Average Error: 0.4 → 0.4
Time: 4.9m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\left(\frac{1 - k}{2}\right)}}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\left(\frac{1 - k}{2}\right)}}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r36593858 = 1.0;
        double r36593859 = k;
        double r36593860 = sqrt(r36593859);
        double r36593861 = r36593858 / r36593860;
        double r36593862 = 2.0;
        double r36593863 = atan2(1.0, 0.0);
        double r36593864 = r36593862 * r36593863;
        double r36593865 = n;
        double r36593866 = r36593864 * r36593865;
        double r36593867 = r36593858 - r36593859;
        double r36593868 = r36593867 / r36593862;
        double r36593869 = pow(r36593866, r36593868);
        double r36593870 = r36593861 * r36593869;
        return r36593870;
}
↓
double f(double k, double n) {
        double r36593871 = n;
        double r36593872 = 1.0;
        double r36593873 = k;
        double r36593874 = r36593872 - r36593873;
        double r36593875 = 2.0;
        double r36593876 = r36593874 / r36593875;
        double r36593877 = pow(r36593871, r36593876);
        double r36593878 = sqrt(r36593873);
        double r36593879 = atan2(1.0, 0.0);
        double r36593880 = pow(r36593879, r36593876);
        double r36593881 = r36593878 / r36593880;
        double r36593882 = r36593877 / r36593881;
        double r36593883 = pow(r36593875, r36593876);
        double r36593884 = r36593882 * r36593883;
        return r36593884;
}

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. Simplified0.3

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

  4. Applied unpow-prod-down0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied unpow-prod-down0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied times-frac0.4

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

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

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))