Average Error: 0.4 → 0.5
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\sqrt[3]{\left(\pi \cdot 2\right) \cdot n} \cdot \left(\sqrt[3]{\left(\pi \cdot 2\right) \cdot n} \cdot \sqrt[3]{\left(\pi \cdot 2\right) \cdot n}\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}{\sqrt{k}}\]

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. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied sqr-pow0.5

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

  6. Applied times-frac0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Final simplification0.5

    \[\leadsto {\left(\sqrt[3]{\left(\pi \cdot 2\right) \cdot n} \cdot \left(\sqrt[3]{\left(\pi \cdot 2\right) \cdot n} \cdot \sqrt[3]{\left(\pi \cdot 2\right) \cdot n}\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}{\sqrt{k}}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))