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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

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

4. Applied sub-neg0.4

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

5. Applied unpow-prod-up0.4

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

6. Applied associate-/l*0.4

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

7. Simplified0.4

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

9. Applied associate-/r/0.4

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

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

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

12. Applied sqrt-prod0.5

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

13. Applied times-frac0.5

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

14. Simplified0.5

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

16. Applied add-sqr-sqrt0.4

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

17. Final simplification0.4

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

Runtime

Time bar (total: 39.5s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))