Average Error: 0.5 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\sqrt{k}} \cdot {\left(n \cdot 2\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt{k}}}\]

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 add-sqr-sqrt0.6

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

  5. Applied unpow-prod-down0.6

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

  6. Applied times-frac0.6

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

  8. Applied div-sub0.6

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

  9. Applied pow-sub0.5

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

  10. Applied associate-/l/0.5

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

  11. Final simplification0.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))