Migdal et al, Equation (51)

Average Accuracy: 74.8% → 74.8%

Time: 14.9s

Precision: binary64

Cost: 26240

Math

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

↓

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

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) (pow (* PI (* 2.0 n)) (fma k 0.5 -0.5))))

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) / pow((((double) M_PI) * (2.0 * n)), fma(k, 0.5, -0.5));
}

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) / (Float64(pi * Float64(2.0 * n)) ^ fma(k, 0.5, -0.5)))
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[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(k * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

2. Applied egg-rr 74.8%

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

   [Start] 74.8	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   div-sub [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}} \]
   metadata-eval [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)} \]
   pow-sub [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]
   pow1/2 [<=] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}} \]
   associate-*l* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(\pi \cdot n\right)}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}} \]
   associate-*l* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)}} \]
   div-inv [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(k \cdot \frac{1}{2}\right)}}} \]
   metadata-eval [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot \color{blue}{0.5}\right)}} \]

3. Simplified 74.8%

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

   [Start] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
   associate-*r* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \pi\right) \cdot n}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
   *-commutative [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\color{blue}{\left(\pi \cdot 2\right)} \cdot n}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
   associate-*l* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\color{blue}{\pi \cdot \left(2 \cdot n\right)}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
   associate-*r* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(k \cdot 0.5\right)}} \]
   *-commutative [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(k \cdot 0.5\right)}} \]
   associate-*l* [=>] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(k \cdot 0.5\right)}} \]

4. Applied egg-rr 62.1%

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

   [Start] 74.8	\[ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}} \]
   expm1-log1p-u [=>] 72.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}\right)\right)} \]
   expm1-udef [=>] 62.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}\right)} - 1} \]

5. Simplified 74.8%

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

   [Start] 62.1	\[ e^{\mathsf{log1p}\left(\frac{{k}^{-0.5}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, 0.5, -0.5\right)\right)}}\right)} - 1 \]
   expm1-def [=>] 72.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{-0.5}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, 0.5, -0.5\right)\right)}}\right)\right)} \]
   expm1-log1p [=>] 74.8	\[ \color{blue}{\frac{{k}^{-0.5}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, 0.5, -0.5\right)\right)}}} \]

6. Final simplification 74.8%

   \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, 0.5, -0.5\right)\right)}} \]

Alternatives

Alternative 1
Accuracy	74.7%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 2
Accuracy	74.8%
Cost	20032

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

Alternative 3
Accuracy	74.7%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4
Accuracy	74.8%
Cost	19904

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

Alternative 5
Accuracy	53.5%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+170}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 6
Accuracy	49.2%
Cost	19584

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

Alternative 7
Accuracy	37.0%
Cost	13184

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

Alternative 8
Accuracy	37.0%
Cost	13184

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

Alternative 9
Accuracy	37.0%
Cost	13184

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

Alternative 10
Accuracy	0.0%
Cost	13056

\[\sqrt{\frac{\frac{\pi}{0}}{k}} \]

Reproduce

herbie shell --seed 2023153 
(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))))