Migdal et al, Equation (51)

Average Error: 0.4 → 0.5

Time: 30.3s

Precision: 64

Math

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

↓

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

C

double f(double k, double n) {
        double r4596778 = 1.0;
        double r4596779 = k;
        double r4596780 = sqrt(r4596779);
        double r4596781 = r4596778 / r4596780;
        double r4596782 = 2.0;
        double r4596783 = atan2(1.0, 0.0);
        double r4596784 = r4596782 * r4596783;
        double r4596785 = n;
        double r4596786 = r4596784 * r4596785;
        double r4596787 = r4596778 - r4596779;
        double r4596788 = r4596787 / r4596782;
        double r4596789 = pow(r4596786, r4596788);
        double r4596790 = r4596781 * r4596789;
        return r4596790;
}

↓

double f(double k, double n) {
        double r4596791 = 2.0;
        double r4596792 = 0.5;
        double r4596793 = k;
        double r4596794 = r4596793 / r4596791;
        double r4596795 = r4596792 - r4596794;
        double r4596796 = pow(r4596791, r4596795);
        double r4596797 = sqrt(r4596793);
        double r4596798 = atan2(1.0, 0.0);
        double r4596799 = pow(r4596798, r4596795);
        double r4596800 = r4596797 / r4596799;
        double r4596801 = r4596796 / r4596800;
        double r4596802 = n;
        double r4596803 = pow(r4596802, r4596795);
        double r4596804 = r4596801 * r4596803;
        return r4596804;
}

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 0.3

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

4. Applied unpow-prod-down 0.4

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

5. Applied associate-/l* 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied unpow-prod-down 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied sqrt-prod 0.4

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

11. Applied times-frac 0.4

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

12. Applied unpow-prod-down 0.5

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

13. Applied times-frac 0.5

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

14. Simplified 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019168 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))