Average Error: 0.4 → 0.5
Time: 4.5m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\frac{1}{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\frac{1}{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r6467711 = 1.0;
        double r6467712 = k;
        double r6467713 = sqrt(r6467712);
        double r6467714 = r6467711 / r6467713;
        double r6467715 = 2.0;
        double r6467716 = atan2(1.0, 0.0);
        double r6467717 = r6467715 * r6467716;
        double r6467718 = n;
        double r6467719 = r6467717 * r6467718;
        double r6467720 = r6467711 - r6467712;
        double r6467721 = r6467720 / r6467715;
        double r6467722 = pow(r6467719, r6467721);
        double r6467723 = r6467714 * r6467722;
        return r6467723;
}
↓
double f(double k, double n) {
        double r6467724 = 1.0;
        double r6467725 = k;
        double r6467726 = sqrt(r6467725);
        double r6467727 = 2.0;
        double r6467728 = 0.5;
        double r6467729 = r6467725 / r6467727;
        double r6467730 = r6467728 - r6467729;
        double r6467731 = pow(r6467727, r6467730);
        double r6467732 = r6467726 / r6467731;
        double r6467733 = n;
        double r6467734 = pow(r6467733, r6467730);
        double r6467735 = r6467724 / r6467734;
        double r6467736 = atan2(1.0, 0.0);
        double r6467737 = pow(r6467736, r6467730);
        double r6467738 = r6467735 / r6467737;
        double r6467739 = r6467732 * r6467738;
        double r6467740 = r6467724 / r6467739;
        return r6467740;
}

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

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

  4. Applied clear-num0.4

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

  6. Applied unpow-prod-down0.4

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

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

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

  8. Applied times-frac0.4

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

  10. Applied unpow-prod-down0.5

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

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

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

  12. Applied sqrt-prod0.5

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

  13. Applied times-frac0.5

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

  14. Applied associate-*r*0.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))