Average Error: 0.4 → 0.5
Time: 15.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r157346 = 1.0;
        double r157347 = k;
        double r157348 = sqrt(r157347);
        double r157349 = r157346 / r157348;
        double r157350 = 2.0;
        double r157351 = atan2(1.0, 0.0);
        double r157352 = r157350 * r157351;
        double r157353 = n;
        double r157354 = r157352 * r157353;
        double r157355 = r157346 - r157347;
        double r157356 = r157355 / r157350;
        double r157357 = pow(r157354, r157356);
        double r157358 = r157349 * r157357;
        return r157358;
}
↓
double f(double k, double n) {
        double r157359 = 1.0;
        double r157360 = k;
        double r157361 = sqrt(r157360);
        double r157362 = sqrt(r157361);
        double r157363 = r157359 / r157362;
        double r157364 = r157363 / r157362;
        double r157365 = 2.0;
        double r157366 = atan2(1.0, 0.0);
        double r157367 = r157365 * r157366;
        double r157368 = n;
        double r157369 = r157367 * r157368;
        double r157370 = r157359 - r157360;
        double r157371 = r157370 / r157365;
        double r157372 = pow(r157369, r157371);
        double r157373 = r157364 * r157372;
        return r157373;
}

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 add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))