Migdal et al, Equation (51)

Average Accuracy: 99.2% → 99.4%

Time: 13.4s

Precision: binary64

Cost: 32896

Math

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

↓

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

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

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

3. Applied egg-rr 99.4%

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

   [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
   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	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]
   pow-sub [=>] 99.4	\[ \frac{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.5}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}}{\sqrt{k}} \]
   pow1/2 [<=] 99.4	\[ \frac{\frac{\color{blue}{\sqrt{\pi \cdot \left(2 \cdot n\right)}}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [=>] 2.6	\[ \frac{\frac{\sqrt{\pi \cdot \color{blue}{\log \left(e^{2 \cdot n}\right)}}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   *-commutative [=>] 2.6	\[ \frac{\frac{\sqrt{\pi \cdot \log \left(e^{\color{blue}{n \cdot 2}}\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   exp-lft-sqr [=>] 2.6	\[ \frac{\frac{\sqrt{\pi \cdot \log \color{blue}{\left(e^{n} \cdot e^{n}\right)}}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   log-prod [=>] 2.7	\[ \frac{\frac{\sqrt{\pi \cdot \color{blue}{\left(\log \left(e^{n}\right) + \log \left(e^{n}\right)\right)}}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [<=] 7.9	\[ \frac{\frac{\sqrt{\pi \cdot \left(\color{blue}{n} + \log \left(e^{n}\right)\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [<=] 99.4	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + \color{blue}{n}\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [=>] 37.0	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \color{blue}{\log \left(e^{2 \cdot n}\right)}\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   *-commutative [=>] 37.0	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \log \left(e^{\color{blue}{n \cdot 2}}\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   exp-lft-sqr [=>] 36.9	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \log \color{blue}{\left(e^{n} \cdot e^{n}\right)}\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   log-prod [=>] 37.0	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \color{blue}{\left(\log \left(e^{n}\right) + \log \left(e^{n}\right)\right)}\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [<=] 66.4	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \left(\color{blue}{n} + \log \left(e^{n}\right)\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]
   add-log-exp [<=] 99.4	\[ \frac{\frac{\sqrt{\pi \cdot \left(n + n\right)}}{{\left(\pi \cdot \left(n + \color{blue}{n}\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \]

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	26432

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

Alternative 2
Accuracy	98.7%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;k \leq 8.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	19904

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

Alternative 5
Accuracy	71.1%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 6
Accuracy	64.2%
Cost	19584

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

Alternative 7
Accuracy	48.5%
Cost	13248

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

Alternative 8
Accuracy	47.5%
Cost	13184

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

Reproduce

herbie shell --seed 2023146 
(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))))