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

Percentage Accurate: 44.1% → 56.7%

Time: 21.7s

Alternatives: 9

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\sqrt[3]{\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)}}\right)}^{3} \cdot \sqrt[3]{{\cos \left(\frac{0.5}{\frac{y_m}{x_m}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x_m}{y_m}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \end{array} \]

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+45)
   (/
    1.0
    (*
     (pow (cbrt (cbrt (cos (* 0.5 (/ x_m y_m))))) 3.0)
     (cbrt (pow (cos (/ 0.5 (/ y_m x_m))) 2.0))))
   (/ 1.0 (expm1 (fma (pow (/ x_m y_m) 2.0) -0.0625 (log 2.0))))))

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+45) {
		tmp = 1.0 / (pow(cbrt(cbrt(cos((0.5 * (x_m / y_m))))), 3.0) * cbrt(pow(cos((0.5 / (y_m / x_m))), 2.0)));
	} else {
		tmp = 1.0 / expm1(fma(pow((x_m / y_m), 2.0), -0.0625, log(2.0)));
	}
	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+45)
		tmp = Float64(1.0 / Float64((cbrt(cbrt(cos(Float64(0.5 * Float64(x_m / y_m))))) ^ 3.0) * cbrt((cos(Float64(0.5 / Float64(y_m / x_m))) ^ 2.0))));
	else
		tmp = Float64(1.0 / expm1(fma((Float64(x_m / y_m) ^ 2.0), -0.0625, log(2.0))));
	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+45], N[(1.0 / N[(N[Power[N[Power[N[Power[N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Power[N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(Exp[N[(N[Power[N[(x$95$m / y$95$m), $MachinePrecision], 2.0], $MachinePrecision] * -0.0625 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5e45

   1. Initial program 57.3%

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

   3. Taylor expanded in x around inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

      1. add-cube-cbrt 71.0%

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

      2. associate-*l* 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-/l* 71.0%

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

      5. div-inv 71.0%

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

      6. metadata-eval 71.0%

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

      7. *-un-lft-identity 71.0%

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

      8. *-commutative 71.0%

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

      9. times-frac 71.0%

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

      10. metadata-eval 71.0%

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

      11. cbrt-unprod 71.0%

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

      12. pow2 71.0%

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

      13. *-un-lft-identity 71.0%

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

      14. times-frac 71.0%

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

      15. metadata-eval 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. associate-*r/ 71.0%

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

      2. associate-/l* 70.8%

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

      3. associate-*r/ 70.8%

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

      4. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

       1. add-cube-cbrt 70.8%

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

       2. pow3 70.8%

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

       3. div-inv 70.8%

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

       4. clear-num 71.1%

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

   11. Applied egg-rr 71.1%

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

   if 5e45 < (/.f64 x (*.f64 y 2))

   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. Taylor expanded in x around inf 6.8%

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

      1. associate-*r/ 6.8%

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

   5. Simplified 6.8%

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

      1. *-commutative 6.8%

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

      2. associate-/l* 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

      5. expm1-log1p-u 6.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      6. *-un-lft-identity 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)\right)} \]

      7. *-commutative 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      8. times-frac 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)} \]

      9. metadata-eval 6.8%

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

   7. Applied egg-rr 6.8%

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

   8. Taylor expanded in x around 0 10.8%

      \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{y}^{2}} + \log 2}\right)} \]

      2. *-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625} + \log 2\right)} \]

      3. fma-def 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.0625, \log 2\right)}\right)} \]

      4. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.0625, \log 2\right)\right)} \]

      5. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.0625, \log 2\right)\right)} \]

      6. times-frac 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.0625, \log 2\right)\right)} \]

      7. unpow2 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -0.0625, \log 2\right)\right)} \]

   10. Simplified 11.5%

       \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\sqrt[3]{\cos \left(0.5 \cdot \frac{x}{y}\right)}}\right)}^{3} \cdot \sqrt[3]{{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\cos \left(\frac{0.5}{\frac{y_m}{x_m}}\right)}^{2}} \cdot \sqrt[3]{\cos \left(\frac{x_m \cdot 0.5}{y_m}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x_m}{y_m}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \end{array} \]

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+45)
   (/
    1.0
    (*
     (cbrt (pow (cos (/ 0.5 (/ y_m x_m))) 2.0))
     (cbrt (cos (/ (* x_m 0.5) y_m)))))
   (/ 1.0 (expm1 (fma (pow (/ x_m y_m) 2.0) -0.0625 (log 2.0))))))

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+45) {
		tmp = 1.0 / (cbrt(pow(cos((0.5 / (y_m / x_m))), 2.0)) * cbrt(cos(((x_m * 0.5) / y_m))));
	} else {
		tmp = 1.0 / expm1(fma(pow((x_m / y_m), 2.0), -0.0625, log(2.0)));
	}
	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+45)
		tmp = Float64(1.0 / Float64(cbrt((cos(Float64(0.5 / Float64(y_m / x_m))) ^ 2.0)) * cbrt(cos(Float64(Float64(x_m * 0.5) / y_m)))));
	else
		tmp = Float64(1.0 / expm1(fma((Float64(x_m / y_m) ^ 2.0), -0.0625, log(2.0))));
	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+45], N[(1.0 / N[(N[Power[N[Power[N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Cos[N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(Exp[N[(N[Power[N[(x$95$m / y$95$m), $MachinePrecision], 2.0], $MachinePrecision] * -0.0625 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5e45

