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

Error

Bits error versus k

Bits error versus n

Try it out

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

    \[\leadsto \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
  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 unpow-prod-down0.6

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

  7. Applied *-un-lft-identity0.6

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

  8. Applied times-frac0.5

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

  9. Applied associate-/r*0.5

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

  11. Applied sub-neg0.5

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

  12. Applied unpow-prod-up0.5

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

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

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

  14. Applied times-frac0.5

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

  15. Applied add-sqr-sqrt0.5

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

  16. Applied times-frac0.5

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

  17. Simplified0.5

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

  18. Simplified0.5

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

  19. Final simplification0.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))