Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.8s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \frac{{k}^{-0.5} \cdot \sqrt{t\_0}}{{t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (/ (* (pow k -0.5) (sqrt t_0)) (pow t_0 (* k 0.5)))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	return (pow(k, -0.5) * sqrt(t_0)) / pow(t_0, (k * 0.5));
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	return (Math.pow(k, -0.5) * Math.sqrt(t_0)) / Math.pow(t_0, (k * 0.5));
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	return (math.pow(k, -0.5) * math.sqrt(t_0)) / math.pow(t_0, (k * 0.5))

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	return Float64(Float64((k ^ -0.5) * sqrt(t_0)) / (t_0 ^ Float64(k * 0.5)))
end

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = ((k ^ -0.5) * sqrt(t_0)) / (t_0 ^ (k * 0.5));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \]

   2. div-sub 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}} \]

   3. metadata-eval 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)} \]

   4. pow-div 99.6%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{0.5}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}} \]

   5. pow1/2 99.6%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\pi \cdot n\right)}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   6. associate-*r/ 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{k}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}} \]

   7. pow1/2 99.6%

      \[\leadsto \frac{\frac{1}{\color{blue}{{k}^{0.5}}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   8. pow-flip 99.7%

      \[\leadsto \frac{\color{blue}{{k}^{\left(-0.5\right)}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   9. metadata-eval 99.7%

      \[\leadsto \frac{{k}^{\color{blue}{-0.5}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   10. div-inv 99.7%

       \[\leadsto \frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(k \cdot \frac{1}{2}\right)}}} \]

   11. metadata-eval 99.7%

       \[\leadsto \frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot \color{blue}{0.5}\right)}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k \cdot {t\_0}^{k}}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n)))) (/ (sqrt t_0) (sqrt (* k (pow t_0 k))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	return sqrt(t_0) / sqrt((k * pow(t_0, k)));
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	return Math.sqrt(t_0) / Math.sqrt((k * Math.pow(t_0, k)));
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	return math.sqrt(t_0) / math.sqrt((k * math.pow(t_0, k)))

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	return Float64(sqrt(t_0) / sqrt(Float64(k * (t_0 ^ k))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = sqrt(t_0) / sqrt((k * (t_0 ^ k)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(k * N[Power[t$95$0, k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \]

   2. div-sub 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}} \]

   3. metadata-eval 99.4%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)} \]

   4. pow-div 99.6%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{0.5}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}} \]

   5. pow1/2 99.6%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\pi \cdot n\right)}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   6. associate-*r/ 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{k}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}} \]

   7. pow1/2 99.6%

      \[\leadsto \frac{\frac{1}{\color{blue}{{k}^{0.5}}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   8. pow-flip 99.7%

      \[\leadsto \frac{\color{blue}{{k}^{\left(-0.5\right)}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   9. metadata-eval 99.7%

      \[\leadsto \frac{{k}^{\color{blue}{-0.5}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \]

   10. div-inv 99.7%

       \[\leadsto \frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(k \cdot \frac{1}{2}\right)}}} \]

   11. metadata-eval 99.7%

       \[\leadsto \frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot \color{blue}{0.5}\right)}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}} \]
5. Step-by-step derivation

   1. pow1 99.7%

      \[\leadsto \frac{\color{blue}{{\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)}^{1}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   2. *-commutative 99.7%

      \[\leadsto \frac{{\color{blue}{\left(\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\right)}}^{1}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   3. associate-*r* 99.7%

      \[\leadsto \frac{{\left(\sqrt{\color{blue}{\left(2 \cdot \pi\right) \cdot n}} \cdot {k}^{-0.5}\right)}^{1}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   4. *-commutative 99.7%

      \[\leadsto \frac{{\left(\sqrt{\color{blue}{n \cdot \left(2 \cdot \pi\right)}} \cdot {k}^{-0.5}\right)}^{1}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot {k}^{-0.5}\right)}^{1}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
7. Step-by-step derivation

   1. unpow1 99.7%

      \[\leadsto \frac{\color{blue}{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot {k}^{-0.5}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   2. exp-to-pow 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \color{blue}{e^{\log k \cdot -0.5}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   3. *-commutative 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot e^{\color{blue}{-0.5 \cdot \log k}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   4. metadata-eval 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot e^{\color{blue}{\left(-0.5\right)} \cdot \log k}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   5. distribute-lft-neg-in 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot e^{\color{blue}{-0.5 \cdot \log k}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   6. *-commutative 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot e^{-\color{blue}{\log k \cdot 0.5}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   7. exp-neg 96.1%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \color{blue}{\frac{1}{e^{\log k \cdot 0.5}}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   8. exp-to-pow 99.6%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \frac{1}{\color{blue}{{k}^{0.5}}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   9. unpow1/2 99.6%

      \[\leadsto \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \frac{1}{\color{blue}{\sqrt{k}}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   10. associate-/l* 99.7%

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot 1}{\sqrt{k}}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   11. *-rgt-identity 99.7%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{n \cdot \left(2 \cdot \pi\right)}}}{\sqrt{k}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

