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

Percentage Accurate: 43.9% → 56.6%

Time: 20.9s

Alternatives: 10

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: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 2e+57) (cbrt (pow (cos (* x (/ 0.5 y))) -3.0)) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+57) {
		tmp = cbrt(pow(cos((x * (0.5 / y))), -3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+57) {
		tmp = Math.cbrt(Math.pow(Math.cos((x * (0.5 / y))), -3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 2e+57)
		tmp = cbrt((cos(Float64(x * Float64(0.5 / y))) ^ -3.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 2e+57], N[Power[N[Power[N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 2.0000000000000001e57

   1. Initial program 50.1%

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

      1. add-cbrt-cube 50.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}} \]

      2. pow3 50.1%

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

      3. clear-num 50.1%

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

      4. inv-pow 50.1%

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

      5. metadata-eval 50.1%

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

      6. pow-pow 50.1%

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

   3. Applied egg-rr 66.4%

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

   if 2.0000000000000001e57 < (/.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. Taylor expanded in x around 0 10.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\sqrt[3]{{\cos \left(x \cdot \frac{0.5}{y}\right)}^{-3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 54.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{t_0 \cdot {t_0}^{2}}\right)}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (log (* 0.5 (/ x y))))))
   (log1p (expm1 (/ 1.0 (cos (exp (* t_0 (pow t_0 2.0)))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = cbrt(log((0.5 * (x / y))));
	return log1p(expm1((1.0 / cos(exp((t_0 * pow(t_0, 2.0)))))));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = Math.cbrt(Math.log((0.5 * (x / y))));
	return Math.log1p(Math.expm1((1.0 / Math.cos(Math.exp((t_0 * Math.pow(t_0, 2.0)))))));
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = cbrt(log(Float64(0.5 * Float64(x / y))))
	return log1p(expm1(Float64(1.0 / cos(exp(Float64(t_0 * (t_0 ^ 2.0)))))))
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[Power[N[Log[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[Exp[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}}\right)\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 35.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)} \cdot \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right) \cdot \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

   2. pow3 35.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

7. Applied egg-rr 35.3%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]
8. Step-by-step derivation

   1. add-cube-cbrt 35.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}}\right)}\right)\right) \]

   2. rem-cbrt-cube 34.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\left(\color{blue}{\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

   3. rem-cbrt-cube 35.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)} \cdot \color{blue}{\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

   4. pow2 35.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

   5. rem-cbrt-cube 35.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

9. Applied egg-rr 35.2%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{2} \cdot \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

10. Final simplification 35.2%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)} \cdot {\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{2}}\right)}\right)\right) \]

Alternative 3: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\sqrt[3]{{\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}}}\right)}\right)\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (exp (cbrt (pow (log (* 0.5 (/ x y))) 3.0))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return log1p(expm1((1.0 / cos(exp(cbrt(pow(log((0.5 * (x / y))), 3.0)))))));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return Math.log1p(Math.expm1((1.0 / Math.cos(Math.exp(Math.cbrt(Math.pow(Math.log((0.5 * (x / y))), 3.0)))))));
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return log1p(expm1(Float64(1.0 / cos(exp(cbrt((log(Float64(0.5 * Float64(x / y))) ^ 3.0)))))))
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[Exp[N[Power[N[Power[N[Log[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}}\right)\right) \]
6. Step-by-step derivation

   1. add-cbrt-cube 34.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\sqrt[3]{\left(\log \left(0.5 \cdot \frac{x}{y}\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

   2. pow3 34.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\sqrt[3]{\color{blue}{{\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}}}}\right)}\right)\right) \]

7. Applied egg-rr 34.6%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\sqrt[3]{{\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}}}}\right)}\right)\right) \]

8. Final simplification 34.6%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\sqrt[3]{{\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}}}\right)}\right)\right) \]

