Average Error: 0.4 → 0.5
Time: 9.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r137967 = 1.0;
        double r137968 = k;
        double r137969 = sqrt(r137968);
        double r137970 = r137967 / r137969;
        double r137971 = 2.0;
        double r137972 = atan2(1.0, 0.0);
        double r137973 = r137971 * r137972;
        double r137974 = n;
        double r137975 = r137973 * r137974;
        double r137976 = r137967 - r137968;
        double r137977 = r137976 / r137971;
        double r137978 = pow(r137975, r137977);
        double r137979 = r137970 * r137978;
        return r137979;
}
↓
double f(double k, double n) {
        double r137980 = 1.0;
        double r137981 = k;
        double r137982 = sqrt(r137981);
        double r137983 = r137980 / r137982;
        double r137984 = 2.0;
        double r137985 = atan2(1.0, 0.0);
        double r137986 = r137984 * r137985;
        double r137987 = n;
        double r137988 = r137986 * r137987;
        double r137989 = r137980 - r137981;
        double r137990 = r137989 / r137984;
        double r137991 = 2.0;
        double r137992 = r137990 / r137991;
        double r137993 = r137992 / r137991;
        double r137994 = pow(r137988, r137993);
        double r137995 = r137994 * r137994;
        double r137996 = r137983 * r137995;
        double r137997 = pow(r137988, r137992);
        double r137998 = r137996 * r137997;
        return r137998;
}

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-pow0.4

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

  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 sqr-pow0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))