inv-pow 63.5%

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

       2. metadata-eval 63.5%

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

       3. div-inv 63.5%

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

       4. div-inv 63.7%

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

       5. clear-num 63.7%

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

   11. Applied egg-rr 63.7%

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

       1. unpow-1 63.7%

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

   13. Simplified 63.7%

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

       1. associate-*r/ 63.5%

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

       2. clear-num 63.2%

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

       3. add-sqr-sqrt 34.1%

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

       4. frac-times 34.4%

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

       5. add-cube-cbrt 33.9%

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

       6. associate-*r* 34.2%

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

       7. cbrt-unprod 34.6%

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

       8. div-inv 34.6%

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

       9. metadata-eval 34.6%

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

       10. div-inv 34.6%

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

       11. metadata-eval 34.6%

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

       12. swap-sqr 34.6%

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

       13. add-sqr-sqrt 34.4%

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

       14. metadata-eval 34.4%

           \[\leadsto \frac{1}{\cos \left(\left(\frac{\sqrt{x}}{y} \cdot \sqrt[3]{x \cdot \color{blue}{0.25}}\right) \cdot \sqrt[3]{\frac{\sqrt{x}}{-2}}\right)} \]

       15. div-inv 34.4%

           \[\leadsto \frac{1}{\cos \left(\left(\frac{\sqrt{x}}{y} \cdot \sqrt[3]{x \cdot 0.25}\right) \cdot \sqrt[3]{\color{blue}{\sqrt{x} \cdot \frac{1}{-2}}}\right)} \]

       16. metadata-eval 34.4%

           \[\leadsto \frac{1}{\cos \left(\left(\frac{\sqrt{x}}{y} \cdot \sqrt[3]{x \cdot 0.25}\right) \cdot \sqrt[3]{\sqrt{x} \cdot \color{blue}{-0.5}}\right)} \]

   15. Applied egg-rr 34.4%

       \[\leadsto \frac{1}{\cos \color{blue}{\left(\left(\frac{\sqrt{x}}{y} \cdot \sqrt[3]{x \cdot 0.25}\right) \cdot \sqrt[3]{\sqrt{x} \cdot -0.5}\right)}} \]
   16. Step-by-step derivation

       1. associate-*l* 34.7%

          \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\sqrt{x}}{y} \cdot \left(\sqrt[3]{x \cdot 0.25} \cdot \sqrt[3]{\sqrt{x} \cdot -0.5}\right)\right)}} \]

       2. *-commutative 34.7%

          \[\leadsto \frac{1}{\cos \left(\frac{\sqrt{x}}{y} \cdot \left(\sqrt[3]{x \cdot 0.25} \cdot \sqrt[3]{\color{blue}{-0.5 \cdot \sqrt{x}}}\right)\right)} \]

   17. Simplified 34.7%

       \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\sqrt{x}}{y} \cdot \left(\sqrt[3]{x \cdot 0.25} \cdot \sqrt[3]{-0.5 \cdot \sqrt{x}}\right)\right)}} \]

   if 4.00000000000000016e192 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.9%

