Average Error: 0.4 → 0.4
Time: 26.7s
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 r64656 = 1.0;
        double r64657 = k;
        double r64658 = sqrt(r64657);
        double r64659 = r64656 / r64658;
        double r64660 = 2.0;
        double r64661 = atan2(1.0, 0.0);
        double r64662 = r64660 * r64661;
        double r64663 = n;
        double r64664 = r64662 * r64663;
        double r64665 = r64656 - r64657;
        double r64666 = r64665 / r64660;
        double r64667 = pow(r64664, r64666);
        double r64668 = r64659 * r64667;
        return r64668;
}
↓
double f(double k, double n) {
        double r64669 = 1.0;
        double r64670 = k;
        double r64671 = sqrt(r64670);
        double r64672 = r64669 / r64671;
        double r64673 = 2.0;
        double r64674 = atan2(1.0, 0.0);
        double r64675 = r64673 * r64674;
        double r64676 = n;
        double r64677 = r64675 * r64676;
        double r64678 = r64669 - r64670;
        double r64679 = r64678 / r64673;
        double r64680 = 2.0;
        double r64681 = r64679 / r64680;
        double r64682 = pow(r64677, r64681);
        double r64683 = r64672 * r64682;
        double r64684 = pow(r64673, r64679);
        double r64685 = r64674 * r64676;
        double r64686 = pow(r64685, r64679);
        double r64687 = r64684 * r64686;
        double r64688 = 0.5;
        double r64689 = pow(r64687, r64688);
        double r64690 = r64683 * r64689;
        return r64690;
}

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))))