Average Error: 0.5 → 0.5
Time: 12.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{n}\right)}^{\left(\frac{1}{2}\right)}\right)}{{\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{k}{2}\right)} \cdot {\left(\sqrt{n}\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{n}\right)}^{\left(\frac{1}{2}\right)}\right)}{{\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{k}{2}\right)} \cdot {\left(\sqrt{n}\right)}^{\left(\frac{k}{2}\right)}}
double code(double k, double n) {
	return ((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0)));
}
↓
double code(double k, double n) {
	return (((1.0 / sqrt(k)) * (pow(((2.0 * ((double) M_PI)) * sqrt(n)), (1.0 / 2.0)) * pow(sqrt(n), (1.0 / 2.0)))) / (pow(((2.0 * ((double) M_PI)) * sqrt(n)), (k / 2.0)) * pow(sqrt(n), (k / 2.0))));
}

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 add-sqr-sqrt0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.6

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

  8. Applied div-sub0.6

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

  9. Applied pow-sub0.5

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

  10. Applied div-sub0.5

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

  11. Applied pow-sub0.5

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

  12. Applied frac-times0.5

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

  13. Applied associate-*r/0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))