Average Error: 0.4 → 0.5
Time: 8.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r145249 = 1.0;
        double r145250 = k;
        double r145251 = sqrt(r145250);
        double r145252 = r145249 / r145251;
        double r145253 = 2.0;
        double r145254 = atan2(1.0, 0.0);
        double r145255 = r145253 * r145254;
        double r145256 = n;
        double r145257 = r145255 * r145256;
        double r145258 = r145249 - r145250;
        double r145259 = r145258 / r145253;
        double r145260 = pow(r145257, r145259);
        double r145261 = r145252 * r145260;
        return r145261;
}
↓
double f(double k, double n) {
        double r145262 = 1.0;
        double r145263 = k;
        double r145264 = sqrt(r145263);
        double r145265 = r145262 / r145264;
        double r145266 = sqrt(r145265);
        double r145267 = 2.0;
        double r145268 = atan2(1.0, 0.0);
        double r145269 = r145267 * r145268;
        double r145270 = n;
        double r145271 = r145269 * r145270;
        double r145272 = r145262 - r145263;
        double r145273 = r145272 / r145267;
        double r145274 = pow(r145271, r145273);
        double r145275 = r145266 * r145274;
        double r145276 = r145266 * r145275;
        return r145276;
}

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. Applied associate-*l*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))