Average Error: 0.4 → 0.5
Time: 7.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\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 n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r128892 = 1.0;
        double r128893 = k;
        double r128894 = sqrt(r128893);
        double r128895 = r128892 / r128894;
        double r128896 = 2.0;
        double r128897 = atan2(1.0, 0.0);
        double r128898 = r128896 * r128897;
        double r128899 = n;
        double r128900 = r128898 * r128899;
        double r128901 = r128892 - r128893;
        double r128902 = r128901 / r128896;
        double r128903 = pow(r128900, r128902);
        double r128904 = r128895 * r128903;
        return r128904;
}
↓
double f(double k, double n) {
        double r128905 = 1.0;
        double r128906 = k;
        double r128907 = sqrt(r128906);
        double r128908 = r128905 / r128907;
        double r128909 = 2.0;
        double r128910 = atan2(1.0, 0.0);
        double r128911 = r128909 * r128910;
        double r128912 = n;
        double r128913 = r128911 * r128912;
        double r128914 = r128905 - r128906;
        double r128915 = r128914 / r128909;
        double r128916 = 2.0;
        double r128917 = r128915 / r128916;
        double r128918 = pow(r128913, r128917);
        double r128919 = r128918 * r128918;
        double r128920 = r128908 * r128919;
        return r128920;
}

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 sqr-pow0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))