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

Error

Bits error versus k

Bits error versus n

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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

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

  4. Applied sub-neg0.5

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

  5. Applied unpow-prod-up0.4

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

  6. Applied associate-/l*0.4

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

  7. Simplified0.4

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

  9. Applied pow-neg0.4

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

  10. Applied associate-/r/0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))