Average Error: 0.4 → 0.5
Time: 11.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left(\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r134619 = 1.0;
        double r134620 = k;
        double r134621 = sqrt(r134620);
        double r134622 = r134619 / r134621;
        double r134623 = 2.0;
        double r134624 = atan2(1.0, 0.0);
        double r134625 = r134623 * r134624;
        double r134626 = n;
        double r134627 = r134625 * r134626;
        double r134628 = r134619 - r134620;
        double r134629 = r134628 / r134623;
        double r134630 = pow(r134627, r134629);
        double r134631 = r134622 * r134630;
        return r134631;
}
↓
double f(double k, double n) {
        double r134632 = 1.0;
        double r134633 = k;
        double r134634 = sqrt(r134633);
        double r134635 = r134632 / r134634;
        double r134636 = 2.0;
        double r134637 = atan2(1.0, 0.0);
        double r134638 = r134636 * r134637;
        double r134639 = n;
        double r134640 = r134638 * r134639;
        double r134641 = r134632 - r134633;
        double r134642 = r134641 / r134636;
        double r134643 = 2.0;
        double r134644 = r134642 / r134643;
        double r134645 = r134644 / r134643;
        double r134646 = pow(r134640, r134645);
        double r134647 = r134646 * r134646;
        double r134648 = pow(r134640, r134644);
        double r134649 = r134647 * r134648;
        double r134650 = r134635 * r134649;
        return r134650;
}

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 sqr-pow0.4

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

  5. Applied sqr-pow0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))