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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 pow-sub0.4

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

6. Applied add-sqr-sqrt0.4

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

7. Applied unpow-prod-down0.5

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

8. Applied times-frac0.5

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

9. Applied associate-/l*0.5

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

10. Simplified0.5

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

12. Applied add-cube-cbrt0.5

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

13. Applied unpow-prod-down0.4

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

14. Applied times-frac0.4

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

15. Applied associate-/r*0.4

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

16. Simplified0.4

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

17. Final simplification0.4

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

Runtime

Time bar (total: 8.8m)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))