Migdal et al, Equation (51)

Average Accuracy: 99.2% → 98.9%

Time: 16.6s

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \]

FPCore

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

↓

(FPCore (k n)
 :precision binary64
 (if (<= k 2e-52)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (* (pow (* n (* 2.0 PI)) (- 1.0 k)) (/ 1.0 k)))))

C

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

↓

double code(double k, double n) {
	double tmp;
	if (k <= 2e-52) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

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));
}

↓

public static double code(double k, double n) {
	double tmp;
	if (k <= 2e-52) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

Python

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

↓

def code(k, n):
	tmp = 0
	if k <= 2e-52:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) * (1.0 / k)))
	return tmp

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

↓

function code(k, n)
	tmp = 0.0
	if (k <= 2e-52)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) * Float64(1.0 / k)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2e-52)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) * (1.0 / k)));
	end
	tmp_2 = tmp;
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]

↓

code[k_, n_] := If[LessEqual[k, 2e-52], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e-52

   1. Initial program 99.2%

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

   2. Simplified 99.3%

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

      [Start] 99.2	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [=>] 99.0	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [<=] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 99.3	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 99.3	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      div-sub [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
      metadata-eval [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 69.9%

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

      [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]
      add-sqr-sqrt [=>] 98.9	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}}} \]
      sqrt-unprod [=>] 69.7	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}}} \]
      frac-times [=>] 69.6	\[ \sqrt{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k} \cdot \sqrt{k}}}} \]
      add-sqr-sqrt [<=] 69.8	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\color{blue}{k}}} \]
      pow-sqr [=>] 69.9	\[ \sqrt{\frac{\color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}{k}} \]
      sub-neg [=>] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(0.5 + \left(-\frac{k}{2}\right)\right)}\right)}}{k}} \]
      div-inv [=>] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-\color{blue}{k \cdot \frac{1}{2}}\right)\right)\right)}}{k}} \]
      metadata-eval [=>] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-k \cdot \color{blue}{0.5}\right)\right)\right)}}{k}} \]
      distribute-rgt-neg-in [=>] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)\right)}}{k}} \]
      metadata-eval [=>] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot \color{blue}{-0.5}\right)\right)}}{k}} \]

   4. Simplified 69.9%

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

      [Start] 69.9	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      associate-*r* [=>] 69.9	\[ \sqrt{\frac{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [=>] 69.9	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [<=] 69.9	\[ \sqrt{\frac{{\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 69.9	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(0.5 \cdot 2 + \left(k \cdot -0.5\right) \cdot 2\right)}}}{k}} \]
      metadata-eval [=>] 69.9	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\color{blue}{1} + \left(k \cdot -0.5\right) \cdot 2\right)}}{k}} \]
      associate-*l* [=>] 69.9	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \color{blue}{k \cdot \left(-0.5 \cdot 2\right)}\right)}}{k}} \]
      metadata-eval [=>] 69.9	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + k \cdot \color{blue}{-1}\right)}}{k}} \]

   5. Taylor expanded in k around 0 69.9%

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

   6. Simplified 69.9%

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

      [Start] 69.9	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k}} \]
      associate-/l* [=>] 69.9	\[ \sqrt{2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}} \]
      associate-/r/ [=>] 69.9	\[ \sqrt{2 \cdot \color{blue}{\left(\frac{n}{k} \cdot \pi\right)}} \]

   7. Taylor expanded in n around 0 69.9%

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

   8. Simplified 69.9%

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

      [Start] 69.9	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k}} \]
      associate-*r/ [<=] 69.9	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \frac{\pi}{k}\right)}} \]

   9. Applied egg-rr 99.3%

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

      [Start] 69.9	\[ \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)} \]
      associate-*r* [=>] 69.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \frac{\pi}{k}}} \]
      sqrt-prod [=>] 99.2	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}} \]
      clear-num [=>] 99.2	\[ \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{\frac{1}{\frac{k}{\pi}}}} \]
      sqrt-div [=>] 99.2	\[ \sqrt{2 \cdot n} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{k}{\pi}}}} \]
      metadata-eval [=>] 99.2	\[ \sqrt{2 \cdot n} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{k}{\pi}}} \]
      associate-*r/ [=>] 99.3	\[ \color{blue}{\frac{\sqrt{2 \cdot n} \cdot 1}{\sqrt{\frac{k}{\pi}}}} \]

