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

Percentage Accurate: 44.4% → 56.5%

Time: 20.9s

Alternatives: 8

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.4% 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: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 2.4:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{\sqrt{x}}{\sqrt{y \cdot 2}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625} + -1}\\ \end{array} \end{array} \]

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 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.4)
     (/ 1.0 (cos (pow (/ (sqrt x) (sqrt (* y 2.0))) 2.0)))
     (/ 1.0 (+ (exp (+ (log 2.0) (* (/ (* x x) (* y y)) -0.0625))) -1.0)))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 2.4) {
		tmp = 1.0 / cos(pow((sqrt(x) / sqrt((y * 2.0))), 2.0));
	} else {
		tmp = 1.0 / (exp((log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -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) :: t_0
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 2.4d0) then
        tmp = 1.0d0 / cos(((sqrt(x) / sqrt((y * 2.0d0))) ** 2.0d0))
    else
        tmp = 1.0d0 / (exp((log(2.0d0) + (((x * x) / (y * y)) * (-0.0625d0)))) + (-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 t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 2.4) {
		tmp = 1.0 / Math.cos(Math.pow((Math.sqrt(x) / Math.sqrt((y * 2.0))), 2.0));
	} else {
		tmp = 1.0 / (Math.exp((Math.log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -1.0);
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	t_0 = x / (y * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 2.4:
		tmp = 1.0 / math.cos(math.pow((math.sqrt(x) / math.sqrt((y * 2.0))), 2.0))
	else:
		tmp = 1.0 / (math.exp((math.log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -1.0)
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 2.4)
		tmp = Float64(1.0 / cos((Float64(sqrt(x) / sqrt(Float64(y * 2.0))) ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64(exp(Float64(log(2.0) + Float64(Float64(Float64(x * x) / Float64(y * y)) * -0.0625))) + -1.0));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 2.4)
		tmp = 1.0 / cos(((sqrt(x) / sqrt((y * 2.0))) ^ 2.0));
	else
		tmp = 1.0 / (exp((log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -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_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.4], N[(1.0 / N[Cos[N[Power[N[(N[Sqrt[x], $MachinePrecision] / N[Sqrt[N[(y * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 2.39999999999999991

   1. Initial program 64.2%

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

      1. add-log-exp 64.2%

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

      2. *-un-lft-identity 64.2%

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

      3. log-prod 64.2%

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

      4. metadata-eval 64.2%

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

      5. add-log-exp 64.2%

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

      6. div-inv 63.5%

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

      7. tan-quot 63.5%

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

      8. associate-*l/ 63.5%

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

      9. pow1 63.5%

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

      10. inv-pow 63.5%

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

      11. pow-prod-up 64.2%

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

      12. metadata-eval 64.2%

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

      13. metadata-eval 64.2%

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

      14. div-inv 64.1%

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

      15. *-commutative 64.1%

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

      16. associate-/r* 64.1%

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

      17. metadata-eval 64.1%

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

   3. Applied egg-rr 64.1%

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

      1. clear-num 64.1%

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

      2. div-inv 64.1%

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

      3. metadata-eval 64.1%

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

      4. div-inv 64.2%

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

      5. *-un-lft-identity 64.2%

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

      6. *-commutative 64.2%

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

      7. times-frac 64.2%

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

      8. metadata-eval 64.2%

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

      9. add-sqr-sqrt 26.6%

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

      10. pow2 26.6%

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

   5. Applied egg-rr 26.6%

      \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)}^{2}\right)}} \]
   6. Step-by-step derivation

      1. metadata-eval 26.6%

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

      2. associate-/r/ 26.7%

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

      3. clear-num 26.6%

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

      4. associate-/l/ 26.6%

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

      5. sqrt-div 11.7%

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

   7. Applied egg-rr 11.7%

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

   if 2.39999999999999991 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 1.5%

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

   2. Taylor expanded in x around inf 48.7%

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

      1. expm1-log1p-u 48.7%

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

      2. expm1-udef 48.7%

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

   4. Applied egg-rr 48.7%

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

   5. Taylor expanded in x around 0 54.2%

      \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}} - 1} \]
   6. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto \frac{1}{e^{\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625}} - 1} \]

      2. unpow2 54.2%

         \[\leadsto \frac{1}{e^{\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.0625} - 1} \]

