Average Error: 0.5 → 0.5
Time: 13.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\left(1 \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1}{2}\right)}\right)\right) \cdot {n}^{\left(\frac{1}{2}\right)}}{\left(\sqrt{k} \cdot {\pi}^{\left(\frac{k}{2}\right)}\right) \cdot {n}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\left(1 \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1}{2}\right)}\right)\right) \cdot {n}^{\left(\frac{1}{2}\right)}}{\left(\sqrt{k} \cdot {\pi}^{\left(\frac{k}{2}\right)}\right) \cdot {n}^{\left(\frac{k}{2}\right)}}
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 (((1.0 * (pow(2.0, ((1.0 - k) / 2.0)) * pow(((double) M_PI), (1.0 / 2.0)))) * pow(n, (1.0 / 2.0))) / ((sqrt(k) * pow(((double) M_PI), (k / 2.0))) * pow(n, (k / 2.0))));
}

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. Using strategy rm

  3. Applied unpow-prod-down0.7

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

  4. Applied associate-*r*0.7

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

  6. Applied unpow-prod-down0.6

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

  8. Applied div-sub0.6

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

  9. Applied pow-sub0.5

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

  10. Applied div-sub0.5

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

  11. Applied pow-sub0.5

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

  12. Applied associate-*r/0.5

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

  13. Applied frac-times0.5

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

  14. Applied frac-times0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))