Average Error: 0.5 → 0.4
Time: 11.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot \left({\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{n}\right)}^{\left(\frac{1}{2}\right)}\right)}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot \left({\left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{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)}}
double f(double k, double n) {
        double r99321 = 1.0;
        double r99322 = k;
        double r99323 = sqrt(r99322);
        double r99324 = r99321 / r99323;
        double r99325 = 2.0;
        double r99326 = atan2(1.0, 0.0);
        double r99327 = r99325 * r99326;
        double r99328 = n;
        double r99329 = r99327 * r99328;
        double r99330 = r99321 - r99322;
        double r99331 = r99330 / r99325;
        double r99332 = pow(r99329, r99331);
        double r99333 = r99324 * r99332;
        return r99333;
}
↓
double f(double k, double n) {
        double r99334 = 1.0;
        double r99335 = 2.0;
        double r99336 = atan2(1.0, 0.0);
        double r99337 = r99335 * r99336;
        double r99338 = n;
        double r99339 = sqrt(r99338);
        double r99340 = r99337 * r99339;
        double r99341 = r99334 / r99335;
        double r99342 = pow(r99340, r99341);
        double r99343 = pow(r99339, r99341);
        double r99344 = r99342 * r99343;
        double r99345 = r99334 * r99344;
        double r99346 = k;
        double r99347 = sqrt(r99346);
        double r99348 = r99337 * r99338;
        double r99349 = r99346 / r99335;
        double r99350 = pow(r99348, r99349);
        double r99351 = r99347 * r99350;
        double r99352 = r99345 / r99351;
        return r99352;
}

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-sqrt0.4

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

  8. Applied associate-*r*0.4

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

  10. Applied unpow-prod-down0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))