Average Error: 0.4 → 0.3
Time: 42.1s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{k}^{\frac{-1}{2}}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{-1}{2} + \frac{k}{2}\right)}}\]

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 clear-num0.4

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

  6. Applied div-inv0.4

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

  7. Applied associate-/r*0.4

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

  9. Applied pow1/20.4

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

  10. Applied pow-flip0.4

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

  11. Simplified0.4

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

  13. Applied pow-flip0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019089 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))