Alternative 4: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}\right)}\right)\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (exp (pow (cbrt (log (* 0.5 (/ x y)))) 3.0)))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return log1p(expm1((1.0 / cos(exp(pow(cbrt(log((0.5 * (x / y)))), 3.0))))));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return Math.log1p(Math.expm1((1.0 / Math.cos(Math.exp(Math.pow(Math.cbrt(Math.log((0.5 * (x / y)))), 3.0))))));
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return log1p(expm1(Float64(1.0 / cos(exp((cbrt(log(Float64(0.5 * Float64(x / y)))) ^ 3.0))))))
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[Exp[N[Power[N[Power[N[Log[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}}\right)\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 35.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)} \cdot \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right) \cdot \sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

   2. pow3 35.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

7. Applied egg-rr 35.3%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}}\right)}\right)\right) \]

8. Final simplification 35.3%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{{\left(\sqrt[3]{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}^{3}}\right)}\right)\right) \]

Alternative 5: 54.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(\frac{\sqrt{x}}{\frac{y \cdot 2}{\sqrt{x}}}\right)}\right)\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (/ (sqrt x) (/ (* y 2.0) (sqrt x))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return log1p(expm1((1.0 / cos((sqrt(x) / ((y * 2.0) / sqrt(x)))))));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return Math.log1p(Math.expm1((1.0 / Math.cos((Math.sqrt(x) / ((y * 2.0) / Math.sqrt(x)))))));
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return math.log1p(math.expm1((1.0 / math.cos((math.sqrt(x) / ((y * 2.0) / math.sqrt(x)))))))

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return log1p(expm1(Float64(1.0 / cos(Float64(sqrt(x) / Float64(Float64(y * 2.0) / sqrt(x)))))))
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(N[Sqrt[x], $MachinePrecision] / N[(N[(y * 2.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}}\right)\right) \]
6. Step-by-step derivation

   1. add-exp-log 54.2%

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

   2. metadata-eval 54.2%

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

   3. times-frac 54.2%

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

   4. *-un-lft-identity 54.2%

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

   5. *-commutative 54.2%

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

   6. add-sqr-sqrt 22.8%

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

   7. associate-/l* 22.8%

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

7. Applied egg-rr 22.8%

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

8. Final simplification 22.8%

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

Alternative 6: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}\right)\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (exp (log (* 0.5 (/ x y)))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return log1p(expm1((1.0 / cos(exp(log((0.5 * (x / y))))))));
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	return Math.log1p(Math.expm1((1.0 / Math.cos(Math.exp(Math.log((0.5 * (x / y))))))));
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return math.log1p(math.expm1((1.0 / math.cos(math.exp(math.log((0.5 * (x / y))))))))

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return log1p(expm1(Float64(1.0 / cos(exp(log(Float64(0.5 * Float64(x / y))))))))
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[Exp[N[Log[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

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

6. Final simplification 34.5%

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

Alternative 7: 54.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ 1.0 (cos (exp (- (log (* 0.5 x)) (log y))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos(exp((log((0.5 * x)) - log(y))));
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos(exp((log((0.5d0 * x)) - log(y))))
end function

Java

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

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0 / math.cos(math.exp((math.log((0.5 * x)) - math.log(y))))

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[(1.0 / N[Cos[N[Exp[N[(N[Log[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

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

   1. log1p-expm1-u 41.3%

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

   2. div-inv 40.1%

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

   3. tan-quot 40.1%

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

   4. associate-*l/ 40.1%

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

   5. pow1 40.1%

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

   6. inv-pow 40.1%

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

   7. pow-prod-up 54.2%

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

   8. metadata-eval 54.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.2%

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

   10. div-inv 54.1%

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

   11. *-commutative 54.1%

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

   12. associate-/r* 54.1%

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

   13. metadata-eval 54.1%

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

3. Applied egg-rr 54.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 54.1%

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

   2. div-inv 54.1%

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

   3. metadata-eval 54.1%

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

   4. div-inv 54.2%

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

   5. associate-/r* 54.2%

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

   6. div-inv 54.2%

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

   7. metadata-eval 54.2%

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

   8. *-commutative 54.2%

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

   9. add-exp-log 34.5%

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

5. Applied egg-rr 34.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(e^{\log \left(0.5 \cdot \frac{x}{y}\right)}\right)}}\right)\right) \]
6. Step-by-step derivation