      3. unpow2 54.2%

         \[\leadsto \frac{1}{e^{\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.0625} - 1} \]

   7. Simplified 54.2%

      \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625}} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2.4:\\ \;\;\;\;\frac{1}{\cos \left({\left(\frac{\sqrt{x}}{\sqrt{y \cdot 2}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625} + -1}\\ \end{array} \]

Alternative 2: 55.2% accurate, 0.7× speedup

Math

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

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 (pow (/ 1.0 (sqrt (* 2.0 (/ y x)))) 2.0))))

C

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

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 / cos(((1.0d0 / sqrt((2.0d0 * (y / x)))) ** 2.0d0))
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.pow((1.0 / Math.sqrt((2.0 * (y / x)))), 2.0));
}

Python

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

Julia

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

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos(((1.0 / sqrt((2.0 * (y / x)))) ^ 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_] := N[(1.0 / N[Cos[N[Power[N[(1.0 / N[Sqrt[N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $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 46.5%

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

   7. tan-quot 46.5%

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

   8. associate-*l/ 46.5%

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

   9. pow1 46.5%

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

   10. inv-pow 46.5%

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

   11. pow-prod-up 59.9%

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

   12. metadata-eval 59.9%

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

   13. metadata-eval 59.9%

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

   14. div-inv 59.9%

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

   15. *-commutative 59.9%

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

   16. associate-/r* 59.9%

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

   17. metadata-eval 59.9%

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

3. Applied egg-rr 59.9%

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

   1. clear-num 59.9%

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

   2. div-inv 59.9%

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

   3. metadata-eval 59.9%

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

   4. div-inv 59.9%

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

   5. *-un-lft-identity 59.9%

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

   6. *-commutative 59.9%

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

   7. times-frac 59.9%

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

   8. metadata-eval 59.9%

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

   9. add-sqr-sqrt 32.4%

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

   10. pow2 32.4%

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

5. Applied egg-rr 32.4%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)}^{2}\right)}} \]
6. Step-by-step derivation

   1. metadata-eval 32.4%

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

   2. associate-/r/ 32.5%

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

   3. clear-num 32.5%

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

   4. sqrt-div 27.7%

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

   5. metadata-eval 27.7%

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

   6. /-rgt-identity 27.7%

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

   7. associate-/r/ 27.7%

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

   8. *-commutative 27.7%

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

7. Applied egg-rr 27.7%

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

   1. *-commutative 27.7%

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

   2. associate-*l/ 27.9%

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

   3. associate-*r/ 27.9%

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

9. Simplified 27.9%

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

10. Final simplification 27.9%

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

Alternative 3: 55.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \frac{1}{\cos \left({\left(\sqrt{x \cdot \frac{0.5}{y}}\right)}^{2}\right)} \end{array} \]

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 (pow (sqrt (* x (/ 0.5 y))) 2.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	return 1.0 / cos(pow(sqrt((x * (0.5 / y))), 2.0));
}

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 / cos((sqrt((x * (0.5d0 / y))) ** 2.0d0))
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.pow(Math.sqrt((x * (0.5 / y))), 2.0));
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return 1.0 / math.cos(math.pow(math.sqrt((x * (0.5 / y))), 2.0))

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	return Float64(1.0 / cos((sqrt(Float64(x * Float64(0.5 / y))) ^ 2.0)))
end

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((sqrt((x * (0.5 / y))) ^ 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_] := N[(1.0 / N[Cos[N[Power[N[Sqrt[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $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 46.5%

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

   7. tan-quot 46.5%

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

   8. associate-*l/ 46.5%

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

   9. pow1 46.5%

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

   10. inv-pow 46.5%

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

   11. pow-prod-up 59.9%

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

   12. metadata-eval 59.9%

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

   13. metadata-eval 59.9%

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

   14. div-inv 59.9%

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

   15. *-commutative 59.9%

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

   16. associate-/r* 59.9%

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

   17. metadata-eval 59.9%

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

3. Applied egg-rr 59.9%

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

   1. clear-num 59.9%

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

   2. div-inv 59.9%

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

   3. metadata-eval 59.9%

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

   4. div-inv 59.9%

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

   5. *-un-lft-identity 59.9%

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

   6. *-commutative 59.9%

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

   7. times-frac 59.9%

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

   8. metadata-eval 59.9%

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

   9. add-sqr-sqrt 32.4%

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

   10. pow2 32.4%

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

5. Applied egg-rr 32.4%

   \[\leadsto 0 + \frac{1}{\cos \color{blue}{\left({\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)}^{2}\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 32.0%

      \[\leadsto 0 + \frac{1}{\cos \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)\right)\right)}}^{2}\right)} \]

