Average Error: 0.6 → 0.6
Time: 11.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right) \cdot \sqrt{n}\right)}^{\left(\frac{1 - k}{2}\right)}\]
\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 \sqrt{n}\right) \cdot \sqrt{n}\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r168326 = 1.0;
        double r168327 = k;
        double r168328 = sqrt(r168327);
        double r168329 = r168326 / r168328;
        double r168330 = 2.0;
        double r168331 = atan2(1.0, 0.0);
        double r168332 = r168330 * r168331;
        double r168333 = n;
        double r168334 = r168332 * r168333;
        double r168335 = r168326 - r168327;
        double r168336 = r168335 / r168330;
        double r168337 = pow(r168334, r168336);
        double r168338 = r168329 * r168337;
        return r168338;
}
↓
double f(double k, double n) {
        double r168339 = 1.0;
        double r168340 = k;
        double r168341 = sqrt(r168340);
        double r168342 = r168339 / r168341;
        double r168343 = 2.0;
        double r168344 = atan2(1.0, 0.0);
        double r168345 = r168343 * r168344;
        double r168346 = n;
        double r168347 = sqrt(r168346);
        double r168348 = r168345 * r168347;
        double r168349 = r168348 * r168347;
        double r168350 = r168339 - r168340;
        double r168351 = r168350 / r168343;
        double r168352 = pow(r168349, r168351);
        double r168353 = r168342 * r168352;
        return r168353;
}

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 0.6

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

  3. Applied add-sqr-sqrt0.6

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

  4. Applied associate-*r*0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))