Average Error: 0.4 → 0.5
Time: 23.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r96960 = 1.0;
        double r96961 = k;
        double r96962 = sqrt(r96961);
        double r96963 = r96960 / r96962;
        double r96964 = 2.0;
        double r96965 = atan2(1.0, 0.0);
        double r96966 = r96964 * r96965;
        double r96967 = n;
        double r96968 = r96966 * r96967;
        double r96969 = r96960 - r96961;
        double r96970 = r96969 / r96964;
        double r96971 = pow(r96968, r96970);
        double r96972 = r96963 * r96971;
        return r96972;
}
↓
double f(double k, double n) {
        double r96973 = 1.0;
        double r96974 = k;
        double r96975 = sqrt(r96974);
        double r96976 = r96973 / r96975;
        double r96977 = 2.0;
        double r96978 = atan2(1.0, 0.0);
        double r96979 = r96977 * r96978;
        double r96980 = n;
        double r96981 = r96979 * r96980;
        double r96982 = r96973 - r96974;
        double r96983 = r96982 / r96977;
        double r96984 = 2.0;
        double r96985 = r96983 / r96984;
        double r96986 = pow(r96981, r96985);
        double r96987 = r96976 * r96986;
        double r96988 = pow(r96979, r96985);
        double r96989 = pow(r96980, r96985);
        double r96990 = r96988 * r96989;
        double r96991 = r96987 * r96990;
        return r96991;
}

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. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))