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Derivation

1. Initial program 0.5

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

2. Initial simplification0.4

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

4. Applied div-inv0.5

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

6. Applied pow1/20.5

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

7. Applied pow-flip0.4

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

9. Applied unpow-prod-down0.6

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

10. Applied associate-*l*0.6

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

12. Applied pow-to-exp0.5

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

13. Final simplification0.5

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

Runtime

Time bar (total: 57.6s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))