Average Error: 0.4 → 0.5
Time: 31.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\sqrt{\frac{1}{\sqrt{k}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right) \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\sqrt{\frac{1}{\sqrt{k}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right) \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r4925281 = 1.0;
        double r4925282 = k;
        double r4925283 = sqrt(r4925282);
        double r4925284 = r4925281 / r4925283;
        double r4925285 = 2.0;
        double r4925286 = atan2(1.0, 0.0);
        double r4925287 = r4925285 * r4925286;
        double r4925288 = n;
        double r4925289 = r4925287 * r4925288;
        double r4925290 = r4925281 - r4925282;
        double r4925291 = r4925290 / r4925285;
        double r4925292 = pow(r4925289, r4925291);
        double r4925293 = r4925284 * r4925292;
        return r4925293;
}
↓
double f(double k, double n) {
        double r4925294 = 1.0;
        double r4925295 = k;
        double r4925296 = sqrt(r4925295);
        double r4925297 = r4925294 / r4925296;
        double r4925298 = sqrt(r4925297);
        double r4925299 = r4925298 * r4925298;
        double r4925300 = n;
        double r4925301 = atan2(1.0, 0.0);
        double r4925302 = 2.0;
        double r4925303 = r4925301 * r4925302;
        double r4925304 = r4925300 * r4925303;
        double r4925305 = r4925294 - r4925295;
        double r4925306 = r4925305 / r4925302;
        double r4925307 = pow(r4925304, r4925306);
        double r4925308 = r4925299 * r4925307;
        return r4925308;
}

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-sqrt0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))