Average Error: 0.5 → 0.6
Time: 1.2m
Precision: 64
Internal Precision: 1408
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{\frac{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}} \cdot \sqrt{\frac{\frac{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}}\]

Error

Bits error versus k

Bits error versus n

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. Applied simplify0.5

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

  4. Applied div-sub0.5

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

  5. Applied pow-sub0.4

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

  7. Applied add-sqr-sqrt0.6

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))