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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 unpow-prod-down0.6

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

5. Applied associate-/l*0.6

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

7. Applied pow-to-exp0.5

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

9. Applied sub-neg0.5

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

10. Applied unpow-prod-up0.4

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

11. Applied add-sqr-sqrt0.6

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

12. Applied times-frac0.5

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

13. Simplified0.5

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

14. Final simplification0.5

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018340 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))