Average Error: 0.4 → 0.5
Time: 9.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)} \cdot \sqrt{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\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)} \cdot \sqrt{\frac{1}{\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 r132267 = 1.0;
        double r132268 = k;
        double r132269 = sqrt(r132268);
        double r132270 = r132267 / r132269;
        double r132271 = 2.0;
        double r132272 = atan2(1.0, 0.0);
        double r132273 = r132271 * r132272;
        double r132274 = n;
        double r132275 = r132273 * r132274;
        double r132276 = r132267 - r132268;
        double r132277 = r132276 / r132271;
        double r132278 = pow(r132275, r132277);
        double r132279 = r132270 * r132278;
        return r132279;
}
↓
double f(double k, double n) {
        double r132280 = 1.0;
        double r132281 = k;
        double r132282 = sqrt(r132281);
        double r132283 = r132280 / r132282;
        double r132284 = 2.0;
        double r132285 = atan2(1.0, 0.0);
        double r132286 = r132284 * r132285;
        double r132287 = n;
        double r132288 = r132286 * r132287;
        double r132289 = r132280 - r132281;
        double r132290 = r132289 / r132284;
        double r132291 = 2.0;
        double r132292 = r132290 / r132291;
        double r132293 = pow(r132288, r132292);
        double r132294 = r132293 * r132293;
        double r132295 = r132283 * r132294;
        double r132296 = sqrt(r132295);
        double r132297 = pow(r132288, r132290);
        double r132298 = r132283 * r132297;
        double r132299 = sqrt(r132298);
        double r132300 = r132296 * r132299;
        return r132300;
}

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

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

  5. Applied sqr-pow0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020081 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))