Average Error: 0.4 → 0.4
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{1}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]

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 0.4

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

  2. Initial simplification0.3

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

  4. Applied clear-num0.4

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

  6. Applied unpow-prod-down0.5

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

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

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

  8. Applied times-frac0.4

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

  9. Final simplification0.4

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018285 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))