Average Error: 0.5 → 0.8
Time: 10.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot {\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\sqrt{1} + \sqrt{k}}{1}\right)}\right)}^{\left(\frac{\sqrt{1} - \sqrt{k}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot {\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\sqrt{1} + \sqrt{k}}{1}\right)}\right)}^{\left(\frac{\sqrt{1} - \sqrt{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) sqrt(k)))) * ((double) pow(((double) pow(((double) (((double) (2.0 * ((double) M_PI))) * n)), ((double) (((double) (((double) sqrt(1.0)) + ((double) sqrt(k)))) / 1.0)))), ((double) (((double) (((double) sqrt(1.0)) - ((double) sqrt(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 *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied add-sqr-sqrt0.6

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

  6. Applied difference-of-squares0.6

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

  7. Applied times-frac0.6

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

  8. Applied pow-unpow0.8

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

  9. Final simplification0.8

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

Reproduce

herbie shell --seed 2020121 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))