   12. *-commutative 99.7%

       \[\leadsto \frac{\frac{\sqrt{n \cdot \color{blue}{\left(\pi \cdot 2\right)}}}{\sqrt{k}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]

8. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. associate-/l/ 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)} \cdot \sqrt{k}}} \]

   2. div-inv 99.6%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)} \cdot \sqrt{k}}} \]

   3. pow-unpow 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{\color{blue}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}\right)}^{0.5}} \cdot \sqrt{k}} \]

   4. pow1/2 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}\right)}^{0.5} \cdot \color{blue}{{k}^{0.5}}} \]

   5. pow-prod-down 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{\color{blue}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]

   6. *-commutative 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left({\color{blue}{\left(\left(\pi \cdot n\right) \cdot 2\right)}}^{k} \cdot k\right)}^{0.5}} \]

   7. *-commutative 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left({\left(\color{blue}{\left(n \cdot \pi\right)} \cdot 2\right)}^{k} \cdot k\right)}^{0.5}} \]

   8. associate-*r* 99.6%

      \[\leadsto \sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left({\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{k} \cdot k\right)}^{0.5}} \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot \frac{1}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]
11. Step-by-step derivation

    1. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)} \cdot 1}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{\color{blue}{\sqrt{n \cdot \left(\pi \cdot 2\right)}}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}} \]

    3. associate-*r* 99.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}} \]

    4. *-commutative 99.7%

       \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}} \]

    5. associate-*r* 99.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}} \]

    6. *-commutative 99.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \pi}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k\right)}^{0.5}} \]

    7. unpow1/2 99.7%

       \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\color{blue}{\sqrt{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k} \cdot k}}} \]

    8. *-commutative 99.7%

       \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{\color{blue}{k \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{k}}}} \]

    9. associate-*r* 99.7%

       \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k \cdot {\color{blue}{\left(\left(n \cdot \pi\right) \cdot 2\right)}}^{k}}} \]

    10. *-commutative 99.7%

        \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k \cdot {\color{blue}{\left(2 \cdot \left(n \cdot \pi\right)\right)}}^{k}}} \]

    11. associate-*r* 99.7%

        \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k \cdot {\color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}^{k}}} \]

    12. *-commutative 99.7%

        \[\leadsto \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k \cdot {\left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}^{k}}} \]

12. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k \cdot {\left(\left(n \cdot 2\right) \cdot \pi\right)}^{k}}}} \]

13. Final simplification 99.7%

    \[\leadsto \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}}} \]
14. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.2e-50)
   (* (sqrt (* PI (* 2.0 n))) (sqrt (/ 1.0 k)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.2e-50) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) * sqrt((1.0 / k));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7.2e-50) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) * Math.sqrt((1.0 / k));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 7.2e-50:
		tmp = math.sqrt((math.pi * (2.0 * n))) * math.sqrt((1.0 / k))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.2e-50)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) * sqrt(Float64(1.0 / k)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 7.2e-50)
		tmp = sqrt((pi * (2.0 * n))) * sqrt((1.0 / k));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.2e-50], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999958e-50

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 68.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 68.6%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 68.6%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 68.6%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 68.6%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 68.8%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. metadata-eval 68.8%

         \[\leadsto {\left(\color{blue}{\frac{2}{1}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5} \]

      2. associate-*r/ 68.8%

         \[\leadsto {\left(\frac{2}{1} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{0.5} \]

      3. *-commutative 68.8%

         \[\leadsto {\left(\frac{2}{1} \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{0.5} \]

      4. times-frac 68.8%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(\pi \cdot n\right)}{1 \cdot k}\right)}}^{0.5} \]

      5. associate-*r* 68.8%

         \[\leadsto {\left(\frac{\color{blue}{\left(2 \cdot \pi\right) \cdot n}}{1 \cdot k}\right)}^{0.5} \]

      6. *-commutative 68.8%

         \[\leadsto {\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{1 \cdot k}\right)}^{0.5} \]

      7. *-un-lft-identity 68.8%

         \[\leadsto {\left(\frac{n \cdot \left(2 \cdot \pi\right)}{\color{blue}{k}}\right)}^{0.5} \]

      8. clear-num 68.7%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(2 \cdot \pi\right)}}\right)}}^{0.5} \]

      9. *-commutative 68.7%

         \[\leadsto {\left(\frac{1}{\frac{k}{n \cdot \color{blue}{\left(\pi \cdot 2\right)}}}\right)}^{0.5} \]

   9. Applied egg-rr 68.7%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}}^{0.5} \]
   10. Step-by-step derivation

       1. associate-/r/ 68.7%

          \[\leadsto {\color{blue}{\left(\frac{1}{k} \cdot \left(n \cdot \left(\pi \cdot 2\right)\right)\right)}}^{0.5} \]

       2. unpow-prod-down 99.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{k}\right)}^{0.5} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{0.5}} \]

       3. *-commutative 99.5%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot {\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{0.5} \]

       4. associate-*l* 99.5%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot {\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{0.5} \]

