Average Error: 0.4 → 0.4
Time: 38.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}} \cdot {2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right) \cdot {n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}} \cdot {2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right) \cdot {n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}
double f(double k, double n) {
        double r4165912 = 1.0;
        double r4165913 = k;
        double r4165914 = sqrt(r4165913);
        double r4165915 = r4165912 / r4165914;
        double r4165916 = 2.0;
        double r4165917 = atan2(1.0, 0.0);
        double r4165918 = r4165916 * r4165917;
        double r4165919 = n;
        double r4165920 = r4165918 * r4165919;
        double r4165921 = r4165912 - r4165913;
        double r4165922 = r4165921 / r4165916;
        double r4165923 = pow(r4165920, r4165922);
        double r4165924 = r4165915 * r4165923;
        return r4165924;
}
↓
double f(double k, double n) {
        double r4165925 = atan2(1.0, 0.0);
        double r4165926 = 0.5;
        double r4165927 = k;
        double r4165928 = 2.0;
        double r4165929 = r4165927 / r4165928;
        double r4165930 = r4165926 - r4165929;
        double r4165931 = pow(r4165925, r4165930);
        double r4165932 = sqrt(r4165927);
        double r4165933 = r4165931 / r4165932;
        double r4165934 = pow(r4165928, r4165930);
        double r4165935 = r4165933 * r4165934;
        double r4165936 = n;
        double r4165937 = pow(r4165936, r4165930);
        double r4165938 = r4165935 * r4165937;
        return r4165938;
}

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

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

  5. Applied sqrt-prod0.4

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

  6. Applied unpow-prod-down0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

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

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

  11. Applied sqrt-prod0.5

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

  12. Applied unpow-prod-down0.5

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

  13. Applied times-frac0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019141 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))