Average Error: 0.4 → 0.5
Time: 2.4m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)} \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)} \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r6736920 = 1.0;
        double r6736921 = k;
        double r6736922 = sqrt(r6736921);
        double r6736923 = r6736920 / r6736922;
        double r6736924 = 2.0;
        double r6736925 = atan2(1.0, 0.0);
        double r6736926 = r6736924 * r6736925;
        double r6736927 = n;
        double r6736928 = r6736926 * r6736927;
        double r6736929 = r6736920 - r6736921;
        double r6736930 = r6736929 / r6736924;
        double r6736931 = pow(r6736928, r6736930);
        double r6736932 = r6736923 * r6736931;
        return r6736932;
}
↓
double f(double k, double n) {
        double r6736933 = atan2(1.0, 0.0);
        double r6736934 = 2.0;
        double r6736935 = r6736933 * r6736934;
        double r6736936 = n;
        double r6736937 = r6736935 * r6736936;
        double r6736938 = 0.5;
        double r6736939 = k;
        double r6736940 = r6736939 / r6736934;
        double r6736941 = r6736938 - r6736940;
        double r6736942 = r6736941 / r6736934;
        double r6736943 = pow(r6736937, r6736942);
        double r6736944 = sqrt(r6736939);
        double r6736945 = r6736943 / r6736944;
        double r6736946 = r6736943 * r6736945;
        return r6736946;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied sqr-pow0.5

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

  6. Applied times-frac0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))