Average Error: 0.4 → 0.5
Time: 26.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}
double f(double k, double n) {
        double r5809927 = 1.0;
        double r5809928 = k;
        double r5809929 = sqrt(r5809928);
        double r5809930 = r5809927 / r5809929;
        double r5809931 = 2.0;
        double r5809932 = atan2(1.0, 0.0);
        double r5809933 = r5809931 * r5809932;
        double r5809934 = n;
        double r5809935 = r5809933 * r5809934;
        double r5809936 = r5809927 - r5809928;
        double r5809937 = r5809936 / r5809931;
        double r5809938 = pow(r5809935, r5809937);
        double r5809939 = r5809930 * r5809938;
        return r5809939;
}
↓
double f(double k, double n) {
        double r5809940 = atan2(1.0, 0.0);
        double r5809941 = n;
        double r5809942 = r5809940 * r5809941;
        double r5809943 = 1.0;
        double r5809944 = k;
        double r5809945 = r5809943 - r5809944;
        double r5809946 = 2.0;
        double r5809947 = r5809945 / r5809946;
        double r5809948 = pow(r5809942, r5809947);
        double r5809949 = pow(r5809946, r5809947);
        double r5809950 = r5809948 * r5809949;
        double r5809951 = sqrt(r5809944);
        double r5809952 = r5809943 / r5809951;
        double r5809953 = r5809950 * r5809952;
        return r5809953;
}

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 0.4

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

  2. Taylor expanded around 0 0.4

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*r*0.5

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

  7. Applied unpow-prod-down0.4

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))