Average Error: 0.5 → 0.4
Time: 49.4s
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(n \cdot \pi\right) \cdot 2\right)}^{\left(\frac{1 - k}{2}\right)}}{{k}^{\frac{1}{2}}}\]

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. Using strategy rm

  3. Applied pow1/20.5

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

  4. Applied pow-flip0.4

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

  6. Applied pow-neg0.5

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

  7. Applied associate-*l/0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around -inf 0.4

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

  10. Final simplification0.4

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

Runtime

Time bar (total: 49.4s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))