Average Error: 0.5 → 0.4
Time: 7.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\sqrt{n \cdot \frac{2 \cdot \pi}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{k}}} \cdot {k}^{-0.5} \]
(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (* (sqrt (* n (/ (* 2.0 PI) (pow (* 2.0 (* n PI)) k)))) (pow k -0.5)))
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 sqrt((n * ((2.0 * ((double) M_PI)) / pow((2.0 * (n * ((double) M_PI))), k)))) * pow(k, -0.5);
}

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.sqrt((n * ((2.0 * Math.PI) / Math.pow((2.0 * (n * Math.PI)), k)))) * Math.pow(k, -0.5);
}

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.sqrt((n * ((2.0 * math.pi) / math.pow((2.0 * (n * math.pi)), k)))) * math.pow(k, -0.5)

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(sqrt(Float64(n * Float64(Float64(2.0 * pi) / (Float64(2.0 * Float64(n * pi)) ^ k)))) * (k ^ -0.5))
end

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 = sqrt((n * ((2.0 * pi) / ((2.0 * (n * pi)) ^ k)))) * (k ^ -0.5);
end

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[Sqrt[N[(n * N[(N[(2.0 * Pi), $MachinePrecision] / N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.5

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

  3. Taylor expanded in n around 0 3.5

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

  4. Simplified0.5

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

  5. Applied egg-rr0.4

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

  6. Applied egg-rr0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2022155 
(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))))