       5. pow1/2 99.5%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot \color{blue}{\sqrt{\left(n \cdot 2\right) \cdot \pi}} \]

   11. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{{\left(\frac{1}{k}\right)}^{0.5} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi}} \]
   12. Step-by-step derivation

       1. unpow1/2 99.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi} \]

       2. *-commutative 99.5%

          \[\leadsto \sqrt{\frac{1}{k}} \cdot \sqrt{\color{blue}{\pi \cdot \left(n \cdot 2\right)}} \]

   13. Simplified 99.5%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}} \]

   if 7.19999999999999958e-50 < k

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}}} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \color{blue}{\left(n \cdot \pi\right)}\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]

      2. distribute-lft-in 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\color{blue}{\left(2 \cdot 0.5 + 2 \cdot \left(k \cdot -0.5\right)\right)}}}{k}} \]

      3. metadata-eval 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(\color{blue}{1} + 2 \cdot \left(k \cdot -0.5\right)\right)}}{k}} \]

      4. *-commutative 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 + 2 \cdot \color{blue}{\left(-0.5 \cdot k\right)}\right)}}{k}} \]

      5. associate-*r* 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 + \color{blue}{\left(2 \cdot -0.5\right) \cdot k}\right)}}{k}} \]

      6. metadata-eval 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 + \color{blue}{-1} \cdot k\right)}}{k}} \]

      7. neg-mul-1 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 + \color{blue}{\left(-k\right)}\right)}}{k}} \]

      8. sub-neg 99.6%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\color{blue}{\left(1 - k\right)}}}{k}} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2e+250)
   (* (sqrt (* PI (* 2.0 n))) (sqrt (/ 1.0 k)))
   (cbrt (pow (* n (* PI (/ 2.0 k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.2e+250) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) * sqrt((1.0 / k));
	} else {
		tmp = cbrt(pow((n * (((double) M_PI) * (2.0 / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.2e+250) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) * Math.sqrt((1.0 / k));
	} else {
		tmp = Math.cbrt(Math.pow((n * (Math.PI * (2.0 / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.2e+250)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) * sqrt(Float64(1.0 / k)));
	else
		tmp = cbrt((Float64(n * Float64(pi * Float64(2.0 / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.2e+250], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1999999999999997e250

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 41.3%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 41.3%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 41.3%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 41.3%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 41.3%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 41.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. metadata-eval 41.4%

         \[\leadsto {\left(\color{blue}{\frac{2}{1}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5} \]

      2. associate-*r/ 41.4%

         \[\leadsto {\left(\frac{2}{1} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{0.5} \]

      3. *-commutative 41.4%

         \[\leadsto {\left(\frac{2}{1} \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{0.5} \]

      4. times-frac 41.4%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(\pi \cdot n\right)}{1 \cdot k}\right)}}^{0.5} \]

      5. associate-*r* 41.4%

         \[\leadsto {\left(\frac{\color{blue}{\left(2 \cdot \pi\right) \cdot n}}{1 \cdot k}\right)}^{0.5} \]

      6. *-commutative 41.4%

         \[\leadsto {\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{1 \cdot k}\right)}^{0.5} \]

      7. *-un-lft-identity 41.4%

         \[\leadsto {\left(\frac{n \cdot \left(2 \cdot \pi\right)}{\color{blue}{k}}\right)}^{0.5} \]

      8. clear-num 41.4%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(2 \cdot \pi\right)}}\right)}}^{0.5} \]

      9. *-commutative 41.4%

         \[\leadsto {\left(\frac{1}{\frac{k}{n \cdot \color{blue}{\left(\pi \cdot 2\right)}}}\right)}^{0.5} \]

   9. Applied egg-rr 41.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}}^{0.5} \]
   10. Step-by-step derivation

       1. associate-/r/ 41.3%

          \[\leadsto {\color{blue}{\left(\frac{1}{k} \cdot \left(n \cdot \left(\pi \cdot 2\right)\right)\right)}}^{0.5} \]

       2. unpow-prod-down 56.0%

          \[\leadsto \color{blue}{{\left(\frac{1}{k}\right)}^{0.5} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{0.5}} \]

       3. *-commutative 56.0%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot {\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{0.5} \]

       4. associate-*l* 56.0%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot {\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{0.5} \]

       5. pow1/2 56.0%

          \[\leadsto {\left(\frac{1}{k}\right)}^{0.5} \cdot \color{blue}{\sqrt{\left(n \cdot 2\right) \cdot \pi}} \]

   11. Applied egg-rr 56.0%

       \[\leadsto \color{blue}{{\left(\frac{1}{k}\right)}^{0.5} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi}} \]
   12. Step-by-step derivation

       1. unpow1/2 56.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi} \]

       2. *-commutative 56.0%

          \[\leadsto \sqrt{\frac{1}{k}} \cdot \sqrt{\color{blue}{\pi \cdot \left(n \cdot 2\right)}} \]

   13. Simplified 56.0%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}} \]

   if 3.1999999999999997e250 < k

   1. Initial program 100.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 3.2%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.2%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 3.2%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 3.2%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 3.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   8. Taylor expanded in n around 0 3.2%

