Average Error: 0.5 → 0.5
Time: 9.5s
Precision: binary64
Cost: 33024
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\frac{{\left(2 \cdot \pi\right)}^{\left(0.5 - 0.5 \cdot k\right)}}{\frac{\sqrt{k}}{\sqrt{n} \cdot {n}^{\left(k \cdot -0.5\right)}}} \]
(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 (* 2.0 PI) (- 0.5 (* 0.5 k)))
  (/ (sqrt k) (* (sqrt n) (pow n (* 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 pow((2.0 * ((double) M_PI)), (0.5 - (0.5 * k))) / (sqrt(k) / (sqrt(n) * pow(n, (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.pow((2.0 * Math.PI), (0.5 - (0.5 * k))) / (Math.sqrt(k) / (Math.sqrt(n) * Math.pow(n, (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.pow((2.0 * math.pi), (0.5 - (0.5 * k))) / (math.sqrt(k) / (math.sqrt(n) * math.pow(n, (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((Float64(2.0 * pi) ^ Float64(0.5 - Float64(0.5 * k))) / Float64(sqrt(k) / Float64(sqrt(n) * (n ^ Float64(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 = ((2.0 * pi) ^ (0.5 - (0.5 * k))) / (sqrt(k) / (sqrt(n) * (n ^ (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[Power[N[(2.0 * Pi), $MachinePrecision], N[(0.5 - N[(0.5 * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] / N[(N[Sqrt[n], $MachinePrecision] * N[Power[n, N[(k * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

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. Applied egg-rr0.6

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error0.4
Cost32896
\[\begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \frac{\frac{\sqrt{t_0}}{{t_0}^{\left(0.5 \cdot k\right)}}}{\sqrt{k}} \end{array} \]
Alternative 2
Error0.6
Cost26624
\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot k\\ \frac{{\left(2 \cdot \pi\right)}^{t_0}}{\frac{\sqrt{k}}{{n}^{t_0}}} \end{array} \]
Alternative 3
Error0.5
Cost19972
\[\begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]
Alternative 4
Error0.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
Alternative 5
Error0.5
Cost19904
\[\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]
Alternative 6
Error12.9
Cost19716
\[\begin{array}{l} \mathbf{if}\;k \leq 10000000:\\ \;\;\;\;\sqrt{\pi \cdot n} \cdot \sqrt{\frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + 2 \cdot \frac{\pi \cdot n}{k}\right) + -1}\\ \end{array} \]
Alternative 7
Error12.8
Cost19716
\[\begin{array}{l} \mathbf{if}\;k \leq 10000000:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + 2 \cdot \frac{\pi \cdot n}{k}\right) + -1}\\ \end{array} \]
Alternative 8
Error22.8
Cost13572
\[\begin{array}{l} \mathbf{if}\;k \leq 10000000:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + 2 \cdot \frac{\pi \cdot n}{k}\right) + -1}\\ \end{array} \]
Alternative 9
Error32.4
Cost13312
\[\frac{1}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}} \]
Alternative 10
Error32.9
Cost13184
\[\sqrt{2 \cdot \frac{\pi}{\frac{k}{n}}} \]
Alternative 11
Error32.9
Cost13184
\[\sqrt{\frac{n}{k} \cdot \left(2 \cdot \pi\right)} \]

Error

Reproduce

herbie shell --seed 2022335 
(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))))