Average Error: 0.4 → 0.5
Time: 13.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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(\left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r117710 = 1.0;
        double r117711 = k;
        double r117712 = sqrt(r117711);
        double r117713 = r117710 / r117712;
        double r117714 = 2.0;
        double r117715 = atan2(1.0, 0.0);
        double r117716 = r117714 * r117715;
        double r117717 = n;
        double r117718 = r117716 * r117717;
        double r117719 = r117710 - r117711;
        double r117720 = r117719 / r117714;
        double r117721 = pow(r117718, r117720);
        double r117722 = r117713 * r117721;
        return r117722;
}
↓
double f(double k, double n) {
        double r117723 = 1.0;
        double r117724 = k;
        double r117725 = sqrt(r117724);
        double r117726 = r117723 / r117725;
        double r117727 = 2.0;
        double r117728 = atan2(1.0, 0.0);
        double r117729 = r117727 * r117728;
        double r117730 = n;
        double r117731 = r117729 * r117730;
        double r117732 = r117723 - r117724;
        double r117733 = r117732 / r117727;
        double r117734 = 2.0;
        double r117735 = r117733 / r117734;
        double r117736 = pow(r117731, r117735);
        double r117737 = r117726 * r117736;
        double r117738 = cbrt(r117730);
        double r117739 = r117738 * r117738;
        double r117740 = r117729 * r117739;
        double r117741 = r117740 * r117738;
        double r117742 = pow(r117741, r117735);
        double r117743 = r117737 * r117742;
        return r117743;
}

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 add-cube-cbrt0.5

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

  7. Applied associate-*r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))