Average Error: 0.5 → 0.5
Time: 5.1s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)} \cdot {k}^{-0.5} \]
(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (* (pow (* n (* 2.0 PI)) (fma k -0.5 0.5)) (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 pow((n * (2.0 * ((double) M_PI))), fma(k, -0.5, 0.5)) * 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((Float64(n * Float64(2.0 * pi)) ^ fma(k, -0.5, 0.5)) * (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[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(k * -0.5 + 0.5), $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)}
↓
{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)} \cdot {k}^{-0.5}

Error

Bits error versus k

Bits error versus n

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

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022165 
(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))))