Average Error: 0.4 → 0.5
Time: 24.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right)}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \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)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right)}{\sqrt{k}}
double f(double k, double n) {
        double r81952 = 1.0;
        double r81953 = k;
        double r81954 = sqrt(r81953);
        double r81955 = r81952 / r81954;
        double r81956 = 2.0;
        double r81957 = atan2(1.0, 0.0);
        double r81958 = r81956 * r81957;
        double r81959 = n;
        double r81960 = r81958 * r81959;
        double r81961 = r81952 - r81953;
        double r81962 = r81961 / r81956;
        double r81963 = pow(r81960, r81962);
        double r81964 = r81955 * r81963;
        return r81964;
}
↓
double f(double k, double n) {
        double r81965 = 1.0;
        double r81966 = 2.0;
        double r81967 = atan2(1.0, 0.0);
        double r81968 = r81966 * r81967;
        double r81969 = n;
        double r81970 = r81968 * r81969;
        double r81971 = k;
        double r81972 = r81965 - r81971;
        double r81973 = r81972 / r81966;
        double r81974 = 2.0;
        double r81975 = r81973 / r81974;
        double r81976 = pow(r81970, r81975);
        double r81977 = pow(r81968, r81975);
        double r81978 = pow(r81969, r81975);
        double r81979 = r81977 * r81978;
        double r81980 = r81976 * r81979;
        double r81981 = r81965 * r81980;
        double r81982 = sqrt(r81971);
        double r81983 = r81981 / r81982;
        return r81983;
}

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 associate-*l/0.4

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

  5. Applied sqr-pow0.5

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

  7. Applied unpow-prod-down0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019298 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))