Error

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. Simplified0.3

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

4. Applied *-un-lft-identity0.3

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

5. Applied associate-/l*0.4

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

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

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

8. Applied add-sqr-sqrt0.4

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

9. Applied times-frac0.4

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

10. Simplified0.4

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

11. Simplified0.3

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

12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019100 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))