Average Error: 0.4 → 0.4
Time: 26.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\left(\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot {\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\left(\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot {\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r2209566 = 1.0;
        double r2209567 = k;
        double r2209568 = sqrt(r2209567);
        double r2209569 = r2209566 / r2209568;
        double r2209570 = 2.0;
        double r2209571 = atan2(1.0, 0.0);
        double r2209572 = r2209570 * r2209571;
        double r2209573 = n;
        double r2209574 = r2209572 * r2209573;
        double r2209575 = r2209566 - r2209567;
        double r2209576 = r2209575 / r2209570;
        double r2209577 = pow(r2209574, r2209576);
        double r2209578 = r2209569 * r2209577;
        return r2209578;
}
↓
double f(double k, double n) {
        double r2209579 = atan2(1.0, 0.0);
        double r2209580 = 0.5;
        double r2209581 = k;
        double r2209582 = 2.0;
        double r2209583 = r2209581 / r2209582;
        double r2209584 = r2209580 - r2209583;
        double r2209585 = pow(r2209579, r2209584);
        double r2209586 = sqrt(r2209585);
        double r2209587 = r2209586 * r2209586;
        double r2209588 = n;
        double r2209589 = r2209588 * r2209582;
        double r2209590 = pow(r2209589, r2209584);
        double r2209591 = r2209587 * r2209590;
        double r2209592 = sqrt(r2209581);
        double r2209593 = r2209591 / r2209592;
        return r2209593;
}

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. Simplified0.4

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

  4. Applied unpow-prod-down0.5

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

  6. Applied add-sqr-sqrt0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))