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

Percentage Accurate: 43.8% → 55.3%

Time: 17.5s

Alternatives: 6

Speedup: 211.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 43.8% 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: 55.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(1.0 / N[Cos[N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Exp[N[(2.0 * N[Log[N[Power[N[(2.0 * y), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x / N[(2.0 * y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around inf 53.8%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

4. Applied egg-rr 54.4%

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

   1. div-inv 54.4%

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

   2. metadata-eval 54.4%

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

   3. clear-num 53.8%

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

   4. times-frac 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. *-commutative 53.8%

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

   7. add-cube-cbrt 53.9%

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

   8. add-cube-cbrt 54.4%

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

   9. times-frac 54.5%

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

   10. pow2 54.5%

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

   11. pow2 54.5%

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

   12. cbrt-div 54.7%

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

6. Applied egg-rr 54.7%

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

   1. add-exp-log 55.1%

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

   2. log-pow 26.7%

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

   3. *-commutative 26.7%

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

8. Applied egg-rr 26.7%

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

9. Final simplification 26.7%

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

Alternative 2: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \frac{1}{\cos \left(\sqrt[3]{\frac{x}{2 \cdot y}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{2}}{{\left(\sqrt[3]{2 \cdot y}\right)}^{2}}\right)} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(1.0 / N[Cos[N[(N[Power[N[(x / N[(2.0 * y), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[N[(2.0 * y), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around inf 53.8%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

4. Applied egg-rr 54.4%

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

   1. div-inv 54.4%

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

   2. metadata-eval 54.4%

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

   3. clear-num 53.8%

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

   4. times-frac 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. *-commutative 53.8%

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

   7. add-cube-cbrt 53.9%

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

   8. add-cube-cbrt 54.4%

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

   9. times-frac 54.5%

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

   10. pow2 54.5%

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

   11. pow2 54.5%

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

   12. cbrt-div 54.7%

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

6. Applied egg-rr 54.7%

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

7. Final simplification 54.7%

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

Alternative 3: 55.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around inf 53.8%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

4. Applied egg-rr 54.4%

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

   1. div-inv 54.4%

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

   2. metadata-eval 54.4%

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

   3. clear-num 53.8%

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

   4. times-frac 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. *-commutative 53.8%

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

   7. add-sqr-sqrt 26.5%

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

   8. add-cube-cbrt 26.4%

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

   9. times-frac 26.5%

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

   10. pow2 26.5%

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

6. Applied egg-rr 26.5%

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

   1. associate-*l/ 26.6%

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

8. Simplified 26.6%

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

   1. add-log-exp 26.6%

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

   2. div-inv 26.4%

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

   3. associate-*r/ 26.6%

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

   4. add-sqr-sqrt 54.0%

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

   5. *-commutative 54.0%

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

   6. pow-flip 54.4%

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

   7. *-commutative 54.4%

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

   8. metadata-eval 54.4%

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

10. Applied egg-rr 54.4%

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

11. Final simplification 54.4%

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

Alternative 4: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{2 \cdot y}\\ \frac{1}{\cos \left(\frac{x}{t_0} \cdot {t_0}^{-2}\right)} \end{array} \end{array} \]

FPCore

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

C

y = abs(y);
double code(double x, double y) {
	double t_0 = cbrt((2.0 * y));
	return 1.0 / cos(((x / t_0) * pow(t_0, -2.0)));
}

Java

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

Julia

y = abs(y)
function code(x, y)
	t_0 = cbrt(Float64(2.0 * y))
	return Float64(1.0 / cos(Float64(Float64(x / t_0) * (t_0 ^ -2.0))))
end

Wolfram

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

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around inf 53.8%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

4. Applied egg-rr 54.4%

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

   1. div-inv 54.4%

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

   2. metadata-eval 54.4%

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

   3. clear-num 53.8%

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

   4. times-frac 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. *-commutative 53.8%

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

   7. add-sqr-sqrt 26.5%

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

   8. add-cube-cbrt 26.4%

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

   9. times-frac 26.5%

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

   10. pow2 26.5%

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

6. Applied egg-rr 26.5%

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

   1. associate-*l/ 26.6%

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

8. Simplified 26.6%

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

   1. expm1-log1p-u 26.6%

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

   2. expm1-udef 26.6%

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

10. Applied egg-rr 54.4%

    \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(\frac{x}{\sqrt[3]{2 \cdot y}} \cdot {\left(\sqrt[3]{2 \cdot y}\right)}^{-2}\right)\right)} - 1}} \]
11. Step-by-step derivation

    1. expm1-def 54.4%

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

    2. expm1-log1p 54.4%

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

    3. *-commutative 54.4%

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

12. Simplified 54.4%

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

13. Final simplification 54.4%

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

Alternative 5: 55.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 42.1%

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

2. Taylor expanded in x around inf 53.8%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

4. Applied egg-rr 54.4%

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

5. Final simplification 54.4%

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

Alternative 6: 55.5% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around 0 54.0%

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

3. Final simplification 54.0%

   \[\leadsto 1 \]

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 2023238 
(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)))))