Migdal et al, Equation (51)

Average Error: 0.3 → 0.4

Time: 30.1s

Precision: 64

Math

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

↓

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

C

double f(double k, double n) {
        double r4885544 = 1.0;
        double r4885545 = k;
        double r4885546 = sqrt(r4885545);
        double r4885547 = r4885544 / r4885546;
        double r4885548 = 2.0;
        double r4885549 = atan2(1.0, 0.0);
        double r4885550 = r4885548 * r4885549;
        double r4885551 = n;
        double r4885552 = r4885550 * r4885551;
        double r4885553 = r4885544 - r4885545;
        double r4885554 = r4885553 / r4885548;
        double r4885555 = pow(r4885552, r4885554);
        double r4885556 = r4885547 * r4885555;
        return r4885556;
}

↓

double f(double k, double n) {
        double r4885557 = 2.0;
        double r4885558 = sqrt(r4885557);
        double r4885559 = n;
        double r4885560 = 0.5;
        double r4885561 = k;
        double r4885562 = r4885561 * r4885560;
        double r4885563 = r4885560 - r4885562;
        double r4885564 = pow(r4885559, r4885563);
        double r4885565 = r4885558 * r4885564;
        double r4885566 = atan2(1.0, 0.0);
        double r4885567 = pow(r4885566, r4885563);
        double r4885568 = r4885565 * r4885567;
        double r4885569 = sqrt(r4885561);
        double r4885570 = pow(r4885557, r4885562);
        double r4885571 = r4885569 * r4885570;
        double r4885572 = r4885568 / r4885571;
        return r4885572;
}

Error

Bits error versus k

Bits error versus n

Derivation

1. Initial program 0.3

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

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 *-un-lft-identity 0.3

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

5. Applied sqrt-prod 0.3

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

6. Applied unpow-prod-down 0.4

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

7. Applied times-frac 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

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

11. Applied unpow-prod-down 0.4

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

12. Applied associate-*l* 0.4

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

14. Applied pow-sub 0.4

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

15. Applied frac-times 0.4

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

16. Applied associate-*r/ 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019164 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))