Migdal et al, Equation (51)

Average Error: 0.5 → 0.5

Time: 11.5s

Precision: binary64

Cost: 32768

Math

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

↓

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

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

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

   [Start] 0.5	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   associate-*l/ [=>] 0.5	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
   *-lft-identity [=>] 0.5	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
   sqr-pow [=>] 0.6	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
   sqr-pow [<=] 0.5	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
   *-commutative [=>] 0.5	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
   associate-*l* [=>] 0.5	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
   div-sub [=>] 0.5	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 0.5	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

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

5. Applied egg-rr 0.6

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}} \]

6. Simplified 0.5

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

   [Start] 0.6	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25} \]
   associate-*l/ [=>] 0.6	\[ \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
   pow-sqr [=>] 0.5	\[ \frac{\color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot 0.25\right)}}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}} \]
   metadata-eval [=>] 0.5	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{0.5}}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}} \]
   unpow1/2 [=>] 0.5	\[ \frac{\color{blue}{\sqrt{\pi \cdot \left(2 \cdot n\right)}}}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}} \]
   unpow1/2 [=>] 0.5	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\color{blue}{\sqrt{k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}}}} \]

7. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.7
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	19904

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

Alternative 3
Error	21.8
Cost	19584

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

Alternative 4
Error	21.9
Cost	19584

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

Alternative 5
Error	31.9
Cost	13248

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

Alternative 6
Error	31.9
Cost	13248

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

Alternative 7
Error	32.6
Cost	13184

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

Alternative 8
Error	32.6
Cost	13184

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

Alternative 9
Error	32.6
Cost	13184

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

Alternative 10
Error	32.6
Cost	13184

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

Reproduce

herbie shell --seed 2023047 
(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))))