Average Error: 0.4 → 0.5
Time: 8.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({\left(\frac{1 \cdot 1}{k}\right)}^{\frac{1}{2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({\left(\frac{1 \cdot 1}{k}\right)}^{\frac{1}{2}} \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)}
double f(double k, double n) {
        double r126865 = 1.0;
        double r126866 = k;
        double r126867 = sqrt(r126866);
        double r126868 = r126865 / r126867;
        double r126869 = 2.0;
        double r126870 = atan2(1.0, 0.0);
        double r126871 = r126869 * r126870;
        double r126872 = n;
        double r126873 = r126871 * r126872;
        double r126874 = r126865 - r126866;
        double r126875 = r126874 / r126869;
        double r126876 = pow(r126873, r126875);
        double r126877 = r126868 * r126876;
        return r126877;
}
↓
double f(double k, double n) {
        double r126878 = 1.0;
        double r126879 = r126878 * r126878;
        double r126880 = k;
        double r126881 = r126879 / r126880;
        double r126882 = 0.5;
        double r126883 = pow(r126881, r126882);
        double r126884 = 2.0;
        double r126885 = atan2(1.0, 0.0);
        double r126886 = r126884 * r126885;
        double r126887 = n;
        double r126888 = r126886 * r126887;
        double r126889 = r126878 - r126880;
        double r126890 = r126889 / r126884;
        double r126891 = 2.0;
        double r126892 = r126890 / r126891;
        double r126893 = pow(r126888, r126892);
        double r126894 = r126883 * r126893;
        double r126895 = r126894 * r126893;
        return r126895;
}

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. Using strategy rm

  5. Applied pow1/20.5

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

  6. Applied pow1/20.5

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

  7. Applied pow-prod-down0.4

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

  8. Simplified0.4

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

  10. Applied sqr-pow0.5

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

  11. Applied associate-*r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020033 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))