Average Error: 0.5 → 0.4
Time: 5.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \frac{\sqrt{t_0 \cdot {\left({\left({t_0}^{k}\right)}^{-0.5}\right)}^{2}}}{\sqrt{k}} \end{array} \]
(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 (* n (* PI 2.0))))
   (/ (sqrt (* t_0 (pow (pow (pow t_0 k) -0.5) 2.0))) (sqrt k))))
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 = n * (((double) M_PI) * 2.0);
	return sqrt((t_0 * pow(pow(pow(t_0, k), -0.5), 2.0))) / sqrt(k);
}

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 = n * (Math.PI * 2.0);
	return Math.sqrt((t_0 * Math.pow(Math.pow(Math.pow(t_0, k), -0.5), 2.0))) / Math.sqrt(k);
}

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 = n * (math.pi * 2.0)
	return math.sqrt((t_0 * math.pow(math.pow(math.pow(t_0, k), -0.5), 2.0))) / math.sqrt(k)

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(n * Float64(pi * 2.0))
	return Float64(sqrt(Float64(t_0 * (((t_0 ^ k) ^ -0.5) ^ 2.0))) / sqrt(k))
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)
	t_0 = n * (pi * 2.0);
	tmp = sqrt((t_0 * (((t_0 ^ k) ^ -0.5) ^ 2.0))) / sqrt(k);
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_] := Block[{t$95$0 = N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[N[(t$95$0 * N[Power[N[Power[N[Power[t$95$0, k], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

\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 := n \cdot \left(\pi \cdot 2\right)\\
\frac{\sqrt{t_0 \cdot {\left({\left({t_0}^{k}\right)}^{-0.5}\right)}^{2}}}{\sqrt{k}}
\end{array}

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. Simplified0.4

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

  3. Applied egg-rr0.3

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

  4. Taylor expanded in k around inf 0.4

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022162 
(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))))