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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. Applied simplify0.5

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

4. Applied div-sub0.5

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

5. Applied pow-sub0.4

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

6. Applied associate-/l/0.4

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

8. Applied div-inv0.4

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

9. Taylor expanded around -inf 61.9

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

10. Applied simplify0.4

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))