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

Percentage Accurate: 43.2% → 56.9%

Time: 15.5s

Alternatives: 7

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: 43.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \sqrt{y \cdot 2}\\ \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+53}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{t_0}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{t_0}{\sqrt[3]{x}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* y 2.0))))
   (if (<= (/ x (* y 2.0)) 1e+53)
     (/ 1.0 (cos (/ 1.0 (* (/ t_0 (pow (cbrt x) 2.0)) (/ t_0 (cbrt x))))))
     1.0)))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = sqrt((y * 2.0));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+53) {
		tmp = 1.0 / cos((1.0 / ((t_0 / pow(cbrt(x), 2.0)) * (t_0 / cbrt(x)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = Math.sqrt((y * 2.0));
	double tmp;
	if ((x / (y * 2.0)) <= 1e+53) {
		tmp = 1.0 / Math.cos((1.0 / ((t_0 / Math.pow(Math.cbrt(x), 2.0)) * (t_0 / Math.cbrt(x)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = sqrt(Float64(y * 2.0))
	tmp = 0.0
	if (Float64(x / Float64(y * 2.0)) <= 1e+53)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(Float64(t_0 / (cbrt(x) ^ 2.0)) * Float64(t_0 / cbrt(x))))));
	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_] := Block[{t$95$0 = N[Sqrt[N[(y * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 1e+53], N[(1.0 / N[Cos[N[(1.0 / N[(N[(t$95$0 / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 9.9999999999999999e52

   1. Initial program 56.4%

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

      1. add-log-exp 56.4%

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

      2. *-un-lft-identity 56.4%

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

      3. log-prod 56.4%

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

      4. metadata-eval 56.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

      5. add-log-exp 56.4%

         \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      6. div-inv 56.2%

         \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      7. tan-quot 56.2%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. associate-*l/ 56.2%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

      9. pow1 56.2%

         \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      10. inv-pow 56.2%

          \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      11. pow-prod-up 67.9%

          \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      12. metadata-eval 67.9%

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

      13. metadata-eval 67.9%

          \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      14. div-inv 67.9%

          \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

      15. *-commutative 67.9%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

      16. associate-/r* 67.9%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

      17. metadata-eval 67.9%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

   3. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   4. Step-by-step derivation

      1. clear-num 67.9%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)} \]

      2. div-inv 67.9%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)} \]

      3. metadata-eval 67.9%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)} \]

      4. div-inv 67.9%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}} \]

      5. associate-/r* 67.9%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}} \]

      6. clear-num 68.0%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]

   5. Applied egg-rr 68.0%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]
   6. Step-by-step derivation

      1. clear-num 68.0%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\color{blue}{\frac{1}{\frac{\frac{x}{y}}{2}}}}\right)} \]

      2. associate-/r* 68.0%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{1}{\color{blue}{\frac{x}{y \cdot 2}}}}\right)} \]

      3. clear-num 67.7%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\color{blue}{\frac{y \cdot 2}{x}}}\right)} \]

      4. add-sqr-sqrt 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{\color{blue}{\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}}}{x}}\right)} \]

      5. add-cube-cbrt 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}\right)} \]

      6. times-frac 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\color{blue}{\frac{\sqrt{y \cdot 2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{y \cdot 2}}{\sqrt[3]{x}}}}\right)} \]

      7. *-commutative 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{\sqrt{\color{blue}{2 \cdot y}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{y \cdot 2}}{\sqrt[3]{x}}}\right)} \]

      8. pow2 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{\sqrt{2 \cdot y}}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \cdot \frac{\sqrt{y \cdot 2}}{\sqrt[3]{x}}}\right)} \]

      9. *-commutative 30.6%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\frac{\sqrt{2 \cdot y}}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt{\color{blue}{2 \cdot y}}}{\sqrt[3]{x}}}\right)} \]

   7. Applied egg-rr 30.6%

      \[\leadsto 0 + \frac{1}{\cos \left(\frac{1}{\color{blue}{\frac{\sqrt{2 \cdot y}}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt{2 \cdot y}}{\sqrt[3]{x}}}}\right)} \]

   if 9.9999999999999999e52 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.7%

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

   2. Taylor expanded in x around 0 10.9%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+53}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{\sqrt{y \cdot 2}}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt{y \cdot 2}}{\sqrt[3]{x}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 56.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+173}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+173)
   (/ 1.0 (cos (* (/ x (pow (cbrt y) 2.0)) (/ 0.5 (cbrt y)))))
   1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 1e+173) {
		tmp = 1.0 / cos(((x / pow(cbrt(y), 2.0)) * (0.5 / cbrt(y))));
	} 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)) <= 1e+173) {
		tmp = 1.0 / Math.cos(((x / Math.pow(Math.cbrt(y), 2.0)) * (0.5 / Math.cbrt(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)) <= 1e+173)
		tmp = Float64(1.0 / cos(Float64(Float64(x / (cbrt(y) ^ 2.0)) * Float64(0.5 / cbrt(y)))));
	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], 1e+173], N[(1.0 / N[Cos[N[(N[(x / N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 1e173

