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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. 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 *-un-lft-identity0.4

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

5. Applied associate-/l*0.4

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

7. Applied unpow-prod-down0.5

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

8. Applied *-un-lft-identity0.5

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

9. Applied times-frac0.5

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

11. Applied unpow-prod-down0.6

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

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

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

13. Applied times-frac0.6

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

14. Applied associate-*r*0.6

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

16. Applied frac-times0.5

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

17. Applied frac-times0.5

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

18. Simplified0.5

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

19. Final simplification0.5

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

Runtime

Time bar (total: 52.2s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))