Migdal et al, Equation (51)

Average Error: 0.4 → 0.4

Time: 30.1s

Precision: 64

Math

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

↓

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

C

double f(double k, double n) {
        double r2644860 = 1.0;
        double r2644861 = k;
        double r2644862 = sqrt(r2644861);
        double r2644863 = r2644860 / r2644862;
        double r2644864 = 2.0;
        double r2644865 = atan2(1.0, 0.0);
        double r2644866 = r2644864 * r2644865;
        double r2644867 = n;
        double r2644868 = r2644866 * r2644867;
        double r2644869 = r2644860 - r2644861;
        double r2644870 = r2644869 / r2644864;
        double r2644871 = pow(r2644868, r2644870);
        double r2644872 = r2644863 * r2644871;
        return r2644872;
}

↓

double f(double k, double n) {
        double r2644873 = atan2(1.0, 0.0);
        double r2644874 = n;
        double r2644875 = r2644873 * r2644874;
        double r2644876 = 2.0;
        double r2644877 = r2644875 * r2644876;
        double r2644878 = 1.0;
        double r2644879 = k;
        double r2644880 = r2644878 - r2644879;
        double r2644881 = r2644880 / r2644876;
        double r2644882 = pow(r2644877, r2644881);
        double r2644883 = -0.5;
        double r2644884 = pow(r2644879, r2644883);
        double r2644885 = r2644882 * r2644884;
        return r2644885;
}

Error

Bits error versus k

Bits error versus n

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. Simplified 0.4

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

4. Applied *-un-lft-identity 0.4

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

5. Applied associate-/l* 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied sqrt-prod 0.4

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

10. Applied times-frac 0.4

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

11. Applied associate-/r* 0.4

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

12. Simplified 0.4

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

14. Applied div-inv 0.4

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

15. Applied *-un-lft-identity 0.4

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

16. Applied times-frac 0.4

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

17. Simplified 0.4

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

19. Applied pow1 0.4

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

20. Applied sqrt-pow1 0.4

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

21. Applied pow-flip 0.4

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

22. Simplified 0.4

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

23. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))