      \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 3.2%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

      2. *-commutative 3.2%

         \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

      3. associate-/l* 3.2%

         \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

   10. Simplified 3.2%

       \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 36.7%

          \[\leadsto \color{blue}{\sqrt[3]{\left({\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}\right) \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}}} \]

       2. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(0.5 + 0.5\right)}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       3. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\color{blue}{1}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       4. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(1 + 0.5\right)}}} \]

       5. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\left(n \cdot \pi\right) \cdot 2}{k}\right)}}^{\left(1 + 0.5\right)}} \]

       6. associate-*r* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       7. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       8. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       9. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{\color{blue}{1.5}}} \]

   12. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{1.5}}} \]
   13. Step-by-step derivation

       1. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}\right)}^{1.5}} \]

       2. associate-/l* 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}}^{1.5}} \]

       3. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \frac{\color{blue}{\pi \cdot 2}}{k}\right)}^{1.5}} \]

       4. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \color{blue}{\left(\pi \cdot \frac{2}{k}\right)}\right)}^{1.5}} \]

   14. Simplified 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (sqrt (/ 1.0 k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return sqrt((1.0 / k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((1.0 / k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.sqrt((1.0 / k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(sqrt(Float64(1.0 / k)) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((1.0 / k)) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 99.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Final simplification 99.5%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+250}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.5e+250)
   (* (pow k -0.5) (sqrt (* PI (* 2.0 n))))
   (cbrt (pow (* n (* PI (/ 2.0 k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.5e+250) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = cbrt(pow((n * (((double) M_PI) * (2.0 / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.5e+250) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
	} else {
		tmp = Math.cbrt(Math.pow((n * (Math.PI * (2.0 / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.5e+250)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = cbrt((Float64(n * Float64(pi * Float64(2.0 / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.5e+250], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5000000000000004e250

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 41.3%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 41.3%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 41.3%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 41.3%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 41.3%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 41.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. metadata-eval 41.4%

         \[\leadsto {\left(\color{blue}{\frac{2}{1}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5} \]

      2. associate-*r/ 41.4%

         \[\leadsto {\left(\frac{2}{1} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{0.5} \]

      3. *-commutative 41.4%

         \[\leadsto {\left(\frac{2}{1} \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{0.5} \]

      4. times-frac 41.4%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(\pi \cdot n\right)}{1 \cdot k}\right)}}^{0.5} \]

      5. associate-*r* 41.4%

         \[\leadsto {\left(\frac{\color{blue}{\left(2 \cdot \pi\right) \cdot n}}{1 \cdot k}\right)}^{0.5} \]

      6. *-commutative 41.4%

         \[\leadsto {\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{1 \cdot k}\right)}^{0.5} \]

      7. *-un-lft-identity 41.4%

         \[\leadsto {\left(\frac{n \cdot \left(2 \cdot \pi\right)}{\color{blue}{k}}\right)}^{0.5} \]

      8. clear-num 41.4%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(2 \cdot \pi\right)}}\right)}}^{0.5} \]

      9. *-commutative 41.4%

         \[\leadsto {\left(\frac{1}{\frac{k}{n \cdot \color{blue}{\left(\pi \cdot 2\right)}}}\right)}^{0.5} \]

   9. Applied egg-rr 41.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}}^{0.5} \]
   10. Step-by-step derivation

       1. unpow1/2 41.4%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}}} \]

       2. associate-/r/ 41.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{k} \cdot \left(n \cdot \left(\pi \cdot 2\right)\right)}} \]

       3. sqrt-prod 56.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}} \]

       4. inv-pow 56.0%

          \[\leadsto \sqrt{\color{blue}{{k}^{-1}}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]

       5. metadata-eval 56.0%

          \[\leadsto \sqrt{{k}^{\color{blue}{\left(-0.5 + -0.5\right)}}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]

       6. pow-prod-up 55.9%

          \[\leadsto \sqrt{\color{blue}{{k}^{-0.5} \cdot {k}^{-0.5}}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]

       7. sqrt-unprod 55.8%

          \[\leadsto \color{blue}{\left(\sqrt{{k}^{-0.5}} \cdot \sqrt{{k}^{-0.5}}\right)} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]

       8. add-sqr-sqrt 55.9%

          \[\leadsto \color{blue}{{k}^{-0.5}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]

       9. *-commutative 55.9%

          \[\leadsto {k}^{-0.5} \cdot \sqrt{n \cdot \color{blue}{\left(2 \cdot \pi\right)}} \]

       10. associate-*l* 55.9%

           \[\leadsto {k}^{-0.5} \cdot \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \pi}} \]

   11. Applied egg-rr 55.9%

       \[\leadsto \color{blue}{{k}^{-0.5} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi}} \]

   if 6.5000000000000004e250 < k

   1. Initial program 100.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 3.2%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.2%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 3.2%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 3.2%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 3.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   8. Taylor expanded in n around 0 3.2%

      \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 3.2%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

