Average Error: 0.5 → 0.5
Time: 10.2s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{{n}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}\right)\right)}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{{n}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}\right)\right)}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double code(double k, double n) {
	return ((double) (((double) (1.0 / ((double) sqrt(k)))) * ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (((double) (1.0 - k)) / 2.0))))));
}
↓
double code(double k, double n) {
	return ((double) (((double) (1.0 * ((double) (((double) sqrt(((double) pow(((double) (2.0 * ((double) M_PI))), ((double) (1.0 / 2.0)))))) * ((double) (((double) sqrt(((double) pow(n, ((double) (1.0 / 2.0)))))) * ((double) sqrt(((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (1.0 / 2.0)))))))))))) / ((double) (((double) sqrt(k)) * ((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (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 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. Using strategy rm

  7. Applied add-sqr-sqrt0.6

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

  9. Applied unpow-prod-down0.6

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

  10. Applied sqrt-prod0.5

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

  11. Applied associate-*l*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020161 
(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))))