Average Error: 0.4 → 0.4
Time: 3.6m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({n}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \sqrt{\pi}\right) \cdot \frac{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\left(-\frac{k}{2}\right)}}}\]

Error

Bits error versus k

Bits error versus n

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. Simplified0.4

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

  4. Applied unpow-prod-down0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied sub-neg0.5

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

  8. Applied unpow-prod-up0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied times-frac0.4

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

  11. Applied unpow-prod-down0.5

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

  12. Applied times-frac0.5

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019093 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))