      2. *-commutative 3.2%

         \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

      3. associate-/l* 3.2%

         \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

   10. Simplified 3.2%

       \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 36.7%

          \[\leadsto \color{blue}{\sqrt[3]{\left({\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}\right) \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}}} \]

       2. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(0.5 + 0.5\right)}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       3. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\color{blue}{1}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       4. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(1 + 0.5\right)}}} \]

       5. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\left(n \cdot \pi\right) \cdot 2}{k}\right)}}^{\left(1 + 0.5\right)}} \]

       6. associate-*r* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       7. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       8. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       9. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{\color{blue}{1.5}}} \]

   12. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{1.5}}} \]
   13. Step-by-step derivation

       1. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}\right)}^{1.5}} \]

       2. associate-/l* 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}}^{1.5}} \]

       3. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \frac{\color{blue}{\pi \cdot 2}}{k}\right)}^{1.5}} \]

       4. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \color{blue}{\left(\pi \cdot \frac{2}{k}\right)}\right)}^{1.5}} \]

   14. Simplified 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+250}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{2}{k}\\ \mathbf{if}\;k \leq 2.85 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot t\_0\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (/ 2.0 k))))
   (if (<= k 2.85e+250) (* (sqrt n) (sqrt t_0)) (cbrt (pow (* n t_0) 1.5)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 / k);
	double tmp;
	if (k <= 2.85e+250) {
		tmp = sqrt(n) * sqrt(t_0);
	} else {
		tmp = cbrt(pow((n * t_0), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 / k);
	double tmp;
	if (k <= 2.85e+250) {
		tmp = Math.sqrt(n) * Math.sqrt(t_0);
	} else {
		tmp = Math.cbrt(Math.pow((n * t_0), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 / k))
	tmp = 0.0
	if (k <= 2.85e+250)
		tmp = Float64(sqrt(n) * sqrt(t_0));
	else
		tmp = cbrt((Float64(n * t_0) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.85e+250], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * t$95$0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.84999999999999993e250

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 41.3%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 41.3%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 41.3%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 41.3%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 41.3%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 41.3%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 41.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   8. Taylor expanded in n around 0 41.4%

      \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 41.4%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

      2. *-commutative 41.4%

         \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

      3. associate-/l* 41.3%

         \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

   10. Simplified 41.3%

       \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*l* 41.4%

          \[\leadsto {\color{blue}{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}}^{0.5} \]

       2. unpow-prod-down 55.9%

          \[\leadsto \color{blue}{{n}^{0.5} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5}} \]

       3. pow1/2 55.9%

          \[\leadsto \color{blue}{\sqrt{n}} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5} \]

   12. Applied egg-rr 55.9%

       \[\leadsto \color{blue}{\sqrt{n} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5}} \]
   13. Step-by-step derivation

       1. unpow1/2 55.9%

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\pi \cdot \frac{2}{k}}} \]

   14. Simplified 55.9%

       \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}} \]

   if 2.84999999999999993e250 < k

   1. Initial program 100.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 3.2%

         \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative 3.2%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

      2. pow1/2 3.2%

         \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

      3. pow1/2 3.2%

         \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

      4. pow-prod-down 3.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

   8. Taylor expanded in n around 0 3.2%

      \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 3.2%

         \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

      2. *-commutative 3.2%

         \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

      3. associate-/l* 3.2%

         \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

   10. Simplified 3.2%

       \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 36.7%

          \[\leadsto \color{blue}{\sqrt[3]{\left({\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}\right) \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}}} \]

       2. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(0.5 + 0.5\right)}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       3. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\color{blue}{1}} \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

       4. pow-prod-up 36.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(1 + 0.5\right)}}} \]

       5. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\left(n \cdot \pi\right) \cdot 2}{k}\right)}}^{\left(1 + 0.5\right)}} \]

       6. associate-*r* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       7. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       8. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       9. metadata-eval 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{\color{blue}{1.5}}} \]

   12. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}^{1.5}}} \]
   13. Step-by-step derivation

       1. associate-*l* 36.7%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}\right)}^{1.5}} \]

       2. associate-/l* 36.7%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}}^{1.5}} \]

       3. *-commutative 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \frac{\color{blue}{\pi \cdot 2}}{k}\right)}^{1.5}} \]

       4. associate-*r/ 36.7%

          \[\leadsto \sqrt[3]{{\left(n \cdot \color{blue}{\left(\pi \cdot \frac{2}{k}\right)}\right)}^{1.5}} \]

   14. Simplified 36.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {k}^{-0.5} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (* 2.0 (* PI n)) (+ 0.5 (* k -0.5)))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow((2.0 * (((double) M_PI) * n)), (0.5 + (k * -0.5)));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow((2.0 * (Math.PI * n)), (0.5 + (k * -0.5)));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow((2.0 * (math.pi * n)), (0.5 + (k * -0.5)))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 + Float64(k * -0.5))))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * ((2.0 * (pi * n)) ^ (0.5 + (k * -0.5)));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(k * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 99.5%

      \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

   2. *-lft-identity 99.5%

      \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]

