Average Error: 0.5 → 0.5
Time: 10.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[1 \cdot \frac{{\left(\sqrt{\pi} \cdot n\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{{\left(2 \cdot \sqrt{\pi}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}{{\left(\sqrt{\pi} \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
1 \cdot \frac{{\left(\sqrt{\pi} \cdot n\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{{\left(2 \cdot \sqrt{\pi}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}{{\left(\sqrt{\pi} \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double code(double k, double n) {
	return ((double) (((double) (1.0 / ((double) sqrt(k)))) * ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (((double) (1.0 - k)) / 2.0))))));
}
↓
double code(double k, double n) {
	return ((double) (1.0 * ((double) (((double) (((double) pow(((double) (((double) sqrt(((double) M_PI))) * n)), ((double) (1.0 / 2.0)))) * ((double) (((double) pow(((double) (2.0 * ((double) sqrt(((double) M_PI))))), ((double) (((double) (1.0 - k)) / 2.0)))) / ((double) sqrt(k)))))) / ((double) pow(((double) (((double) sqrt(((double) M_PI))) * n)), ((double) (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 add-sqr-sqrt0.6

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

  4. Applied associate-*r*0.6

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

  5. Applied associate-*l*0.6

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

  6. Applied unpow-prod-down0.6

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

  7. Applied associate-*r*0.6

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

  9. Applied div-inv0.6

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

  10. Applied associate-*l*0.6

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

  11. Applied associate-*l*0.6

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

  12. Simplified0.5

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

  14. Applied div-sub0.5

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

  15. Applied pow-sub0.5

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

  16. Applied associate-*l/0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020114 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))