Average Error: 0.4 → 0.4
Time: 15.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\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 r126616 = 1.0;
        double r126617 = k;
        double r126618 = sqrt(r126617);
        double r126619 = r126616 / r126618;
        double r126620 = 2.0;
        double r126621 = atan2(1.0, 0.0);
        double r126622 = r126620 * r126621;
        double r126623 = n;
        double r126624 = r126622 * r126623;
        double r126625 = r126616 - r126617;
        double r126626 = r126625 / r126620;
        double r126627 = pow(r126624, r126626);
        double r126628 = r126619 * r126627;
        return r126628;
}
↓
double f(double k, double n) {
        double r126629 = 1.0;
        double r126630 = k;
        double r126631 = sqrt(r126630);
        double r126632 = sqrt(r126631);
        double r126633 = r126629 / r126632;
        double r126634 = r126633 / r126632;
        double r126635 = sqrt(r126634);
        double r126636 = r126629 / r126631;
        double r126637 = sqrt(r126636);
        double r126638 = 2.0;
        double r126639 = atan2(1.0, 0.0);
        double r126640 = r126638 * r126639;
        double r126641 = n;
        double r126642 = r126640 * r126641;
        double r126643 = r126629 - r126630;
        double r126644 = r126643 / r126638;
        double r126645 = pow(r126642, r126644);
        double r126646 = r126637 * r126645;
        double r126647 = r126635 * r126646;
        return r126647;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt0.5

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

  7. Applied sqrt-prod0.5

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))