   3. associate-*l* 99.5%

      \[\leadsto \frac{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]

   4. div-sub 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. metadata-eval 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{\frac{1}{2}} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   2. div-sub 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]

   3. associate-*r* 99.5%

      \[\leadsto \frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]

   4. div-inv 99.4%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]

   5. associate-*r* 99.4%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]

   6. div-sub 99.4%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{1}{\sqrt{k}} \]

   7. metadata-eval 99.4%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]

   8. sub-neg 99.4%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(0.5 + \left(-\frac{k}{2}\right)\right)}} \cdot \frac{1}{\sqrt{k}} \]

   9. div-inv 99.4%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + \left(-\color{blue}{k \cdot \frac{1}{2}}\right)\right)} \cdot \frac{1}{\sqrt{k}} \]

   10. metadata-eval 99.4%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + \left(-k \cdot \color{blue}{0.5}\right)\right)} \cdot \frac{1}{\sqrt{k}} \]

   11. distribute-rgt-neg-in 99.4%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)} \cdot \frac{1}{\sqrt{k}} \]

   12. metadata-eval 99.4%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot \color{blue}{-0.5}\right)} \cdot \frac{1}{\sqrt{k}} \]

   13. pow1/2 99.4%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot \frac{1}{\color{blue}{{k}^{0.5}}} \]

   14. pow-flip 99.5%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot \color{blue}{{k}^{\left(-0.5\right)}} \]

   15. metadata-eval 99.5%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot {k}^{\color{blue}{-0.5}} \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot {k}^{-0.5}} \]

7. Final simplification 99.5%

   \[\leadsto {k}^{-0.5} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \]
8. Add Preprocessing

Alternative 9: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* 2.0 (* PI n)) (- 0.5 (/ k 2.0))) (sqrt k)))

C

double code(double k, double n) {
	return pow((2.0 * (((double) M_PI) * n)), (0.5 - (k / 2.0))) / sqrt(k);
}

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 - (k / 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 99.5%

      \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

   2. *-lft-identity 99.5%

      \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]

   3. associate-*l* 99.5%

      \[\leadsto \frac{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]

   4. div-sub 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 10: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (* (sqrt n) (sqrt (* PI (/ 2.0 k)))))

C

double code(double k, double n) {
	return sqrt(n) * sqrt((((double) M_PI) * (2.0 / k)));
}

Java

public static double code(double k, double n) {
	return Math.sqrt(n) * Math.sqrt((Math.PI * (2.0 / k)));
}

Python

def code(k, n):
	return math.sqrt(n) * math.sqrt((math.pi * (2.0 / k)))

Julia

function code(k, n)
	return Float64(sqrt(n) * sqrt(Float64(pi * Float64(2.0 / k))))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(n) * sqrt((pi * (2.0 / k)));
end

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. associate-/l* 38.3%

      \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

   2. pow1/2 38.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

   3. pow1/2 38.3%

      \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

   4. pow-prod-down 38.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

8. Taylor expanded in n around 0 38.4%

   \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
9. Step-by-step derivation

   1. associate-*r/ 38.4%

      \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

   2. *-commutative 38.4%

      \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

   3. associate-/l* 38.4%

      \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

10. Simplified 38.4%

    \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
11. Step-by-step derivation

    1. associate-*l* 38.4%

       \[\leadsto {\color{blue}{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}}^{0.5} \]

    2. unpow-prod-down 51.8%

       \[\leadsto \color{blue}{{n}^{0.5} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5}} \]

    3. pow1/2 51.8%

       \[\leadsto \color{blue}{\sqrt{n}} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5} \]

12. Applied egg-rr 51.8%

    \[\leadsto \color{blue}{\sqrt{n} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{0.5}} \]
13. Step-by-step derivation

    1. unpow1/2 51.8%

       \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\pi \cdot \frac{2}{k}}} \]

14. Simplified 51.8%

    \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}} \]
15. Add Preprocessing

Alternative 11: 39.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{k \cdot 0.5}{\pi \cdot n}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (pow (/ (* k 0.5) (* PI n)) -0.5))

C

double code(double k, double n) {
	return pow(((k * 0.5) / (((double) M_PI) * n)), -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow(((k * 0.5) / (Math.PI * n)), -0.5);
}

Python

def code(k, n):
	return math.pow(((k * 0.5) / (math.pi * n)), -0.5)

Julia

function code(k, n)
	return Float64(Float64(k * 0.5) / Float64(pi * n)) ^ -0.5
end

MATLAB

function tmp = code(k, n)
	tmp = ((k * 0.5) / (pi * n)) ^ -0.5;
end

Wolfram

code[k_, n_] := N[Power[N[(N[(k * 0.5), $MachinePrecision] / N[(Pi * n), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. associate-/l* 38.3%

      \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

   2. pow1/2 38.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

   3. pow1/2 38.3%

      \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

   4. pow-prod-down 38.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]
8. Step-by-step derivation

   1. metadata-eval 38.4%

      \[\leadsto {\left(\color{blue}{\frac{2}{1}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5} \]

