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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 add-sqr-sqrt0.7

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

6. Applied *-un-lft-identity0.7

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

7. Applied unpow-prod-down0.7

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

8. Applied times-frac0.6

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

9. Applied sqrt-prod0.6

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

10. Applied associate-*r*0.6

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

12. Applied unpow-prod-down0.6

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

13. Applied associate-/l*0.6

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

14. Final simplification0.6

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

Runtime

Time bar (total: 57.5s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))