Average Error: 0.4 → 0.4
Time: 18.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\sqrt{\pi} \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot {\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
\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)} \cdot {\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r1642622 = 1.0;
        double r1642623 = k;
        double r1642624 = sqrt(r1642623);
        double r1642625 = r1642622 / r1642624;
        double r1642626 = 2.0;
        double r1642627 = atan2(1.0, 0.0);
        double r1642628 = r1642626 * r1642627;
        double r1642629 = n;
        double r1642630 = r1642628 * r1642629;
        double r1642631 = r1642622 - r1642623;
        double r1642632 = r1642631 / r1642626;
        double r1642633 = pow(r1642630, r1642632);
        double r1642634 = r1642625 * r1642633;
        return r1642634;
}
↓
double f(double k, double n) {
        double r1642635 = atan2(1.0, 0.0);
        double r1642636 = sqrt(r1642635);
        double r1642637 = n;
        double r1642638 = 2.0;
        double r1642639 = r1642637 * r1642638;
        double r1642640 = r1642636 * r1642639;
        double r1642641 = 0.5;
        double r1642642 = k;
        double r1642643 = r1642642 / r1642638;
        double r1642644 = r1642641 - r1642643;
        double r1642645 = pow(r1642640, r1642644);
        double r1642646 = pow(r1642636, r1642644);
        double r1642647 = r1642645 * r1642646;
        double r1642648 = sqrt(r1642642);
        double r1642649 = r1642647 / r1642648;
        return r1642649;
}

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. Using strategy rm

  7. Applied unpow-prod-down0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))