Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 11.6s

Precision: binary64

Cost: 32832

Math

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

↓

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

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
 (* (pow k -0.5) (sqrt (* (/ n (pow n k)) (pow (* 2.0 PI) (- 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) {
	return pow(k, -0.5) * sqrt(((n / pow(n, k)) * pow((2.0 * ((double) M_PI)), (1.0 - 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.pow(k, -0.5) * Math.sqrt(((n / Math.pow(n, k)) * Math.pow((2.0 * Math.PI), (1.0 - 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.pow(k, -0.5) * math.sqrt(((n / math.pow(n, k)) * math.pow((2.0 * math.pi), (1.0 - 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((k ^ -0.5) * sqrt(Float64(Float64(n / (n ^ k)) * (Float64(2.0 * pi) ^ Float64(1.0 - 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 = (k ^ -0.5) * sqrt(((n / (n ^ k)) * ((2.0 * pi) ^ (1.0 - 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[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(N[(n / N[Power[n, k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 0.5

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

3. Simplified 0.4

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

   [Start] 0.5	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{n} \cdot {\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}{{n}^{\left(k \cdot 0.5\right)}} \]
   associate-/l* [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{n}}{\frac{{n}^{\left(k \cdot 0.5\right)}}{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}}} \]
   *-commutative [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{n}}{\frac{{n}^{\color{blue}{\left(0.5 \cdot k\right)}}}{{\left(2 \cdot \pi\right)}^{\left(0.5 - k \cdot 0.5\right)}}} \]
   *-commutative [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{n}}{\frac{{n}^{\left(0.5 \cdot k\right)}}{{\left(2 \cdot \pi\right)}^{\left(0.5 - \color{blue}{0.5 \cdot k}\right)}}} \]

4. Applied egg-rr 0.4

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

5. Simplified 0.4

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

   [Start] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}} \]
   distribute-lft-in [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\color{blue}{\left(2 \cdot 0.5 + 2 \cdot \left(k \cdot -0.5\right)\right)}}} \]
   metadata-eval [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(\color{blue}{1} + 2 \cdot \left(k \cdot -0.5\right)\right)}} \]
   *-commutative [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 + 2 \cdot \color{blue}{\left(-0.5 \cdot k\right)}\right)}} \]
   associate-*r* [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 + \color{blue}{\left(2 \cdot -0.5\right) \cdot k}\right)}} \]
   metadata-eval [=>] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 + \color{blue}{-1} \cdot k\right)}} \]
   neg-mul-1 [<=] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 + \color{blue}{\left(-k\right)}\right)}} \]
   sub-neg [<=] 0.4	\[ \frac{1}{\sqrt{k}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\color{blue}{\left(1 - k\right)}}} \]

6. Applied egg-rr 3.3

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 - k\right)}} \]

7. Simplified 0.4

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

   [Start] 3.3	\[ \left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right) \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 - k\right)}} \]
   expm1-def [=>] 3.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 - k\right)}} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{{k}^{-0.5}} \cdot \sqrt{\frac{n}{{n}^{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(1 - k\right)}} \]

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.3
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	19904

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

Alternative 3
Error	32.9
Cost	19584

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

Alternative 4
Error	32.9
Cost	19584

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

Alternative 5
Error	22.5
Cost	19584

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

Alternative 6
Error	22.5
Cost	19584

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

Alternative 7
Error	22.5
Cost	19584

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

Alternative 8
Error	32.9
Cost	13184

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

Alternative 9
Error	32.9
Cost	13184

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

Alternative 10
Error	32.9
Cost	13184

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

Alternative 11
Error	32.9
Cost	13184

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

Reproduce

herbie shell --seed 2023010 
(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))))