Average Error: 0.5 → 0.4
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\pi \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k}} \cdot \frac{{2}^{\left(\frac{1}{2}\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied div-sub0.5

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

  4. Applied pow-sub0.4

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

  5. Applied frac-times0.4

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

  6. Applied simplify0.4

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

  8. Applied unpow-prod-down0.4

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

  9. Applied times-frac0.4

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018207 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))