Average Error: 0.4 → 0.4
Time: 27.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}
double f(double k, double n) {
        double r64609 = 1.0;
        double r64610 = k;
        double r64611 = sqrt(r64610);
        double r64612 = r64609 / r64611;
        double r64613 = 2.0;
        double r64614 = atan2(1.0, 0.0);
        double r64615 = r64613 * r64614;
        double r64616 = n;
        double r64617 = r64615 * r64616;
        double r64618 = r64609 - r64610;
        double r64619 = r64618 / r64613;
        double r64620 = pow(r64617, r64619);
        double r64621 = r64612 * r64620;
        return r64621;
}
↓
double f(double k, double n) {
        double r64622 = 1.0;
        double r64623 = k;
        double r64624 = sqrt(r64623);
        double r64625 = r64622 / r64624;
        double r64626 = 2.0;
        double r64627 = atan2(1.0, 0.0);
        double r64628 = r64626 * r64627;
        double r64629 = n;
        double r64630 = r64628 * r64629;
        double r64631 = r64622 - r64623;
        double r64632 = r64631 / r64626;
        double r64633 = 2.0;
        double r64634 = r64632 / r64633;
        double r64635 = pow(r64630, r64634);
        double r64636 = r64625 * r64635;
        double r64637 = pow(r64626, r64632);
        double r64638 = r64627 * r64629;
        double r64639 = pow(r64638, r64632);
        double r64640 = r64637 * r64639;
        double r64641 = 0.5;
        double r64642 = pow(r64640, r64641);
        double r64643 = r64636 * r64642;
        return r64643;
}

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. Applied associate-*r*0.5

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

  6. Applied div-inv0.5

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

  7. Applied pow-unpow0.5

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

  8. Simplified0.5

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*l*0.5

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

  13. Applied unpow-prod-down0.5

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))