Average Error: 0.4 → 0.4
Time: 9.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1 \cdot 1}{k}} \cdot \left(\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1 \cdot 1}{k}} \cdot \left(\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r184773 = 1.0;
        double r184774 = k;
        double r184775 = sqrt(r184774);
        double r184776 = r184773 / r184775;
        double r184777 = 2.0;
        double r184778 = atan2(1.0, 0.0);
        double r184779 = r184777 * r184778;
        double r184780 = n;
        double r184781 = r184779 * r184780;
        double r184782 = r184773 - r184774;
        double r184783 = r184782 / r184777;
        double r184784 = pow(r184781, r184783);
        double r184785 = r184776 * r184784;
        return r184785;
}
↓
double f(double k, double n) {
        double r184786 = 1.0;
        double r184787 = r184786 * r184786;
        double r184788 = k;
        double r184789 = r184787 / r184788;
        double r184790 = sqrt(r184789);
        double r184791 = 2.0;
        double r184792 = r184786 - r184788;
        double r184793 = r184792 / r184791;
        double r184794 = pow(r184791, r184793);
        double r184795 = atan2(1.0, 0.0);
        double r184796 = pow(r184795, r184793);
        double r184797 = r184794 * r184796;
        double r184798 = n;
        double r184799 = pow(r184798, r184793);
        double r184800 = r184797 * r184799;
        double r184801 = r184790 * r184800;
        return r184801;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.5

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

  5. Applied sqrt-unprod0.4

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

  6. Simplified0.4

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

  8. Applied unpow-prod-down0.5

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

  10. Applied unpow-prod-down0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))