Average Error: 0.5 → 0.6
Time: 56.3s
Precision: 64
Internal Precision: 1408
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(n \cdot \sqrt{\pi + \pi}\right) \cdot \sqrt{\pi + \pi}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]

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

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied associate-*r*0.6

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

Runtime

Time bar (total: 56.3s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))