Average Error: 0.5 → 0.4
Time: 9.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ {t_0}^{\left(k \cdot -0.5\right)} \cdot \left(\sqrt{t_0} \cdot {k}^{-0.5}\right) \end{array} \]
(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 (* (* 2.0 PI) n)))
   (* (pow t_0 (* k -0.5)) (* (sqrt t_0) (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) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	return pow(t_0, (k * -0.5)) * (sqrt(t_0) * 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) {
	double t_0 = (2.0 * Math.PI) * n;
	return Math.pow(t_0, (k * -0.5)) * (Math.sqrt(t_0) * 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):
	t_0 = (2.0 * math.pi) * n
	return math.pow(t_0, (k * -0.5)) * (math.sqrt(t_0) * 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)
	t_0 = Float64(Float64(2.0 * pi) * n)
	return Float64((t_0 ^ Float64(k * -0.5)) * Float64(sqrt(t_0) * (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)
	t_0 = (2.0 * pi) * n;
	tmp = (t_0 ^ (k * -0.5)) * (sqrt(t_0) * (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_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $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 := \left(2 \cdot \pi\right) \cdot n\\
{t_0}^{\left(k \cdot -0.5\right)} \cdot \left(\sqrt{t_0} \cdot {k}^{-0.5}\right)
\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. Simplified0.5

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022139 
(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))))