Migdal et al, Equation (51)

Average Accuracy: 99.2% → 99.3%

Time: 13.3s

Precision: binary64

Cost: 32768

Math

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

↓

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

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

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

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

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

MATLAB

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

↓

function tmp = code(k, n)
	tmp = sqrt(((((2.0 * pi) ^ (1.0 - k)) * n) / (n ^ k))) / sqrt(k);
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_] := N[(N[Sqrt[N[(N[(N[Power[N[(2.0 * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / N[Power[n, k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

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 99.3%

   \[\leadsto \frac{\color{blue}{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}}}{\sqrt{k}} \]
   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 [=>] 99.0	\[ \frac{\color{blue}{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}} \cdot \sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}}}{\sqrt{k}} \]
   sqrt-unprod [=>] 99.3	\[ \frac{\color{blue}{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}}}{\sqrt{k}} \]
   pow-sqr [=>] 99.3	\[ \frac{\sqrt{\color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}}{\sqrt{k}} \]
   sub-neg [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(0.5 + \left(-\frac{k}{2}\right)\right)}\right)}}}{\sqrt{k}} \]
   div-inv [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-\color{blue}{k \cdot \frac{1}{2}}\right)\right)\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \left(-k \cdot \color{blue}{0.5}\right)\right)\right)}}}{\sqrt{k}} \]
   distribute-rgt-neg-in [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot \color{blue}{-0.5}\right)\right)}}}{\sqrt{k}} \]

4. Simplified 99.3%

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

   [Start] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}}{\sqrt{k}} \]
   distribute-rgt-in [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(0.5 \cdot 2 + \left(k \cdot -0.5\right) \cdot 2\right)}}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{1} + \left(k \cdot -0.5\right) \cdot 2\right)}}}{\sqrt{k}} \]
   associate-*l* [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 + \color{blue}{k \cdot \left(-0.5 \cdot 2\right)}\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.3	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 + k \cdot \color{blue}{-1}\right)}}}{\sqrt{k}} \]

5. Taylor expanded in k around inf 99.3%

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

6. Simplified 99.3%

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

   [Start] 99.3	\[ \frac{\sqrt{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}}{\sqrt{k}} \]
   *-commutative [<=] 99.3	\[ \frac{\sqrt{{\left(2 \cdot \color{blue}{\left(\pi \cdot n\right)}\right)}^{\left(1 - k\right)}}}{\sqrt{k}} \]
   associate-*r* [=>] 99.3	\[ \frac{\sqrt{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}}{\sqrt{k}} \]

7. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \frac{\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}}{\sqrt{k}} \]
   unpow-prod-down [=>] 99.1	\[ \frac{\sqrt{\color{blue}{{\left(2 \cdot \pi\right)}^{\left(1 - k\right)} \cdot {n}^{\left(1 - k\right)}}}}{\sqrt{k}} \]
   pow-sub [=>] 99.3	\[ \frac{\sqrt{{\left(2 \cdot \pi\right)}^{\left(1 - k\right)} \cdot \color{blue}{\frac{{n}^{1}}{{n}^{k}}}}}{\sqrt{k}} \]
   pow1 [<=] 99.3	\[ \frac{\sqrt{{\left(2 \cdot \pi\right)}^{\left(1 - k\right)} \cdot \frac{\color{blue}{n}}{{n}^{k}}}}{\sqrt{k}} \]
   associate-*r/ [=>] 99.3	\[ \frac{\sqrt{\color{blue}{\frac{{\left(2 \cdot \pi\right)}^{\left(1 - k\right)} \cdot n}{{n}^{k}}}}}{\sqrt{k}} \]

8. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	19968

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

Alternative 2
Accuracy	97.9%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 4.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3
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 4
Accuracy	65.1%
Cost	19584

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

Alternative 5
Accuracy	65.2%
Cost	19584

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

Alternative 6
Accuracy	48.5%
Cost	13184

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

Alternative 7
Accuracy	48.5%
Cost	13184

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

Reproduce

herbie shell --seed 2023131 
(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))))