   2. associate-*r/ 38.4%

      \[\leadsto {\left(\frac{2}{1} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{0.5} \]

   3. *-commutative 38.4%

      \[\leadsto {\left(\frac{2}{1} \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{0.5} \]

   4. times-frac 38.4%

      \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(\pi \cdot n\right)}{1 \cdot k}\right)}}^{0.5} \]

   5. associate-*r* 38.4%

      \[\leadsto {\left(\frac{\color{blue}{\left(2 \cdot \pi\right) \cdot n}}{1 \cdot k}\right)}^{0.5} \]

   6. *-commutative 38.4%

      \[\leadsto {\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{1 \cdot k}\right)}^{0.5} \]

   7. *-un-lft-identity 38.4%

      \[\leadsto {\left(\frac{n \cdot \left(2 \cdot \pi\right)}{\color{blue}{k}}\right)}^{0.5} \]

   8. clear-num 38.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(2 \cdot \pi\right)}}\right)}}^{0.5} \]

   9. *-commutative 38.4%

      \[\leadsto {\left(\frac{1}{\frac{k}{n \cdot \color{blue}{\left(\pi \cdot 2\right)}}}\right)}^{0.5} \]

9. Applied egg-rr 38.4%

   \[\leadsto {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}}^{0.5} \]
10. Step-by-step derivation

    1. *-un-lft-identity 38.4%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}^{0.5}} \]

    2. inv-pow 38.4%

       \[\leadsto 1 \cdot {\color{blue}{\left({\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{-1}\right)}}^{0.5} \]

    3. pow-pow 39.2%

       \[\leadsto 1 \cdot \color{blue}{{\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{\left(-1 \cdot 0.5\right)}} \]

    4. *-un-lft-identity 39.2%

       \[\leadsto 1 \cdot {\left(\frac{\color{blue}{1 \cdot k}}{n \cdot \left(\pi \cdot 2\right)}\right)}^{\left(-1 \cdot 0.5\right)} \]

    5. associate-*r* 39.2%

       \[\leadsto 1 \cdot {\left(\frac{1 \cdot k}{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}\right)}^{\left(-1 \cdot 0.5\right)} \]

    6. *-commutative 39.2%

       \[\leadsto 1 \cdot {\left(\frac{1 \cdot k}{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}\right)}^{\left(-1 \cdot 0.5\right)} \]

    7. times-frac 39.2%

       \[\leadsto 1 \cdot {\color{blue}{\left(\frac{1}{2} \cdot \frac{k}{n \cdot \pi}\right)}}^{\left(-1 \cdot 0.5\right)} \]

    8. metadata-eval 39.2%

       \[\leadsto 1 \cdot {\left(\color{blue}{0.5} \cdot \frac{k}{n \cdot \pi}\right)}^{\left(-1 \cdot 0.5\right)} \]

    9. metadata-eval 39.2%

       \[\leadsto 1 \cdot {\left(0.5 \cdot \frac{k}{n \cdot \pi}\right)}^{\color{blue}{-0.5}} \]

11. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{1 \cdot {\left(0.5 \cdot \frac{k}{n \cdot \pi}\right)}^{-0.5}} \]
12. Step-by-step derivation

    1. *-lft-identity 39.2%

       \[\leadsto \color{blue}{{\left(0.5 \cdot \frac{k}{n \cdot \pi}\right)}^{-0.5}} \]

    2. associate-*r/ 39.2%

       \[\leadsto {\color{blue}{\left(\frac{0.5 \cdot k}{n \cdot \pi}\right)}}^{-0.5} \]

    3. *-commutative 39.2%

       \[\leadsto {\left(\frac{\color{blue}{k \cdot 0.5}}{n \cdot \pi}\right)}^{-0.5} \]

13. Simplified 39.2%

    \[\leadsto \color{blue}{{\left(\frac{k \cdot 0.5}{n \cdot \pi}\right)}^{-0.5}} \]

14. Final simplification 39.2%

    \[\leadsto {\left(\frac{k \cdot 0.5}{\pi \cdot n}\right)}^{-0.5} \]
15. Add Preprocessing

Alternative 12: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (pow (* 2.0 (* n (/ PI k))) 0.5))

C

double code(double k, double n) {
	return pow((2.0 * (n * (((double) M_PI) / k))), 0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (n * (Math.PI / k))), 0.5);
}

Python

def code(k, n):
	return math.pow((2.0 * (n * (math.pi / k))), 0.5)

Julia

function code(k, n)
	return Float64(2.0 * Float64(n * Float64(pi / k))) ^ 0.5
end

MATLAB

function tmp = code(k, n)
	tmp = (2.0 * (n * (pi / k))) ^ 0.5;
end

Wolfram

code[k_, n_] := N[Power[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. associate-/l* 38.3%

      \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

   2. pow1/2 38.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

   3. pow1/2 38.3%

      \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

   4. pow-prod-down 38.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]
8. Add Preprocessing

Alternative 13: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\pi \cdot \left(2 \cdot n\right)}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ (* PI (* 2.0 n)) k)))

