Average Error: 0.5 → 0.4
Time: 13.9s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double code(double k, double n) {
	return ((double) ((1.0 / ((double) sqrt(k))) * ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), (((double) (1.0 - k)) / 2.0)))));
}
↓
double code(double k, double n) {
	return (((double) (1.0 * (((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), (1.0 / 2.0))) / ((double) sqrt(k))))) / ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * 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 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 associate-*r/0.4

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

  7. Applied div-inv0.4

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

  8. Applied associate-*l*0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))