Average Error: 0.5 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\sqrt{\left(\pi \cdot 2\right) \cdot n}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]

Error

Bits error versus k

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied div-sub0.4

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

  5. Applied pow-sub0.4

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

  6. Taylor expanded around inf 3.5

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))