   1. Initial program 52.2%

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

      1. add-log-exp 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. log-prod 52.2%

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

      4. metadata-eval 52.2%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

      5. add-log-exp 52.2%

         \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      6. div-inv 52.1%

         \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      7. tan-quot 52.1%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. associate-*l/ 52.1%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

      9. pow1 52.1%

         \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      10. inv-pow 52.1%

          \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      11. pow-prod-up 62.7%

          \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      12. metadata-eval 62.7%

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

      13. metadata-eval 62.7%

          \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      14. div-inv 62.6%

          \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

      15. *-commutative 62.6%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

      16. associate-/r* 62.6%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

      17. metadata-eval 62.6%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

   3. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   4. Step-by-step derivation

      1. clear-num 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)} \]

      2. div-inv 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)} \]

      3. metadata-eval 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)} \]

      4. div-inv 62.7%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}} \]

      5. associate-/r* 62.7%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}} \]

      6. clear-num 62.8%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]

   5. Applied egg-rr 62.8%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]
   6. Step-by-step derivation

      1. clear-num 62.7%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}} \]

      2. associate-/r* 62.7%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}} \]

      3. div-inv 62.6%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

      4. metadata-eval 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}}\right)} \]

      5. div-inv 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}}\right)} \]

      6. clear-num 62.6%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{0.5}{y}}\right)} \]

      7. associate-*r/ 62.7%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x \cdot 0.5}{y}\right)}} \]

      8. add-cube-cbrt 62.5%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{x \cdot 0.5}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)} \]

      9. times-frac 62.5%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)}} \]

      10. pow2 62.5%

          \[\leadsto 0 + \frac{1}{\cos \left(\frac{x}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)} \]

   7. Applied egg-rr 62.5%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)}} \]

   if 1e173 < (/.f64 x (*.f64 y 2))

   1. Initial program 3.5%

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

   2. Taylor expanded in x around 0 10.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+173}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{y}}{\frac{2}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+52) (/ 1.0 (cos (/ (/ 1.0 y) (/ 2.0 x)))) 1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+52) {
		tmp = 1.0 / cos(((1.0 / y) / (2.0 / x)));
	} 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+52) then
        tmp = 1.0d0 / cos(((1.0d0 / y) / (2.0d0 / x)))
    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+52) {
		tmp = 1.0 / Math.cos(((1.0 / y) / (2.0 / x)));
	} 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+52:
		tmp = 1.0 / math.cos(((1.0 / y) / (2.0 / x)))
	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+52)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / y) / Float64(2.0 / x))));
	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+52)
		tmp = 1.0 / cos(((1.0 / y) / (2.0 / x)));
	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+52], N[(1.0 / N[Cos[N[(N[(1.0 / y), $MachinePrecision] / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 2e52

   1. Initial program 56.6%

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

      1. add-log-exp 56.6%

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

      2. *-un-lft-identity 56.6%

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

      3. log-prod 56.6%

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

      4. metadata-eval 56.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

      5. add-log-exp 56.6%

         \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      6. div-inv 56.5%

         \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

      7. tan-quot 56.5%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. associate-*l/ 56.5%

         \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

      9. pow1 56.5%

         \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      10. inv-pow 56.5%

          \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      11. pow-prod-up 68.2%

          \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      12. metadata-eval 68.2%

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

      13. metadata-eval 68.2%

          \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

      14. div-inv 68.2%

          \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

      15. *-commutative 68.2%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

      16. associate-/r* 68.2%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

      17. metadata-eval 68.2%

          \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

   3. Applied egg-rr 68.2%

      \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
   4. Step-by-step derivation

      1. clear-num 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)} \]

      2. div-inv 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)} \]

      3. metadata-eval 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)} \]

      4. div-inv 68.2%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}} \]

      5. associate-/r* 68.2%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}} \]

      6. clear-num 68.3%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]

   5. Applied egg-rr 68.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]
   6. Step-by-step derivation

      1. inv-pow 68.3%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\frac{2}{\frac{x}{y}}\right)}^{-1}\right)}} \]

      2. associate-/r/ 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left({\color{blue}{\left(\frac{2}{x} \cdot y\right)}}^{-1}\right)} \]

