Average Error: 0.4 → 0.5
Time: 23.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\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(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)
double f(double k, double n) {
        double r84677 = 1.0;
        double r84678 = k;
        double r84679 = sqrt(r84678);
        double r84680 = r84677 / r84679;
        double r84681 = 2.0;
        double r84682 = atan2(1.0, 0.0);
        double r84683 = r84681 * r84682;
        double r84684 = n;
        double r84685 = r84683 * r84684;
        double r84686 = r84677 - r84678;
        double r84687 = r84686 / r84681;
        double r84688 = pow(r84685, r84687);
        double r84689 = r84680 * r84688;
        return r84689;
}
↓
double f(double k, double n) {
        double r84690 = 1.0;
        double r84691 = k;
        double r84692 = sqrt(r84691);
        double r84693 = r84690 / r84692;
        double r84694 = 2.0;
        double r84695 = atan2(1.0, 0.0);
        double r84696 = r84694 * r84695;
        double r84697 = n;
        double r84698 = r84696 * r84697;
        double r84699 = r84690 - r84691;
        double r84700 = r84699 / r84694;
        double r84701 = 2.0;
        double r84702 = r84700 / r84701;
        double r84703 = pow(r84698, r84702);
        double r84704 = r84703 * r84703;
        double r84705 = r84693 * r84704;
        return r84705;
}

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-pow0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))