Average Error: 0.3 → 0.4
Time: 36.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{1}{2} \cdot k\right)} \cdot \frac{e^{\log \pi \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{1}{2} \cdot k\right)} \cdot \frac{e^{\log \pi \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r4529845 = 1.0;
        double r4529846 = k;
        double r4529847 = sqrt(r4529846);
        double r4529848 = r4529845 / r4529847;
        double r4529849 = 2.0;
        double r4529850 = atan2(1.0, 0.0);
        double r4529851 = r4529849 * r4529850;
        double r4529852 = n;
        double r4529853 = r4529851 * r4529852;
        double r4529854 = r4529845 - r4529846;
        double r4529855 = r4529854 / r4529849;
        double r4529856 = pow(r4529853, r4529855);
        double r4529857 = r4529848 * r4529856;
        return r4529857;
}
↓
double f(double k, double n) {
        double r4529858 = 2.0;
        double r4529859 = n;
        double r4529860 = r4529858 * r4529859;
        double r4529861 = 0.5;
        double r4529862 = k;
        double r4529863 = r4529861 * r4529862;
        double r4529864 = r4529861 - r4529863;
        double r4529865 = pow(r4529860, r4529864);
        double r4529866 = atan2(1.0, 0.0);
        double r4529867 = log(r4529866);
        double r4529868 = r4529867 * r4529864;
        double r4529869 = exp(r4529868);
        double r4529870 = sqrt(r4529862);
        double r4529871 = r4529869 / r4529870;
        double r4529872 = r4529865 * r4529871;
        return r4529872;
}

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 0.3

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

  2. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied sqrt-prod0.3

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

  6. Applied unpow-prod-down0.4

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

  7. Applied times-frac0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied add-exp-log0.4

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

  12. Applied pow-exp0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))