Average Error: 0.5 → 0.5
Time: 46.0s
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\frac{\frac{1}{2}}{\pi \cdot n}}}}{\frac{\sqrt{k}}{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(-\frac{k}{2}\right)}}}\]

Error

Bits error versus k

Bits error versus n

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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 simplification0.5

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

  4. Applied clear-num0.5

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

  6. Applied sub-neg0.5

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

  7. Applied unpow-prod-up0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  10. Applied associate-/r*0.5

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

  11. Taylor expanded around 0 3.6

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Runtime

Time bar (total: 46.0s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))