   1. Initial program 57.3%

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

   3. Taylor expanded in x around inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

      1. add-cube-cbrt 71.0%

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

      2. associate-*l* 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-/l* 71.0%

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

      5. div-inv 71.0%

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

      6. metadata-eval 71.0%

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

      7. *-un-lft-identity 71.0%

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

      8. *-commutative 71.0%

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

      9. times-frac 71.0%

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

      10. metadata-eval 71.0%

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

      11. cbrt-unprod 71.0%

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

      12. pow2 71.0%

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

      13. *-un-lft-identity 71.0%

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

      14. times-frac 71.0%

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

      15. metadata-eval 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. associate-*r/ 71.0%

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

      2. associate-/l* 70.8%

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

      3. associate-*r/ 70.8%

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

      4. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

   10. Taylor expanded in y around 0 71.0%

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

       1. *-commutative 71.0%

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

       2. associate-*l/ 71.0%

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

   12. Simplified 71.0%

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

   if 5e45 < (/.f64 x (*.f64 y 2))

   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. Taylor expanded in x around inf 6.8%

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

      1. associate-*r/ 6.8%

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

   5. Simplified 6.8%

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

      1. *-commutative 6.8%

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

      2. associate-/l* 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

      5. expm1-log1p-u 6.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      6. *-un-lft-identity 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)\right)} \]

      7. *-commutative 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      8. times-frac 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)} \]

      9. metadata-eval 6.8%

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

   7. Applied egg-rr 6.8%

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

   8. Taylor expanded in x around 0 10.8%

      \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{y}^{2}} + \log 2}\right)} \]

      2. *-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625} + \log 2\right)} \]

      3. fma-def 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.0625, \log 2\right)}\right)} \]

      4. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.0625, \log 2\right)\right)} \]

      5. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.0625, \log 2\right)\right)} \]

      6. times-frac 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.0625, \log 2\right)\right)} \]

      7. unpow2 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -0.0625, \log 2\right)\right)} \]

   10. Simplified 11.5%

       \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}^{2}} \cdot \sqrt[3]{\cos \left(\frac{x \cdot 0.5}{y}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x_m}{y_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x_m}{y_m}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \end{array} \]

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+45)
   (/ 1.0 (expm1 (log1p (cos (* 0.5 (/ x_m y_m))))))
   (/ 1.0 (expm1 (fma (pow (/ x_m y_m) 2.0) -0.0625 (log 2.0))))))

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+45) {
		tmp = 1.0 / expm1(log1p(cos((0.5 * (x_m / y_m)))));
	} else {
		tmp = 1.0 / expm1(fma(pow((x_m / y_m), 2.0), -0.0625, log(2.0)));
	}
	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+45)
		tmp = Float64(1.0 / expm1(log1p(cos(Float64(0.5 * Float64(x_m / y_m))))));
	else
		tmp = Float64(1.0 / expm1(fma((Float64(x_m / y_m) ^ 2.0), -0.0625, log(2.0))));
	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+45], N[(1.0 / N[(Exp[N[Log[1 + N[Cos[N[(0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(Exp[N[(N[Power[N[(x$95$m / y$95$m), $MachinePrecision], 2.0], $MachinePrecision] * -0.0625 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 5e45

   1. Initial program 57.3%

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

   3. Taylor expanded in x around inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-/l* 71.0%

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

      3. div-inv 71.0%

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

      4. metadata-eval 71.0%

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

      5. expm1-log1p-u 71.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      6. *-un-lft-identity 71.0%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)\right)} \]

      7. *-commutative 71.0%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      8. times-frac 71.0%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)} \]

      9. metadata-eval 71.0%

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

   7. Applied egg-rr 71.0%

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

   if 5e45 < (/.f64 x (*.f64 y 2))

   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. Taylor expanded in x around inf 6.8%

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

      1. associate-*r/ 6.8%

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

   5. Simplified 6.8%

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

      1. *-commutative 6.8%

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

      2. associate-/l* 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

      5. expm1-log1p-u 6.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)\right)}} \]

      6. *-un-lft-identity 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\color{blue}{1 \cdot x}}{y \cdot 2}\right)\right)\right)} \]

      7. *-commutative 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1 \cdot x}{\color{blue}{2 \cdot y}}\right)\right)\right)} \]

      8. times-frac 6.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)} \]

      9. metadata-eval 6.8%

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

   7. Applied egg-rr 6.8%

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

   8. Taylor expanded in x around 0 10.8%

      \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{y}^{2}} + \log 2}\right)} \]

      2. *-commutative 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625} + \log 2\right)} \]

      3. fma-def 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.0625, \log 2\right)}\right)} \]

      4. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.0625, \log 2\right)\right)} \]

      5. unpow2 10.8%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.0625, \log 2\right)\right)} \]

