Average Error: 0.4 → 0.4
Time: 52.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\sqrt{\pi} \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\sqrt{\pi} \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r4826448 = 1.0;
        double r4826449 = k;
        double r4826450 = sqrt(r4826449);
        double r4826451 = r4826448 / r4826450;
        double r4826452 = 2.0;
        double r4826453 = atan2(1.0, 0.0);
        double r4826454 = r4826452 * r4826453;
        double r4826455 = n;
        double r4826456 = r4826454 * r4826455;
        double r4826457 = r4826448 - r4826449;
        double r4826458 = r4826457 / r4826452;
        double r4826459 = pow(r4826456, r4826458);
        double r4826460 = r4826451 * r4826459;
        return r4826460;
}
↓
double f(double k, double n) {
        double r4826461 = atan2(1.0, 0.0);
        double r4826462 = sqrt(r4826461);
        double r4826463 = n;
        double r4826464 = 2.0;
        double r4826465 = r4826463 * r4826464;
        double r4826466 = r4826462 * r4826465;
        double r4826467 = 0.5;
        double r4826468 = k;
        double r4826469 = r4826468 / r4826464;
        double r4826470 = r4826467 - r4826469;
        double r4826471 = pow(r4826466, r4826470);
        double r4826472 = sqrt(r4826468);
        double r4826473 = pow(r4826462, r4826470);
        double r4826474 = r4826472 / r4826473;
        double r4826475 = r4826471 / r4826474;
        return r4826475;
}

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. Simplified0.3

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.4

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

  6. Applied unpow-prod-down0.4

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

  7. Applied associate-/l*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))