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

      1. remove-double-neg 7.9%

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

      2. distribute-frac-neg 7.9%

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

      3. tan-neg 7.9%

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

      4. distribute-frac-neg2 7.9%

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

      5. distribute-lft-neg-out 7.9%

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

      6. distribute-frac-neg2 7.9%

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

      7. distribute-lft-neg-out 7.9%

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

      8. distribute-frac-neg2 7.9%

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

      9. distribute-frac-neg 7.9%

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

      10. neg-mul-1 7.9%

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

      11. *-commutative 7.9%

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

      12. associate-/l* 8.4%

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

      13. *-commutative 8.4%

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

      14. associate-/r* 8.4%

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

      15. metadata-eval 8.4%

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

      16. sin-neg 8.4%

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

      17. distribute-frac-neg 8.4%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 11.4%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt{x}}{y} \cdot \left(\sqrt[3]{x \cdot 0.25} \cdot \sqrt[3]{\sqrt{x} \cdot -0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\sqrt{x\_m}}{y\_m} \cdot \frac{\sqrt{x\_m}}{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 4e+192)
   (/ 1.0 (cos (* (/ (sqrt x_m) y_m) (/ (sqrt x_m) -2.0))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+192) {
		tmp = 1.0 / cos(((sqrt(x_m) / y_m) * (sqrt(x_m) / -2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 4d+192) then
        tmp = 1.0d0 / cos(((sqrt(x_m) / y_m) * (sqrt(x_m) / (-2.0d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 4e+192) {
		tmp = 1.0 / Math.cos(((Math.sqrt(x_m) / y_m) * (Math.sqrt(x_m) / -2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 4e+192:
		tmp = 1.0 / math.cos(((math.sqrt(x_m) / y_m) * (math.sqrt(x_m) / -2.0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 4e+192)
		tmp = Float64(1.0 / cos(Float64(Float64(sqrt(x_m) / y_m) * Float64(sqrt(x_m) / -2.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 4e+192)
		tmp = 1.0 / cos(((sqrt(x_m) / y_m) * (sqrt(x_m) / -2.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 4e+192], N[(1.0 / N[Cos[N[(N[(N[Sqrt[x$95$m], $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[Sqrt[x$95$m], $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.00000000000000016e192

   1. Initial program 51.4%

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

      1. remove-double-neg 51.4%

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

      2. distribute-frac-neg 51.4%

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

      3. tan-neg 51.4%

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

      4. distribute-frac-neg2 51.4%

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

      5. distribute-lft-neg-out 51.4%

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

      6. distribute-frac-neg2 51.4%

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

      7. distribute-lft-neg-out 51.4%

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

      8. distribute-frac-neg2 51.4%

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

      9. distribute-frac-neg 51.4%

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

      10. neg-mul-1 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-/l* 50.7%

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

      13. *-commutative 50.7%

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

      14. associate-/r* 50.7%

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

      15. metadata-eval 50.7%

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

      16. sin-neg 50.7%

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

      17. distribute-frac-neg 50.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 63.2%

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

      1. associate-*r/ 63.2%

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

      2. *-commutative 63.2%

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

      3. associate-*r/ 63.3%

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

   7. Simplified 63.3%

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

      1. associate-*r/ 63.2%

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

      2. clear-num 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. clear-num 63.2%

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

       2. metadata-eval 63.2%

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

       3. div-inv 63.2%

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

       4. associate-/l/ 63.2%

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

       5. add-sqr-sqrt 34.1%

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

       6. times-frac 34.4%

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

   11. Applied egg-rr 34.4%

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

   if 4.00000000000000016e192 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.9%

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

      1. remove-double-neg 7.9%

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

      2. distribute-frac-neg 7.9%

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

      3. tan-neg 7.9%

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

      4. distribute-frac-neg2 7.9%

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

      5. distribute-lft-neg-out 7.9%

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

      6. distribute-frac-neg2 7.9%

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

      7. distribute-lft-neg-out 7.9%

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

      8. distribute-frac-neg2 7.9%

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

      9. distribute-frac-neg 7.9%

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

      10. neg-mul-1 7.9%

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

      11. *-commutative 7.9%

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

      12. associate-/l* 8.4%

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

      13. *-commutative 8.4%

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

      14. associate-/r* 8.4%

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

      15. metadata-eval 8.4%

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

      16. sin-neg 8.4%

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

      17. distribute-frac-neg 8.4%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 11.4%

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

Alternative 3: 56.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt[3]{{\left(x\_m \cdot \frac{-0.5}{y\_m}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+100)
   (/ 1.0 (cos (cbrt (pow (* x_m (/ -0.5 y_m)) 3.0))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+100) {
		tmp = 1.0 / cos(cbrt(pow((x_m * (-0.5 / y_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 tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+100) {
		tmp = 1.0 / Math.cos(Math.cbrt(Math.pow((x_m * (-0.5 / y_m)), 3.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+100)
		tmp = Float64(1.0 / cos(cbrt((Float64(x_m * Float64(-0.5 / y_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_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+100], N[(1.0 / N[Cos[N[Power[N[Power[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e100

   1. Initial program 53.3%

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

      1. remove-double-neg 53.3%

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

      2. distribute-frac-neg 53.3%

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

      3. tan-neg 53.3%

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

      4. distribute-frac-neg2 53.3%

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

      5. distribute-lft-neg-out 53.3%

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

      6. distribute-frac-neg2 53.3%

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

      7. distribute-lft-neg-out 53.3%

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

      8. distribute-frac-neg2 53.3%

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

      9. distribute-frac-neg 53.3%

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

      10. neg-mul-1 53.3%

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

      11. *-commutative 53.3%

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

      12. associate-/l* 52.7%

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

      13. *-commutative 52.7%

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

      14. associate-/r* 52.7%

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

      15. metadata-eval 52.7%

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

      16. sin-neg 52.7%

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

      17. distribute-frac-neg 52.7%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in x around inf 65.7%