      3. unpow-prod-down 68.2%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\frac{2}{x}\right)}^{-1} \cdot {y}^{-1}\right)}} \]

      4. inv-pow 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left({\left(\frac{2}{x}\right)}^{-1} \cdot \color{blue}{\frac{1}{y}}\right)} \]

   7. Applied egg-rr 68.2%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\frac{2}{x}\right)}^{-1} \cdot \frac{1}{y}\right)}} \]
   8. Step-by-step derivation

      1. unpow-1 68.2%

         \[\leadsto 0 + \frac{1}{\cos \left(\color{blue}{\frac{1}{\frac{2}{x}}} \cdot \frac{1}{y}\right)} \]

      2. associate-*l/ 68.3%

         \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1 \cdot \frac{1}{y}}{\frac{2}{x}}\right)}} \]

      3. *-lft-identity 68.3%

         \[\leadsto 0 + \frac{1}{\cos \left(\frac{\color{blue}{\frac{1}{y}}}{\frac{2}{x}}\right)} \]

   9. Simplified 68.3%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{1}{y}}{\frac{2}{x}}\right)}} \]

   if 2e52 < (/.f64 x (*.f64 y 2))

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 10.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{\frac{1}{y}}{\frac{2}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 55.1% accurate, 1.9× speedup

Math

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

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 (/ 1.0 (/ 2.0 (/ x y))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos((1.0 / (2.0 / (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((1.0d0 / (2.0d0 / (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((1.0 / (2.0 / (x / y))));
}

Python

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

Julia

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

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((1.0 / (2.0 / (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[(1.0 / N[(2.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

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

   1. add-log-exp 47.0%

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

   2. *-un-lft-identity 47.0%

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

   3. log-prod 47.0%

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

   4. metadata-eval 47.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

   5. add-log-exp 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   6. div-inv 47.0%

      \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   7. tan-quot 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. associate-*l/ 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

   9. pow1 47.0%

      \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   10. inv-pow 47.0%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   11. pow-prod-up 56.4%

       \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   12. metadata-eval 56.4%

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

   13. metadata-eval 56.4%

       \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   14. div-inv 56.4%

       \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

   15. *-commutative 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

   16. associate-/r* 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

   17. metadata-eval 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

3. Applied egg-rr 56.4%

   \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]
4. Step-by-step derivation

   1. clear-num 56.4%

      \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{1}{\frac{y}{0.5}}}\right)} \]

   2. div-inv 56.4%

      \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{y \cdot \frac{1}{0.5}}}\right)} \]

   3. metadata-eval 56.4%

      \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{y \cdot \color{blue}{2}}\right)} \]

   4. div-inv 56.4%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{x}{y \cdot 2}\right)}} \]

   5. associate-/r* 56.4%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}} \]

   6. clear-num 56.6%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]

5. Applied egg-rr 56.6%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}} \]

6. Final simplification 56.6%

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

Alternative 5: 55.1% 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 47.0%

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

2. Taylor expanded in x around inf 56.4%

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

3. Final simplification 56.4%

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

Alternative 6: 55.2% accurate, 2.0× speedup

Math

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

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 (* x (/ 0.5 y)))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos((x * (0.5 / 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((x * (0.5d0 / 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((x * (0.5 / y)));
}

Python

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

Julia

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

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((x * (0.5 / 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[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

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

   1. add-log-exp 47.0%

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

   2. *-un-lft-identity 47.0%

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

   3. log-prod 47.0%

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

   4. metadata-eval 47.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \]

   5. add-log-exp 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   6. div-inv 47.0%

      \[\leadsto 0 + \color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}} \]

   7. tan-quot 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. associate-*l/ 47.0%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}} \]

   9. pow1 47.0%

      \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   10. inv-pow 47.0%

       \[\leadsto 0 + \frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   11. pow-prod-up 56.4%

       \[\leadsto 0 + \frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   12. metadata-eval 56.4%

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

   13. metadata-eval 56.4%

       \[\leadsto 0 + \frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)} \]

   14. div-inv 56.4%

       \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}} \]

   15. *-commutative 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)} \]

   16. associate-/r* 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)} \]

   17. metadata-eval 56.4%

       \[\leadsto 0 + \frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)} \]

3. Applied egg-rr 56.4%

   \[\leadsto \color{blue}{0 + \frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}} \]

4. Final simplification 56.4%

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

Alternative 7: 55.5% 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 47.0%

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

2. Taylor expanded in x around 0 55.8%

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

3. Final simplification 55.8%

   \[\leadsto 1 \]

Developer target: 55.5% 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 2023257 
(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)))))