Average Error: 0.5 → 0.5
Time: 48.1s
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\sqrt{2 \cdot n}}{\frac{\sqrt{k}}{{\left(2 \cdot n\right)}^{\left(-\frac{k}{2}\right)}}} \cdot {\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\]

Error

Bits error versus k

Bits error versus n

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Results

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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 *-un-lft-identity0.5

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

  5. Applied unpow-prod-down0.6

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

  6. Applied times-frac0.6

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

  7. Simplified0.6

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

  9. Applied sub-neg0.6

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

  10. Applied unpow-prod-up0.5

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

  11. Applied associate-/l*0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Runtime

Time bar (total: 48.1s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))