Average Error: 0.4 → 0.4
Time: 4.9m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{1}{\sqrt{k}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{1}{\sqrt{k}}\right)
double f(double k, double n) {
        double r55902723 = 1.0;
        double r55902724 = k;
        double r55902725 = sqrt(r55902724);
        double r55902726 = r55902723 / r55902725;
        double r55902727 = 2.0;
        double r55902728 = atan2(1.0, 0.0);
        double r55902729 = r55902727 * r55902728;
        double r55902730 = n;
        double r55902731 = r55902729 * r55902730;
        double r55902732 = r55902723 - r55902724;
        double r55902733 = r55902732 / r55902727;
        double r55902734 = pow(r55902731, r55902733);
        double r55902735 = r55902726 * r55902734;
        return r55902735;
}
↓
double f(double k, double n) {
        double r55902736 = 2.0;
        double r55902737 = atan2(1.0, 0.0);
        double r55902738 = r55902736 * r55902737;
        double r55902739 = 1.0;
        double r55902740 = k;
        double r55902741 = r55902739 - r55902740;
        double r55902742 = r55902741 / r55902736;
        double r55902743 = r55902742 / r55902736;
        double r55902744 = pow(r55902738, r55902743);
        double r55902745 = n;
        double r55902746 = pow(r55902745, r55902742);
        double r55902747 = r55902744 * r55902746;
        double r55902748 = sqrt(r55902740);
        double r55902749 = r55902739 / r55902748;
        double r55902750 = r55902744 * r55902749;
        double r55902751 = r55902747 * r55902750;
        return r55902751;
}

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 unpow-prod-down0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied sqr-pow0.5

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

  7. Applied associate-*r*0.4

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

  9. Applied associate-*l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))