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

      1. associate-*r/ 65.7%

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

      2. *-commutative 65.7%

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

      3. associate-*r/ 65.8%

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

   7. Simplified 65.8%

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

      1. add-cbrt-cube 65.4%

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

      2. pow3 65.5%

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

   9. Applied egg-rr 65.5%

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

   if 4.9999999999999999e100 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.5%

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

      1. remove-double-neg 8.5%

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

      2. distribute-frac-neg 8.5%

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

      3. tan-neg 8.5%

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

      4. distribute-frac-neg2 8.5%

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

      5. distribute-lft-neg-out 8.5%

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

      6. distribute-frac-neg2 8.5%

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

      7. distribute-lft-neg-out 8.5%

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

      8. distribute-frac-neg2 8.5%

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

      9. distribute-frac-neg 8.5%

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

      10. neg-mul-1 8.5%

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

      11. *-commutative 8.5%

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

      12. associate-/l* 7.8%

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

      13. *-commutative 7.8%

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

      14. associate-/r* 7.8%

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

      15. metadata-eval 7.8%

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

      16. sin-neg 7.8%

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

      17. distribute-frac-neg 7.8%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 11.5%

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

Alternative 4: 56.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{-0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+74) (/ 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) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+74) {
		tmp = 1.0 / cos((x_m * (-0.5 / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 2d+74) then
        tmp = 1.0d0 / cos((x_m * ((-0.5d0) / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+74) {
		tmp = 1.0 / Math.cos((x_m * (-0.5 / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+74:
		tmp = 1.0 / math.cos((x_m * (-0.5 / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+74)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(-0.5 / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+74)
		tmp = 1.0 / cos((x_m * (-0.5 / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+74], N[(1.0 / N[Cos[N[(x$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e74

   1. Initial program 53.4%

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

      1. remove-double-neg 53.4%

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

      2. distribute-frac-neg 53.4%

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

      3. tan-neg 53.4%

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

      4. distribute-frac-neg2 53.4%

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

      5. distribute-lft-neg-out 53.4%

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

      6. distribute-frac-neg2 53.4%

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

      7. distribute-lft-neg-out 53.4%

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

      8. distribute-frac-neg2 53.4%

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

      9. distribute-frac-neg 53.4%

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

      10. neg-mul-1 53.4%

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

      11. *-commutative 53.4%

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

      12. associate-/l* 52.9%

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

      13. *-commutative 52.9%

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

      14. associate-/r* 52.9%

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

      15. metadata-eval 52.9%

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

      16. sin-neg 52.9%

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

      17. distribute-frac-neg 52.9%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in x around inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*r/ 66.0%

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

   7. Simplified 66.0%

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

   if 1.9999999999999999e74 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 8.8%

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

      1. remove-double-neg 8.8%

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

      2. distribute-frac-neg 8.8%

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

      3. tan-neg 8.8%

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

      4. distribute-frac-neg2 8.8%

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

      5. distribute-lft-neg-out 8.8%

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

      6. distribute-frac-neg2 8.8%

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

      7. distribute-lft-neg-out 8.8%

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

      8. distribute-frac-neg2 8.8%

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

      9. distribute-frac-neg 8.8%

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

      10. neg-mul-1 8.8%

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

      11. *-commutative 8.8%

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

      12. associate-/l* 8.1%

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

      13. *-commutative 8.1%

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

      14. associate-/r* 8.1%

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

      15. metadata-eval 8.1%

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

      16. sin-neg 8.1%

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

      17. distribute-frac-neg 8.1%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in x around 0 12.0%

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

Alternative 5: 55.1% 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 46.6%

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

3. Taylor expanded in x around inf 57.2%

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

Alternative 6: 55.2% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. remove-double-neg 46.6%

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

   2. distribute-frac-neg 46.6%

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

   3. tan-neg 46.6%

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

   4. distribute-frac-neg2 46.6%

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

   5. distribute-lft-neg-out 46.6%

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

   6. distribute-frac-neg2 46.6%

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

   7. distribute-lft-neg-out 46.6%

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

   8. distribute-frac-neg2 46.6%

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

   9. distribute-frac-neg 46.6%

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

   10. neg-mul-1 46.6%

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

   11. *-commutative 46.6%

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

   12. associate-/l* 46.0%

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

   13. *-commutative 46.0%

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

   14. associate-/r* 46.0%

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

   15. metadata-eval 46.0%

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

   16. sin-neg 46.0%

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

   17. distribute-frac-neg 46.0%

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

3. Simplified 46.7%

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

5. Taylor expanded in x around 0 56.7%

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

Developer target: 55.2% 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 2024086 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

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