Average Error: 0.4 → 0.4
Time: 2.9m
Precision: 64
Internal Precision: 1408
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]

Error

Bits error versus k

Bits error versus n

Derivation

  1. Initial program 0.4

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

  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. Removed slow pow expressions.

Runtime

Time bar (total: 2.9m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))