Average Error: 0.4 → 0.4
Time: 28.4s
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 r86862 = 1.0;
        double r86863 = k;
        double r86864 = sqrt(r86863);
        double r86865 = r86862 / r86864;
        double r86866 = 2.0;
        double r86867 = atan2(1.0, 0.0);
        double r86868 = r86866 * r86867;
        double r86869 = n;
        double r86870 = r86868 * r86869;
        double r86871 = r86862 - r86863;
        double r86872 = r86871 / r86866;
        double r86873 = pow(r86870, r86872);
        double r86874 = r86865 * r86873;
        return r86874;
}
↓
double f(double k, double n) {
        double r86875 = 1.0;
        double r86876 = k;
        double r86877 = sqrt(r86876);
        double r86878 = r86875 / r86877;
        double r86879 = 2.0;
        double r86880 = atan2(1.0, 0.0);
        double r86881 = r86879 * r86880;
        double r86882 = n;
        double r86883 = r86881 * r86882;
        double r86884 = r86875 - r86876;
        double r86885 = r86884 / r86879;
        double r86886 = 2.0;
        double r86887 = r86885 / r86886;
        double r86888 = pow(r86883, r86887);
        double r86889 = r86878 * r86888;
        double r86890 = pow(r86879, r86885);
        double r86891 = r86880 * r86882;
        double r86892 = pow(r86891, r86885);
        double r86893 = r86890 * r86892;
        double r86894 = 0.5;
        double r86895 = pow(r86893, r86894);
        double r86896 = r86889 * r86895;
        return r86896;
}

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