   1. add-cbrt-cube 34.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{\sqrt[3]{\left(\log \left(0.5 \cdot \frac{x}{y}\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)}}}\right)}\right)\right) \]

   2. pow1/3 3.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left(\left(\log \left(0.5 \cdot \frac{x}{y}\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)\right) \cdot \log \left(0.5 \cdot \frac{x}{y}\right)\right)}^{0.3333333333333333}}}\right)}\right)\right) \]

   3. pow3 3.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{{\color{blue}{\left({\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}}^{0.3333333333333333}}\right)}\right)\right) \]

7. Applied egg-rr 3.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(e^{\color{blue}{{\left({\log \left(0.5 \cdot \frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}}}\right)}\right)\right) \]

8. Taylor expanded in y around 0 13.2%

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

9. Final simplification 13.2%

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

Alternative 8: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (/ x (* y 2.0)) 2e+57) (/ 1.0 (cos (* x (/ 0.5 y)))) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+57) {
		tmp = 1.0 / cos((x * (0.5 / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x / (y * 2.0d0)) <= 2d+57) then
        tmp = 1.0d0 / cos((x * (0.5d0 / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+57) {
		tmp = 1.0 / Math.cos((x * (0.5 / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 2e+57:
		tmp = 1.0 / math.cos((x * (0.5 / y)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 2e+57)
		tmp = Float64(1.0 / cos(Float64(x * Float64(0.5 / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x / (y * 2.0)) <= 2e+57)
		tmp = 1.0 / cos((x * (0.5 / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 2e+57], N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 2.0000000000000001e57

   1. Initial program 50.1%

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

      1. add-log-exp 8.5%

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

      2. add-sqr-sqrt 8.4%

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

      3. log-prod 8.4%

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

      4. div-inv 8.2%

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

      5. *-commutative 8.2%

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

      6. associate-/r* 8.2%

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

      7. metadata-eval 8.2%

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

      8. div-inv 7.9%

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

      9. *-commutative 7.9%

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

      10. associate-/r* 7.9%

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

      11. metadata-eval 7.9%

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

   3. Applied egg-rr 7.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right) + \log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)}} \]
   4. Step-by-step derivation

      1. count-2 7.9%

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

   5. Simplified 7.9%

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{2 \cdot \log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 7.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)}\right)}} \]

      2. pow3 7.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{2 \cdot \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{e^{\sin \left(x \cdot \frac{0.5}{y}\right)}}\right)}\right)}^{3}}} \]

      3. pow1/2 7.9%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{2 \cdot {\left(\sqrt[3]{\log \color{blue}{\left({\left(e^{\sin \left(x \cdot \frac{0.5}{y}\right)}\right)}^{0.5}\right)}}\right)}^{3}} \]

      4. log-pow 8.0%

         \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{2 \cdot {\left(\sqrt[3]{\color{blue}{0.5 \cdot \log \left(e^{\sin \left(x \cdot \frac{0.5}{y}\right)}\right)}}\right)}^{3}} \]

      5. add-log-exp 48.1%

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

      6. clear-num 48.1%

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

      7. div-inv 48.1%

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

      8. metadata-eval 48.1%

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

      9. div-inv 48.6%

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

      10. *-un-lft-identity 48.6%

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

      11. *-commutative 48.6%

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

      12. times-frac 48.6%

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

      13. metadata-eval 48.6%

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

   7. Applied egg-rr 48.6%

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

   8. Taylor expanded in x around inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.3%

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

      3. associate-*r/ 66.4%

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

   10. Simplified 66.4%

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

   if 2.0000000000000001e57 < (/.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. Taylor expanded in x around 0 10.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 54.8% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (/ 1.0 (cos (* 0.5 (/ x y)))))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos((0.5d0 * (x / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := N[(1.0 / N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

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

2. Taylor expanded in x around inf 54.2%

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

3. Final simplification 54.2%

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

Alternative 10: 54.6% accurate, 211.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ 1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 1.0)

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

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

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := 1.0

Derivation

1. Initial program 41.3%

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

2. Taylor expanded in x around 0 53.4%

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

3. Final simplification 53.4%

   \[\leadsto 1 \]

Developer target: 54.6% 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 2023290 
(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)))))