Average Error: 0.4 → 0.3
Time: 3.1m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(n \cdot \pi\right) \cdot 2\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]

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. 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

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

  5. Applied associate-/l*0.4

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

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

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

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

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

  9. Applied sqrt-prod0.4

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

  10. Applied times-frac0.4

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

  11. Applied add-cube-cbrt0.4

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))