Average Error: 0.4 → 0.5
Time: 30.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\sqrt{k}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\sqrt{k}}}
double f(double k, double n) {
        double r3276509 = 1.0;
        double r3276510 = k;
        double r3276511 = sqrt(r3276510);
        double r3276512 = r3276509 / r3276511;
        double r3276513 = 2.0;
        double r3276514 = atan2(1.0, 0.0);
        double r3276515 = r3276513 * r3276514;
        double r3276516 = n;
        double r3276517 = r3276515 * r3276516;
        double r3276518 = r3276509 - r3276510;
        double r3276519 = r3276518 / r3276513;
        double r3276520 = pow(r3276517, r3276519);
        double r3276521 = r3276512 * r3276520;
        return r3276521;
}
↓
double f(double k, double n) {
        double r3276522 = atan2(1.0, 0.0);
        double r3276523 = 2.0;
        double r3276524 = r3276522 * r3276523;
        double r3276525 = n;
        double r3276526 = r3276524 * r3276525;
        double r3276527 = 1.0;
        double r3276528 = k;
        double r3276529 = r3276527 - r3276528;
        double r3276530 = 0.5;
        double r3276531 = r3276529 * r3276530;
        double r3276532 = pow(r3276526, r3276531);
        double r3276533 = sqrt(r3276528);
        double r3276534 = r3276532 / r3276533;
        double r3276535 = sqrt(r3276534);
        double r3276536 = r3276535 * r3276535;
        return r3276536;
}

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

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

  4. Applied add-sqr-sqrt0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))