Migdal et al, Equation (51)

Average Error: 0.5 → 0.5

Time: 13.0s

Precision: binary64

Cost: 19904

Math

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

↓

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

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

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

   [Start] 0.5	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   rational.json-simplify-50 [=>] 0.5	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]
   rational.json-simplify-27 [<=] 0.5	\[ \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{1}}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \frac{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{1}} \]
   rational.json-simplify-24 [=>] 0.5	\[ \frac{{\color{blue}{\left(2 \cdot \left(n \cdot \pi\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{1}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \frac{{\left(2 \cdot \color{blue}{\left(\pi \cdot n\right)}\right)}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{1}} \]
   rational.json-simplify-21 [=>] 0.5	\[ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\frac{\sqrt{k}}{1}} \]
   metadata-eval [=>] 0.5	\[ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{1}} \]
   rational.json-simplify-26 [=>] 0.5	\[ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\color{blue}{\sqrt{k}}} \]

3. Taylor expanded in k around inf 0.5

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

4. Simplified 0.5

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

   [Start] 0.5	\[ \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \frac{{\left(2 \cdot \color{blue}{\left(\pi \cdot n\right)}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-26 [<=] 0.5	\[ \frac{{\left(2 \cdot \left(\pi \cdot \color{blue}{\frac{n}{1}}\right)\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-27 [<=] 0.5	\[ \frac{{\left(2 \cdot \color{blue}{\frac{\pi}{\frac{1}{n}}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-67 [=>] 0.5	\[ \frac{{\left(2 \cdot \color{blue}{\left(\left(\pi + \pi\right) \cdot \frac{0.5}{\frac{1}{n}}\right)}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-24 [=>] 0.5	\[ \frac{{\color{blue}{\left(\left(\pi + \pi\right) \cdot \left(2 \cdot \frac{0.5}{\frac{1}{n}}\right)\right)}}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-13 [=>] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \left(2 \cdot \color{blue}{\frac{n}{\frac{1}{0.5}}}\right)\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   metadata-eval [=>] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \left(2 \cdot \frac{n}{\color{blue}{2}}\right)\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-27 [<=] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \color{blue}{\frac{2}{\frac{2}{n}}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-13 [=>] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \color{blue}{\frac{n}{\frac{2}{2}}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   metadata-eval [=>] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \frac{n}{\color{blue}{1}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   metadata-eval [<=] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \frac{n}{\color{blue}{\frac{0.5}{0.5}}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-13 [<=] 0.5	\[ \frac{{\left(\left(\pi + \pi\right) \cdot \color{blue}{\frac{0.5}{\frac{0.5}{n}}}\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]
   rational.json-simplify-67 [<=] 0.5	\[ \frac{{\color{blue}{\left(\frac{\pi}{\frac{0.5}{n}}\right)}}^{\left(0.5 - 0.5 \cdot k\right)}}{\sqrt{k}} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	19904

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

Alternative 2
Error	21.8
Cost	19840

\[\frac{1}{\sqrt{k}} \cdot \sqrt{2 \cdot \frac{\pi}{\frac{1}{n}}} \]

Alternative 3
Error	21.7
Cost	19584

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

Reproduce

herbie shell --seed 2023073 
(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))))