Average Error: 0.4 → 0.5
Time: 26.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r95577 = 1.0;
        double r95578 = k;
        double r95579 = sqrt(r95578);
        double r95580 = r95577 / r95579;
        double r95581 = 2.0;
        double r95582 = atan2(1.0, 0.0);
        double r95583 = r95581 * r95582;
        double r95584 = n;
        double r95585 = r95583 * r95584;
        double r95586 = r95577 - r95578;
        double r95587 = r95586 / r95581;
        double r95588 = pow(r95585, r95587);
        double r95589 = r95580 * r95588;
        return r95589;
}
↓
double f(double k, double n) {
        double r95590 = 1.0;
        double r95591 = k;
        double r95592 = sqrt(r95591);
        double r95593 = r95590 / r95592;
        double r95594 = sqrt(r95593);
        double r95595 = 2.0;
        double r95596 = atan2(1.0, 0.0);
        double r95597 = r95595 * r95596;
        double r95598 = n;
        double r95599 = r95597 * r95598;
        double r95600 = r95590 - r95591;
        double r95601 = r95600 / r95595;
        double r95602 = pow(r95599, r95601);
        double r95603 = r95594 * r95602;
        double r95604 = r95594 * r95603;
        return r95604;
}

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. Applied associate-*l*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019208 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))