Average Error: 0.4 → 0.4
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\pi}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)}}} \cdot {2}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)}\]

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. Simplified0.3

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

  3. Taylor expanded around -inf 62.2

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

  4. Simplified0.4

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

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

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

  7. Applied div-inv0.4

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

  8. Applied times-frac0.4

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

  9. Applied unpow-prod-down0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  14. Applied unpow-prod-down0.4

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

  15. Applied associate-/l*0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019050 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))