Average Error: 0.5 → 0.5
Time: 4.5m
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}}{{\pi}^{\left(\frac{k}{2}\right)}} \cdot \frac{1}{\sqrt{k}}\right) \cdot \frac{1}{{\left(n \cdot 2\right)}^{\left(\frac{k}{2}\right)}}\]

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 simplification0.5

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

  4. Applied div-sub0.5

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

  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. Using strategy rm

  7. Applied add-sqr-sqrt0.6

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

  8. Applied associate-/r*0.6

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

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

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

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

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

  12. Applied unpow-prod-down0.7

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

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

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

  14. Applied times-frac0.7

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

  15. Applied times-frac0.7

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

  16. Applied times-frac0.7

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

  17. Simplified0.7

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

  18. Simplified0.5

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

  20. Applied div-inv0.5

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

  21. Final simplification0.5

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))