Average Error: 0.4 → 0.5
Time: 28.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r102916 = 1.0;
        double r102917 = k;
        double r102918 = sqrt(r102917);
        double r102919 = r102916 / r102918;
        double r102920 = 2.0;
        double r102921 = atan2(1.0, 0.0);
        double r102922 = r102920 * r102921;
        double r102923 = n;
        double r102924 = r102922 * r102923;
        double r102925 = r102916 - r102917;
        double r102926 = r102925 / r102920;
        double r102927 = pow(r102924, r102926);
        double r102928 = r102919 * r102927;
        return r102928;
}
↓
double f(double k, double n) {
        double r102929 = 1.0;
        double r102930 = k;
        double r102931 = sqrt(r102930);
        double r102932 = sqrt(r102931);
        double r102933 = r102929 / r102932;
        double r102934 = r102933 / r102932;
        double r102935 = 2.0;
        double r102936 = atan2(1.0, 0.0);
        double r102937 = r102935 * r102936;
        double r102938 = n;
        double r102939 = r102937 * r102938;
        double r102940 = r102929 - r102930;
        double r102941 = r102940 / r102935;
        double r102942 = pow(r102939, r102941);
        double r102943 = r102934 * r102942;
        return r102943;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019303 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))