Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 18.3s

Alternatives: 8

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, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, 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. *-commutative 99.4%

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

   2. div-inv 99.5%

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

   3. div-sub 99.5%

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

   4. pow-sub 99.7%

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

   5. sqrt-pow1 99.7%

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

   6. pow1 99.7%

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

   7. associate-/l/ 99.7%

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

   8. associate-*l* 99.7%

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

   9. associate-*l* 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{1}{\sqrt{k \cdot {t_0}^{k}}} \cdot \sqrt{t_0} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (* (/ 1.0 (sqrt (* k (pow t_0 k)))) (sqrt t_0))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	return (1.0 / sqrt((k * pow(t_0, k)))) * sqrt(t_0);
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	return (1.0 / Math.sqrt((k * Math.pow(t_0, k)))) * Math.sqrt(t_0);
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	return (1.0 / math.sqrt((k * math.pow(t_0, k)))) * math.sqrt(t_0)

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	return Float64(Float64(1.0 / sqrt(Float64(k * (t_0 ^ k)))) * sqrt(t_0))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = (1.0 / sqrt((k * (t_0 ^ k)))) * sqrt(t_0);
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[N[(k * N[Power[t$95$0, k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$0], $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. *-commutative 99.4%

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

   2. div-inv 99.5%

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

   3. div-sub 99.5%

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

   4. pow-sub 99.7%

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

   5. sqrt-pow1 99.7%

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

   6. pow1 99.7%

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

   7. associate-/l/ 99.7%

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

   8. associate-*l* 99.7%

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

   9. associate-*l* 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / 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. 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. Simplified 99.5%

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

4. Taylor expanded in k around 0 49.4%

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

   1. associate-/l* 49.3%

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

   2. div-inv 49.3%

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

   3. *-commutative 49.3%

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

   4. sqrt-undiv 49.4%

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

6. Applied egg-rr 49.4%

   \[\leadsto \color{blue}{\sqrt{\pi \cdot n} \cdot \frac{1}{\sqrt{\frac{k}{2}}}} \]
7. Step-by-step derivation

   1. associate-*r/ 49.5%

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

   2. *-rgt-identity 49.5%

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

   3. *-commutative 49.5%

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

8. Simplified 49.5%

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

9. Final simplification 49.5%

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

Alternative 5: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * N[(2.0 * n), $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. Simplified 99.5%

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

4. Taylor expanded in k around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. sqrt-prod 49.5%

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

   3. *-commutative 49.5%

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

   4. add-sqr-sqrt 49.2%

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

   5. *-un-lft-identity 49.2%

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

   6. times-frac 49.3%

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

   7. pow1/2 49.3%

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

   8. *-commutative 49.3%

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

   9. sqrt-pow1 49.3%

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

   10. *-commutative 49.3%

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

   11. *-commutative 49.3%

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

   12. associate-*l* 49.3%

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

   13. metadata-eval 49.3%

       \[\leadsto \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{2 \cdot \left(\pi \cdot n\right)}}}{\sqrt{k}} \]

6. Applied egg-rr 49.3%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}}{\sqrt{k}}} \]
7. Step-by-step derivation

   1. associate-*r/ 49.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}}{1} \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}}{\sqrt{k}}} \]

   2. /-rgt-identity 49.3%

      \[\leadsto \frac{\color{blue}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}} \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{0.25}}{\sqrt{k}} \]

   3. pow-sqr 49.5%

      \[\leadsto \frac{\color{blue}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(2 \cdot 0.25\right)}}}{\sqrt{k}} \]

   4. metadata-eval 49.5%

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

   5. unpow1/2 49.5%

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

8. Simplified 49.5%

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

9. Final simplification 49.5%

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

Alternative 6: 38.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(k / n), $MachinePrecision] * N[(0.5 / Pi), $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. 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. Simplified 99.5%

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

4. Taylor expanded in k around 0 49.4%

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

   1. expm1-log1p-u 46.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)\right)} \]

   2. expm1-udef 46.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)} - 1} \]

6. Applied egg-rr 37.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 38.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)\right)} \]

   2. expm1-log1p 40.0%

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

8. Simplified 40.0%

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

9. Taylor expanded in n around 0 40.0%

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

    1. *-commutative 40.0%

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

    2. associate-*l/ 40.0%

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

    3. *-commutative 40.0%

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

    4. associate-*r* 40.0%

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

    5. sqrt-div 49.5%

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

    6. clear-num 49.4%

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

    7. pow1 49.4%

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

    8. pow-flip 49.4%

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

    9. sqrt-undiv 40.6%

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

    10. associate-*r* 40.6%

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

    11. *-commutative 40.6%

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

    12. associate-/l/ 40.6%

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

    13. sqrt-pow2 40.6%

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

    14. div-inv 40.6%

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

    15. times-frac 40.6%

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

    16. metadata-eval 40.6%

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

11. Applied egg-rr 40.6%

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

12. Final simplification 40.6%

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

Alternative 7: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n * N[(Pi / 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. 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. Simplified 99.5%

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

4. Taylor expanded in k around 0 49.4%

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

   1. expm1-log1p-u 46.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)\right)} \]

   2. expm1-udef 46.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)} - 1} \]

6. Applied egg-rr 37.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 38.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)\right)} \]

   2. expm1-log1p 40.0%

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

8. Simplified 40.0%

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

9. Taylor expanded in n around 0 40.0%

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

    1. expm1-log1p-u 38.3%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)\right)}} \]

    2. expm1-udef 37.1%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)} - 1\right)}} \]

    3. *-commutative 37.1%

       \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\pi \cdot n}}{k}\right)} - 1\right)} \]

    4. associate-/l* 37.1%

       \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\pi}{\frac{k}{n}}}\right)} - 1\right)} \]

11. Applied egg-rr 37.1%

    \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{\frac{k}{n}}\right)} - 1\right)}} \]
12. Step-by-step derivation

    1. expm1-def 38.3%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\frac{k}{n}}\right)\right)}} \]

    2. expm1-log1p 40.0%

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

    3. associate-/r/ 39.6%

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

    4. *-commutative 39.6%

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

13. Simplified 39.6%

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

14. Final simplification 39.6%

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

Alternative 8: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(Pi * N[(n / 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. 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. Simplified 99.5%

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

4. Taylor expanded in k around 0 49.4%

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

   1. expm1-log1p-u 46.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)\right)} \]

   2. expm1-udef 46.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{n \cdot \pi} \cdot \sqrt{2}}{\sqrt{k}}\right)} - 1} \]

6. Applied egg-rr 37.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 38.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\right)\right)} \]

   2. expm1-log1p 40.0%

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

8. Simplified 40.0%

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

9. Taylor expanded in n around 0 40.0%

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

    1. associate-/l* 40.0%

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

    2. associate-/r/ 40.0%

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

11. Applied egg-rr 40.0%

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

12. Final simplification 40.0%

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

Reproduce

herbie shell --seed 2023326 
(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))))