Average Error: 0.5 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{k}^{\left(-\frac{1}{2}\right)} \cdot {\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{k}{2}\right)}}\]

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. Applied associate-/l/0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied div-inv0.5

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

  12. Applied pow1/20.5

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

  13. Applied pow-flip0.4

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

  14. Final simplification0.4

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))