Average Error: 0.4 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \left({k}^{\frac{-1}{2}} \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]

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

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

  4. Applied div-inv0.4

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

  6. Applied pow10.4

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

  7. Applied sqrt-pow10.4

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

  8. Applied pow-flip0.4

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

  9. Simplified0.4

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

  11. Applied sqr-pow0.5

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

  12. Applied associate-*l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019090 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))