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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 simplification0.5

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

4. Applied pow-sub0.4

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

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

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

7. Applied div-inv0.4

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

8. Applied times-frac0.4

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

9. Simplified0.4

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

10. Final simplification0.4

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

Runtime

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herbie shell --seed 2018263 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))