Average Error: 0.4 → 0.4
Time: 30.4s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}} \cdot \left({n}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot {\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right)\]

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 0.4

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

  2. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied unpow-prod-down0.4

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

  6. Applied times-frac0.4

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

  7. Simplified0.4

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

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

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

  10. Applied unpow-prod-down0.4

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

  11. Applied times-frac0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))