      6. times-frac 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.0625, \log 2\right)\right)} \]

      7. unpow2 11.5%

         \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -0.0625, \log 2\right)\right)} \]

   10. Simplified 11.5%

       \[\leadsto \frac{1}{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{expm1}\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.0625, \log 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{0.5}{y_m}}\\ \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{\cos \left(t_0 \cdot \left(x_m \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.5 y_m))))
   (if (<= (/ x_m (* y_m 2.0)) 2e+128) (/ 1.0 (cos (* t_0 (* x_m t_0)))) 1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = sqrt((0.5 / y_m));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+128) {
		tmp = 1.0 / cos((t_0 * (x_m * t_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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 / y_m))
    if ((x_m / (y_m * 2.0d0)) <= 2d+128) then
        tmp = 1.0d0 / cos((t_0 * (x_m * t_0)))
    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 t_0 = Math.sqrt((0.5 / y_m));
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+128) {
		tmp = 1.0 / Math.cos((t_0 * (x_m * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.sqrt((0.5 / y_m))
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+128:
		tmp = 1.0 / math.cos((t_0 * (x_m * t_0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = sqrt(Float64(0.5 / y_m))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+128)
		tmp = Float64(1.0 / cos(Float64(t_0 * Float64(x_m * t_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)
	t_0 = sqrt((0.5 / y_m));
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+128)
		tmp = 1.0 / cos((t_0 * (x_m * t_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_] := Block[{t$95$0 = N[Sqrt[N[(0.5 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+128], N[(1.0 / N[Cos[N[(t$95$0 * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 2.0000000000000002e128

   1. Initial program 55.0%

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

   3. Taylor expanded in x around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. associate-/l* 67.8%

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

   8. Simplified 67.8%

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

      1. add-cube-cbrt 67.2%

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

      2. pow3 67.4%

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

      3. div-inv 67.4%

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

      4. clear-num 67.6%

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

   10. Applied egg-rr 67.6%

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

       1. rem-cube-cbrt 68.0%

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

       2. associate-*r/ 68.0%

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

       3. associate-*l/ 67.8%

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

       4. *-commutative 67.8%

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

       5. add-sqr-sqrt 31.7%

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

       6. associate-*r* 31.9%

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

   12. Applied egg-rr 31.9%

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

   if 2.0000000000000002e128 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.9%

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

   3. Taylor expanded in x around 0 11.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt{\frac{0.5}{y}} \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{x_m}{y_m}}\\ \frac{1}{\cos \left(t_0 \cdot \left(0.5 \cdot t_0\right)\right)} \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = sqrt((x_m / y_m));
	return 1.0 / cos((t_0 * (0.5 * t_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
    real(8) :: t_0
    t_0 = sqrt((x_m / y_m))
    code = 1.0d0 / cos((t_0 * (0.5d0 * t_0)))
end function

Java

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

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.sqrt((x_m / y_m))
	return 1.0 / math.cos((t_0 * (0.5 * t_0)))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = sqrt(Float64(x_m / y_m))
	return Float64(1.0 / cos(Float64(t_0 * Float64(0.5 * t_0))))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	t_0 = sqrt((x_m / y_m));
	tmp = 1.0 / cos((t_0 * (0.5 * t_0)));
end

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

5. Simplified 60.0%

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

6. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. associate-/l* 59.6%

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

8. Simplified 59.6%

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

   1. div-inv 59.6%

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

   2. clear-num 60.0%

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

   3. add-sqr-sqrt 36.2%

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

   4. associate-*r* 36.2%

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

10. Applied egg-rr 36.2%

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

11. Final simplification 36.2%

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

Alternative 6: 54.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

5. Simplified 60.0%

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

6. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. associate-/l* 59.6%

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

8. Simplified 59.6%

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

   1. div-inv 59.6%

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

   2. clear-num 60.0%

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

   3. add-exp-log 36.2%

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

10. Applied egg-rr 36.2%

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

11. Final simplification 36.2%

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

Alternative 7: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x_m}{y_m \cdot 2} \leq 10^{+116}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x_m \cdot 0.5}{y_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+116) (/ 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)) <= 1e+116) {
		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)) <= 1d+116) 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)) <= 1e+116) {
		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)) <= 1e+116:
		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)) <= 1e+116)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * 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)) <= 1e+116)
		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], 1e+116], N[(1.0 / N[Cos[N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1.00000000000000002e116

   1. Initial program 55.2%

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

   3. Taylor expanded in x around inf 68.3%

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

      1. associate-*r/ 68.3%

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

   5. Simplified 68.3%

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

   if 1.00000000000000002e116 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0 11.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+116}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x \cdot 0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

5. Simplified 60.0%

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

6. Taylor expanded in x around inf 60.0%

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

   1. associate-*r/ 60.0%

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

   2. associate-/l* 59.6%

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

8. Simplified 59.6%

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

   1. associate-/r/ 59.8%

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

10. Applied egg-rr 59.8%

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

11. Final simplification 59.8%

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

Alternative 9: 54.8% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0 59.0%

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

4. Final simplification 59.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 54.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (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)))))