Migdal et al, Equation (51)

Average Accuracy: 74.2% → 74.0%

Time: 16.8s

Precision: binary64

Cost: 26624

Math

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

↓

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

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
 (let* ((t_0 (+ 0.5 (* k -0.5))))
   (* (pow n t_0) (/ (pow (* 2.0 PI) t_0) (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) {
	double t_0 = 0.5 + (k * -0.5);
	return pow(n, t_0) * (pow((2.0 * ((double) M_PI)), t_0) / 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) {
	double t_0 = 0.5 + (k * -0.5);
	return Math.pow(n, t_0) * (Math.pow((2.0 * Math.PI), t_0) / 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):
	t_0 = 0.5 + (k * -0.5)
	return math.pow(n, t_0) * (math.pow((2.0 * math.pi), t_0) / 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)
	t_0 = Float64(0.5 + Float64(k * -0.5))
	return Float64((n ^ t_0) * Float64((Float64(2.0 * pi) ^ t_0) / 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)
	t_0 = 0.5 + (k * -0.5);
	tmp = (n ^ t_0) * (((2.0 * pi) ^ t_0) / 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_] := Block[{t$95$0 = N[(0.5 + N[(k * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[n, t$95$0], $MachinePrecision] * N[(N[Power[N[(2.0 * Pi), $MachinePrecision], t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.3%

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

2. Applied egg-rr 75.5%

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

   [Start] 75.3	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   associate-*l/ [=>] 75.3	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
   *-un-lft-identity [<=] 75.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
   unpow-prod-down [=>] 75.5	\[ \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
   associate-/l* [=>] 75.5	\[ \color{blue}{\frac{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}}} \]
   div-sub [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}} \]
   metadata-eval [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}} \]
   div-inv [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}} \]
   metadata-eval [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}} \]
   div-sub [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\frac{\sqrt{k}}{{n}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}} \]
   metadata-eval [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\frac{\sqrt{k}}{{n}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}} \]
   div-inv [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\frac{\sqrt{k}}{{n}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}} \]
   metadata-eval [=>] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\frac{\sqrt{k}}{{n}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}} \]

3. Simplified 75.5%

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

   [Start] 75.5	\[ \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\frac{\sqrt{k}}{{n}^{\left(0.5 - k \cdot 0.5\right)}}} \]
   associate-/r/ [=>] 75.5	\[ \color{blue}{\frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}} \cdot {n}^{\left(0.5 - k \cdot 0.5\right)}} \]
   *-commutative [=>] 75.5	\[ \color{blue}{{n}^{\left(0.5 - k \cdot 0.5\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}} \]
   sub-neg [=>] 75.5	\[ {n}^{\color{blue}{\left(0.5 + \left(-k \cdot 0.5\right)\right)}} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}} \]
   distribute-rgt-neg-in [=>] 75.5	\[ {n}^{\left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}} \]
   metadata-eval [=>] 75.5	\[ {n}^{\left(0.5 + k \cdot \color{blue}{-0.5}\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}} \]
   sub-neg [=>] 75.5	\[ {n}^{\left(0.5 + k \cdot -0.5\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\color{blue}{\left(0.5 + \left(-k \cdot 0.5\right)\right)}}}{\sqrt{k}} \]
   distribute-rgt-neg-in [=>] 75.5	\[ {n}^{\left(0.5 + k \cdot -0.5\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 + \color{blue}{k \cdot \left(-0.5\right)}\right)}}{\sqrt{k}} \]
   metadata-eval [=>] 75.5	\[ {n}^{\left(0.5 + k \cdot -0.5\right)} \cdot \frac{{\left(2 \cdot \pi\right)}^{\left(0.5 + k \cdot \color{blue}{-0.5}\right)}}{\sqrt{k}} \]

4. Final simplification 75.5%

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

Alternatives

Alternative 1
Accuracy	74.3%
Cost	19968

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

Alternative 2
Accuracy	74.1%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3
Accuracy	74.3%
Cost	19904

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

Alternative 4
Accuracy	54.5%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+188}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \frac{n}{k}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 5
Accuracy	50.3%
Cost	19584

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

Alternative 6
Accuracy	36.6%
Cost	13184

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

Alternative 7
Accuracy	36.6%
Cost	13184

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

Alternative 8
Accuracy	36.6%
Cost	13184

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

Alternative 9
Accuracy	36.6%
Cost	13184

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

Reproduce

herbie shell --seed 2023157 
(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))))