C

double code(double k, double n) {
	return sqrt(((((double) M_PI) * (2.0 * n)) / k));
}

Java

public static double code(double k, double n) {
	return Math.sqrt(((Math.PI * (2.0 * n)) / k));
}

Python

def code(k, n):
	return math.sqrt(((math.pi * (2.0 * n)) / k))

Julia

function code(k, n)
	return sqrt(Float64(Float64(pi * Float64(2.0 * n)) / k))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((pi * (2.0 * n)) / k));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. associate-/l* 38.3%

      \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

   2. pow1/2 38.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

   3. pow1/2 38.3%

      \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

   4. pow-prod-down 38.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

8. Taylor expanded in n around 0 38.4%

   \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
9. Step-by-step derivation

   1. associate-*r/ 38.4%

      \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

   2. *-commutative 38.4%

      \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

   3. associate-/l* 38.4%

      \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

10. Simplified 38.4%

    \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
11. Step-by-step derivation

    1. unpow1/2 38.4%

       \[\leadsto \color{blue}{\sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}} \]

    2. associate-*r/ 38.4%

       \[\leadsto \sqrt{\color{blue}{\frac{\left(n \cdot \pi\right) \cdot 2}{k}}} \]

    3. associate-*r* 38.4%

       \[\leadsto \sqrt{\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}} \]

    4. *-commutative 38.4%

       \[\leadsto \sqrt{\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}} \]

    5. associate-*l* 38.4%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}} \]

12. Applied egg-rr 38.4%

    \[\leadsto \color{blue}{\sqrt{\frac{\left(n \cdot 2\right) \cdot \pi}{k}}} \]

13. Final simplification 38.4%

    \[\leadsto \sqrt{\frac{\pi \cdot \left(2 \cdot n\right)}{k}} \]
14. Add Preprocessing

Alternative 14: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(\pi \cdot \frac{2}{k}\right)} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* n (* PI (/ 2.0 k)))))

C

double code(double k, double n) {
	return sqrt((n * (((double) M_PI) * (2.0 / k))));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((n * (Math.PI * (2.0 / k))));
}

Python

def code(k, n):
	return math.sqrt((n * (math.pi * (2.0 / k))))

Julia

function code(k, n)
	return sqrt(Float64(n * Float64(pi * Float64(2.0 / k))))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((n * (pi * (2.0 / k))));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. associate-/l* 38.3%

      \[\leadsto \sqrt{\color{blue}{n \cdot \frac{\pi}{k}}} \cdot \sqrt{2} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{n \cdot \frac{\pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \frac{\pi}{k}}} \]

   2. pow1/2 38.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{n \cdot \frac{\pi}{k}} \]

   3. pow1/2 38.3%

      \[\leadsto {2}^{0.5} \cdot \color{blue}{{\left(n \cdot \frac{\pi}{k}\right)}^{0.5}} \]

   4. pow-prod-down 38.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}} \]

8. Taylor expanded in n around 0 38.4%

   \[\leadsto {\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{0.5} \]
9. Step-by-step derivation

   1. associate-*r/ 38.4%

      \[\leadsto {\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.5} \]

   2. *-commutative 38.4%

      \[\leadsto {\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)}^{0.5} \]

   3. associate-/l* 38.4%

      \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]

10. Simplified 38.4%

    \[\leadsto {\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{0.5} \]
11. Step-by-step derivation

    1. *-un-lft-identity 38.4%

       \[\leadsto \color{blue}{1 \cdot {\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}} \]

    2. unpow1/2 38.4%

       \[\leadsto 1 \cdot \color{blue}{\sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}} \]

    3. associate-*r/ 38.4%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{\left(n \cdot \pi\right) \cdot 2}{k}}} \]

    4. associate-*r* 38.4%

       \[\leadsto 1 \cdot \sqrt{\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}} \]

    5. *-commutative 38.4%

       \[\leadsto 1 \cdot \sqrt{\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}} \]

    6. associate-*l* 38.4%

       \[\leadsto 1 \cdot \sqrt{\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}} \]

12. Applied egg-rr 38.4%

    \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{\left(n \cdot 2\right) \cdot \pi}{k}}} \]
13. Step-by-step derivation

    1. *-lft-identity 38.4%

       \[\leadsto \color{blue}{\sqrt{\frac{\left(n \cdot 2\right) \cdot \pi}{k}}} \]

    2. associate-*l* 38.4%

       \[\leadsto \sqrt{\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}} \]

    3. associate-/l* 38.4%

       \[\leadsto \sqrt{\color{blue}{n \cdot \frac{2 \cdot \pi}{k}}} \]

    4. *-commutative 38.4%

       \[\leadsto \sqrt{n \cdot \frac{\color{blue}{\pi \cdot 2}}{k}} \]

    5. associate-*r/ 38.4%

       \[\leadsto \sqrt{n \cdot \color{blue}{\left(\pi \cdot \frac{2}{k}\right)}} \]

14. Simplified 38.4%

    \[\leadsto \color{blue}{\sqrt{n \cdot \left(\pi \cdot \frac{2}{k}\right)}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))