Average Error: 0.5 → 0.4
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{{\left(\sqrt{n + n} \cdot \left(\sqrt{n + n} \cdot \pi\right)\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]

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 simplify0.4

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

  4. Applied div-sub0.4

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

  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-sqrt0.4

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

  8. Applied associate-*l*0.4

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))