Average Error: 0.5 → 0.4
Time: 9.6s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[1 \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{\frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1}{2}\right)}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
1 \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{\frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1}{2}\right)}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\right)
double code(double k, double n) {
	return ((double) ((1.0 / ((double) sqrt(k))) * ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), (((double) (1.0 - k)) / 2.0)))));
}
↓
double code(double k, double n) {
	return ((double) (1.0 * ((double) (((double) pow(2.0, (((double) (1.0 - k)) / 2.0))) * ((((double) (((double) pow(((double) M_PI), (((double) (1.0 - k)) / 2.0))) * ((double) pow(n, (1.0 / 2.0))))) / ((double) pow(n, (k / 2.0)))) / ((double) sqrt(k)))))));
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied sqrt-prod0.4

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

  6. Applied unpow-prod-down0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

  10. Applied unpow-prod-down0.5

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

  12. Applied div-sub0.5

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

  13. Applied pow-sub0.4

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

  14. Applied associate-*r/0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(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))))