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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-sub0.4

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

5. Applied pow-sub0.4

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

7. Applied unpow-prod-down0.5

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

9. Applied pow-to-exp0.4

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

10. Final simplification0.4

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018219 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))