Average Error: 0.5 → 0.4
Time: 1.5m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{1}}{\sqrt{k} \cdot {\left(\sqrt{2 \cdot \pi}\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{\sqrt{1}}{{\left(\sqrt{2 \cdot \pi} \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{1}}{\sqrt{k} \cdot {\left(\sqrt{2 \cdot \pi}\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{\sqrt{1}}{{\left(\sqrt{2 \cdot \pi} \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) (((double) (((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (1.0 / 2.0)))) * ((double) sqrt(1.0)))) / ((double) (((double) sqrt(k)) * ((double) pow(((double) sqrt(((double) (2.0 * ((double) M_PI))))), ((double) (k / 2.0)))))))) * ((double) (((double) sqrt(1.0)) / ((double) pow(((double) (((double) sqrt(((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. Simplified0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-*l*0.4

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

  10. Applied unpow-prod-down0.4

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

  11. Applied associate-*r*0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-*r*0.4

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

  14. Applied times-frac0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020113 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))