Average Error: 0.3 → 0.4
Time: 7.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)
double f(double k, double n) {
        double r130459 = 1.0;
        double r130460 = k;
        double r130461 = sqrt(r130460);
        double r130462 = r130459 / r130461;
        double r130463 = 2.0;
        double r130464 = atan2(1.0, 0.0);
        double r130465 = r130463 * r130464;
        double r130466 = n;
        double r130467 = r130465 * r130466;
        double r130468 = r130459 - r130460;
        double r130469 = r130468 / r130463;
        double r130470 = pow(r130467, r130469);
        double r130471 = r130462 * r130470;
        return r130471;
}
↓
double f(double k, double n) {
        double r130472 = 1.0;
        double r130473 = k;
        double r130474 = sqrt(r130473);
        double r130475 = r130472 / r130474;
        double r130476 = 2.0;
        double r130477 = atan2(1.0, 0.0);
        double r130478 = r130476 * r130477;
        double r130479 = n;
        double r130480 = r130478 * r130479;
        double r130481 = r130472 - r130473;
        double r130482 = r130481 / r130476;
        double r130483 = pow(r130480, r130482);
        double r130484 = sqrt(r130483);
        double r130485 = r130484 * r130484;
        double r130486 = r130475 * r130485;
        return r130486;
}

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 0.3

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

  3. Applied add-sqr-sqrt0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))