Average Error: 0.5 → 0.5
Time: 11.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {2}^{\left(\frac{1}{2}\right)}}{\frac{\sqrt{k}}{1} \cdot {2}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {2}^{\left(\frac{1}{2}\right)}}{\frac{\sqrt{k}}{1} \cdot {2}^{\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 (((pow(((double) M_PI), ((1.0 - k) / 2.0)) * pow(n, ((1.0 - k) / 2.0))) * pow(2.0, (1.0 / 2.0))) / ((sqrt(k) / 1.0) * pow(2.0, (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 associate-*l*0.5

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

  4. Applied unpow-prod-down0.6

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

  5. Applied associate-*r*0.6

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

  7. Applied div-sub0.6

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

  8. Applied pow-sub0.6

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

  9. Applied clear-num0.6

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

  10. Applied frac-times0.5

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

  11. Applied associate-*l/0.5

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

  12. Simplified0.5

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

  14. Applied unpow-prod-down0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))