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. 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. Taylor expanded around inf 17.8

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

4. Simplified0.6

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

6. Applied unpow-prod-down0.5

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

8. Applied associate-/l*0.5

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

9. Final simplification0.5

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

Runtime

Time bar (total: 57.4s)Debug logProfile

herbie shell --seed 2018351 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))