Average Error: 0.5 → 0.5
Time: 17.2s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}}
(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 2.0 (/ (- 1.0 k) 2.0))
  (/ (pow PI (/ (- 1.0 k) 2.0)) (/ (sqrt k) (pow n (/ (- 1.0 k) 2.0))))))
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(2.0, ((1.0 - k) / 2.0)) * (pow(((double) M_PI), ((1.0 - k) / 2.0)) / (sqrt(k) / pow(n, ((1.0 - 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. Simplified0.5

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

  4. Applied unpow-prod-down_binary64_8390.6

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

  5. Applied associate-/l*_binary64_7050.6

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

  7. Applied *-un-lft-identity_binary64_7600.6

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

  8. Applied unpow-prod-down_binary64_8390.6

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

  9. Applied *-un-lft-identity_binary64_7600.6

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

  10. Applied sqrt-prod_binary64_7760.6

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

  11. Applied times-frac_binary64_7660.6

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

  12. Applied unpow-prod-down_binary64_8390.6

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

  13. Applied times-frac_binary64_7660.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020352 
(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))))