   10. Simplified 99.3%

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

       [Start] 99.3	\[ \frac{\sqrt{2 \cdot n} \cdot 1}{\sqrt{\frac{k}{\pi}}} \]
       *-rgt-identity [=>] 99.3	\[ \frac{\color{blue}{\sqrt{2 \cdot n}}}{\sqrt{\frac{k}{\pi}}} \]
       *-commutative [=>] 99.3	\[ \frac{\sqrt{\color{blue}{n \cdot 2}}}{\sqrt{\frac{k}{\pi}}} \]

   if 2e-52 < k

   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. Simplified 99.3%

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

      [Start] 99.3	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [=>] 99.2	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [<=] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 99.3	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 99.3	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      div-sub [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
      metadata-eval [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 98.4%

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

      [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]
      add-sqr-sqrt [=>] 99.2	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}}} \]
      sqrt-unprod [=>] 98.4	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}}} \]
      frac-times [=>] 98.4	\[ \sqrt{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k} \cdot \sqrt{k}}}} \]
      add-sqr-sqrt [<=] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\color{blue}{k}}} \]
      pow-sqr [=>] 98.4	\[ \sqrt{\frac{\color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}{k}} \]
      sub-neg [=>] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(0.5 + \left(-\frac{k}{2}\right)\right)}\right)}}{k}} \]
      div-inv [=>] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-\color{blue}{k \cdot \frac{1}{2}}\right)\right)\right)}}{k}} \]
      metadata-eval [=>] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-k \cdot \color{blue}{0.5}\right)\right)\right)}}{k}} \]
      distribute-rgt-neg-in [=>] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)\right)}}{k}} \]
      metadata-eval [=>] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot \color{blue}{-0.5}\right)\right)}}{k}} \]

   4. Simplified 98.4%

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

      [Start] 98.4	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      associate-*r* [=>] 98.4	\[ \sqrt{\frac{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [=>] 98.4	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [<=] 98.4	\[ \sqrt{\frac{{\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 98.4	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(0.5 \cdot 2 + \left(k \cdot -0.5\right) \cdot 2\right)}}}{k}} \]
      metadata-eval [=>] 98.4	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\color{blue}{1} + \left(k \cdot -0.5\right) \cdot 2\right)}}{k}} \]
      associate-*l* [=>] 98.4	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \color{blue}{k \cdot \left(-0.5 \cdot 2\right)}\right)}}{k}} \]
      metadata-eval [=>] 98.4	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + k \cdot \color{blue}{-1}\right)}}{k}} \]

   5. Taylor expanded in n around 0 97.0%

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

   6. Simplified 98.4%

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

      [Start] 97.0	\[ \sqrt{\frac{e^{\left(1 + -1 \cdot k\right) \cdot \left(\log n + \log \left(2 \cdot \pi\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 97.0	\[ \sqrt{\frac{e^{\color{blue}{\log n \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}}{k}} \]
      remove-double-neg [<=] 97.0	\[ \sqrt{\frac{e^{\color{blue}{\left(-\left(-\log n\right)\right)} \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      log-rec [<=] 97.0	\[ \sqrt{\frac{e^{\left(-\color{blue}{\log \left(\frac{1}{n}\right)}\right) \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      mul-1-neg [<=] 97.0	\[ \sqrt{\frac{e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{n}\right)\right)} \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      distribute-rgt-in [<=] 97.0	\[ \sqrt{\frac{e^{\color{blue}{\left(1 + -1 \cdot k\right) \cdot \left(-1 \cdot \log \left(\frac{1}{n}\right) + \log \left(2 \cdot \pi\right)\right)}}}{k}} \]
      *-commutative [<=] 97.0	\[ \sqrt{\frac{e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{n}\right) + \log \left(2 \cdot \pi\right)\right) \cdot \left(1 + -1 \cdot k\right)}}}{k}} \]

   7. Applied egg-rr 98.4%

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

      [Start] 98.4	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}} \]
      div-inv [=>] 98.4	\[ \sqrt{\color{blue}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	26624

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

Alternative 2
Accuracy	99.0%
Cost	26624

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

Alternative 3
Accuracy	99.3%
Cost	26176

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

Alternative 4
Accuracy	98.9%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	19904

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

Alternative 6
Accuracy	66.4%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 7
Accuracy	66.4%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \]

Alternative 8
Accuracy	65.2%
Cost	19584

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

Alternative 9
Accuracy	65.3%
Cost	19584

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

Alternative 10
Accuracy	48.6%
Cost	13184

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

Alternative 11
Accuracy	48.6%
Cost	13184

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

Alternative 12
Accuracy	48.6%
Cost	13184

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

Alternative 13
Accuracy	48.6%
Cost	13184

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

Reproduce

herbie shell --seed 2023135 
(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))))