Average Error: 0.4 → 0.5
Time: 28.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r74919 = 1.0;
        double r74920 = k;
        double r74921 = sqrt(r74920);
        double r74922 = r74919 / r74921;
        double r74923 = 2.0;
        double r74924 = atan2(1.0, 0.0);
        double r74925 = r74923 * r74924;
        double r74926 = n;
        double r74927 = r74925 * r74926;
        double r74928 = r74919 - r74920;
        double r74929 = r74928 / r74923;
        double r74930 = pow(r74927, r74929);
        double r74931 = r74922 * r74930;
        return r74931;
}
↓
double f(double k, double n) {
        double r74932 = 1.0;
        double r74933 = k;
        double r74934 = sqrt(r74933);
        double r74935 = r74932 / r74934;
        double r74936 = sqrt(r74935);
        double r74937 = 2.0;
        double r74938 = atan2(1.0, 0.0);
        double r74939 = r74937 * r74938;
        double r74940 = n;
        double r74941 = r74939 * r74940;
        double r74942 = r74932 - r74933;
        double r74943 = r74942 / r74937;
        double r74944 = pow(r74941, r74943);
        double r74945 = r74936 * r74944;
        double r74946 = r74936 * r74945;
        return r74946;
}

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-sqrt0.5

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

  4. Applied associate-*l*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019306 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))