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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 simplification0.4

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

4. Applied pow-sub0.3

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

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

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

7. Applied add-sqr-sqrt0.4

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

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

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

9. Applied times-frac0.4

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

10. Applied times-frac0.4

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

11. Simplified0.4

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

12. Simplified0.4

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

13. Final simplification0.4

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

Runtime

Time bar (total: 42.5s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))