   2. expm1-udef 32.1%

      \[\leadsto 0 + \frac{1}{\cos \left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \frac{x}{y}}\right)} - 1\right)}}^{2}\right)} \]

   3. metadata-eval 32.1%

      \[\leadsto 0 + \frac{1}{\cos \left({\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{2}} \cdot \frac{x}{y}}\right)} - 1\right)}^{2}\right)} \]

   4. associate-/r/ 32.1%

      \[\leadsto 0 + \frac{1}{\cos \left({\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{\frac{2}{\frac{x}{y}}}}}\right)} - 1\right)}^{2}\right)} \]

   5. clear-num 32.1%

      \[\leadsto 0 + \frac{1}{\cos \left({\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{\frac{x}{y}}{2}}}\right)} - 1\right)}^{2}\right)} \]

   6. associate-/l/ 32.1%

      \[\leadsto 0 + \frac{1}{\cos \left({\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{x}{2 \cdot y}}}\right)} - 1\right)}^{2}\right)} \]

7. Applied egg-rr 32.1%

   \[\leadsto 0 + \frac{1}{\cos \left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{x}{2 \cdot y}}\right)} - 1\right)}}^{2}\right)} \]
8. Step-by-step derivation

   1. expm1-def 32.0%

      \[\leadsto 0 + \frac{1}{\cos \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x}{2 \cdot y}}\right)\right)\right)}}^{2}\right)} \]

   2. expm1-log1p 32.4%

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

   3. associate-/r* 32.4%

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

   4. metadata-eval 32.4%

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

   5. associate-/l* 32.4%

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

   6. /-rgt-identity 32.4%

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

   7. associate-*r/ 32.5%

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

9. Simplified 32.5%

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

10. Final simplification 32.5%

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

Alternative 4: 56.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625} + -1}\\ \end{array} \end{array} \]

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+140)
   (/ 1.0 (cos (/ 1.0 (/ 2.0 (/ x y)))))
   (/ 1.0 (+ (exp (+ (log 2.0) (* (/ (* x x) (* y y)) -0.0625))) -1.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x / (y * 2.0)) <= 2e+140) {
		tmp = 1.0 / cos((1.0 / (2.0 / (x / y))));
	} else {
		tmp = 1.0 / (exp((log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -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+140) then
        tmp = 1.0d0 / cos((1.0d0 / (2.0d0 / (x / y))))
    else
        tmp = 1.0d0 / (exp((log(2.0d0) + (((x * x) / (y * y)) * (-0.0625d0)))) + (-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+140) {
		tmp = 1.0 / Math.cos((1.0 / (2.0 / (x / y))));
	} else {
		tmp = 1.0 / (Math.exp((Math.log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -1.0);
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if (x / (y * 2.0)) <= 2e+140:
		tmp = 1.0 / math.cos((1.0 / (2.0 / (x / y))))
	else:
		tmp = 1.0 / (math.exp((math.log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -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+140)
		tmp = Float64(1.0 / cos(Float64(1.0 / Float64(2.0 / Float64(x / y)))));
	else
		tmp = Float64(1.0 / Float64(exp(Float64(log(2.0) + Float64(Float64(Float64(x * x) / Float64(y * y)) * -0.0625))) + -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+140)
		tmp = 1.0 / cos((1.0 / (2.0 / (x / y))));
	else
		tmp = 1.0 / (exp((log(2.0) + (((x * x) / (y * y)) * -0.0625))) + -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+140], N[(1.0 / N[Cos[N[(1.0 / N[(2.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y 2)) < 2.00000000000000012e140

