Average Error: 0.4 → 0.5
Time: 8.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\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({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r130462 = 1.0;
        double r130463 = k;
        double r130464 = sqrt(r130463);
        double r130465 = r130462 / r130464;
        double r130466 = 2.0;
        double r130467 = atan2(1.0, 0.0);
        double r130468 = r130466 * r130467;
        double r130469 = n;
        double r130470 = r130468 * r130469;
        double r130471 = r130462 - r130463;
        double r130472 = r130471 / r130466;
        double r130473 = pow(r130470, r130472);
        double r130474 = r130465 * r130473;
        return r130474;
}
↓
double f(double k, double n) {
        double r130475 = 1.0;
        double r130476 = k;
        double r130477 = sqrt(r130476);
        double r130478 = r130475 / r130477;
        double r130479 = 2.0;
        double r130480 = atan2(1.0, 0.0);
        double r130481 = r130479 * r130480;
        double r130482 = n;
        double r130483 = r130481 * r130482;
        double r130484 = r130475 - r130476;
        double r130485 = r130484 / r130479;
        double r130486 = 2.0;
        double r130487 = r130485 / r130486;
        double r130488 = pow(r130483, r130487);
        double r130489 = r130488 * r130488;
        double r130490 = r130478 * r130489;
        return r130490;
}

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. Using strategy rm

  3. Applied sqr-pow0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))