   1. Initial program 52.3%

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

      1. add-log-exp 52.3%

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

      2. *-un-lft-identity 52.3%

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

      3. log-prod 52.3%

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

      4. metadata-eval 52.3%

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

      5. add-log-exp 52.3%

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

      6. div-inv 51.8%

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

      7. tan-quot 51.8%

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

      8. associate-*l/ 51.8%

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

      9. pow1 51.8%

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

      10. inv-pow 51.8%

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

      11. pow-prod-up 66.8%

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

      12. metadata-eval 66.8%

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

      13. metadata-eval 66.8%

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

      14. div-inv 66.9%

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

      15. *-commutative 66.9%

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

      16. associate-/r* 66.9%

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

      17. metadata-eval 66.9%

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

   3. Applied egg-rr 66.9%

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

      1. clear-num 66.9%

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

      2. div-inv 66.9%

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

      3. metadata-eval 66.9%

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

      4. div-inv 66.8%

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

      5. associate-/r* 66.8%

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

      6. clear-num 66.9%

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

   5. Applied egg-rr 66.9%

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

   if 2.00000000000000012e140 < (/.f64 x (*.f64 y 2))

   1. Initial program 5.7%

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

   2. Taylor expanded in x around inf 5.7%

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

      1. expm1-log1p-u 5.7%

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

      2. expm1-udef 5.7%

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

   4. Applied egg-rr 5.7%

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

   5. Taylor expanded in x around 0 9.4%

      \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}} - 1} \]
   6. Step-by-step derivation

      1. *-commutative 9.4%

         \[\leadsto \frac{1}{e^{\log 2 + \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625}} - 1} \]

      2. unpow2 9.4%

         \[\leadsto \frac{1}{e^{\log 2 + \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot -0.0625} - 1} \]

      3. unpow2 9.4%

         \[\leadsto \frac{1}{e^{\log 2 + \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot -0.0625} - 1} \]

   7. Simplified 9.4%

      \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625}} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{\frac{2}{\frac{x}{y}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\log 2 + \frac{x \cdot x}{y \cdot y} \cdot -0.0625} + -1}\\ \end{array} \]

Alternative 5: 55.3% 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 46.5%

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

   7. tan-quot 46.5%

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

   8. associate-*l/ 46.5%

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

   9. pow1 46.5%

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

   10. inv-pow 46.5%

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

   11. pow-prod-up 59.9%

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

   12. metadata-eval 59.9%

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

   13. metadata-eval 59.9%

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

   14. div-inv 59.9%

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

   15. *-commutative 59.9%

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

   16. associate-/r* 59.9%

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

   17. metadata-eval 59.9%

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

3. Applied egg-rr 59.9%

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

   1. clear-num 59.9%

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

   2. div-inv 59.9%

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

   3. metadata-eval 59.9%

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

   4. div-inv 59.9%

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

   5. associate-/r* 59.9%

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

   6. clear-num 60.0%

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

5. Applied egg-rr 60.0%

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

6. Final simplification 60.0%

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

Alternative 6: 55.3% 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 59.9%

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

3. Final simplification 59.9%

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

Alternative 7: 55.3% accurate, 2.0× speedup

Math

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

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 (/ y x)))))

C

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

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 / (y / x)))
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 / (y / x)));
}

Python

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

Julia

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

MATLAB

x = abs(x)
y = abs(y)
function tmp = code(x, y)
	tmp = 1.0 / cos((0.5 / (y / 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[(1.0 / N[Cos[N[(0.5 / N[(y / x), $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 46.5%

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

   7. tan-quot 46.5%

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

   8. associate-*l/ 46.5%

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

   9. pow1 46.5%

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

   10. inv-pow 46.5%

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

   11. pow-prod-up 59.9%

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

   12. metadata-eval 59.9%

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

   13. metadata-eval 59.9%

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

   14. div-inv 59.9%

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

   15. *-commutative 59.9%

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

   16. associate-/r* 59.9%

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

   17. metadata-eval 59.9%

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

3. Applied egg-rr 59.9%

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

   1. clear-num 59.9%

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

   2. div-inv 59.9%

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

   3. metadata-eval 59.9%

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

   4. div-inv 59.9%

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

   5. *-un-lft-identity 59.9%

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

   6. *-commutative 59.9%

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

   7. times-frac 59.9%

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

   8. metadata-eval 59.9%

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

   9. clear-num 60.0%

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

   10. un-div-inv 60.0%

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

5. Applied egg-rr 60.0%

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

6. Final simplification 60.0%

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

Alternative 8: 55.2% 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 59.2%

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

3. Final simplification 59.2%

   \[\leadsto 1 \]

Developer target: 55.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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)))))