Average Error: 0.4 → 0.5
Time: 15.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r148238 = 1.0;
        double r148239 = k;
        double r148240 = sqrt(r148239);
        double r148241 = r148238 / r148240;
        double r148242 = 2.0;
        double r148243 = atan2(1.0, 0.0);
        double r148244 = r148242 * r148243;
        double r148245 = n;
        double r148246 = r148244 * r148245;
        double r148247 = r148238 - r148239;
        double r148248 = r148247 / r148242;
        double r148249 = pow(r148246, r148248);
        double r148250 = r148241 * r148249;
        return r148250;
}
↓
double f(double k, double n) {
        double r148251 = 1.0;
        double r148252 = k;
        double r148253 = sqrt(r148252);
        double r148254 = r148251 / r148253;
        double r148255 = 2.0;
        double r148256 = atan2(1.0, 0.0);
        double r148257 = r148255 * r148256;
        double r148258 = r148251 - r148252;
        double r148259 = r148258 / r148255;
        double r148260 = 2.0;
        double r148261 = r148259 / r148260;
        double r148262 = pow(r148257, r148261);
        double r148263 = n;
        double r148264 = pow(r148263, r148261);
        double r148265 = r148262 * r148264;
        double r148266 = r148254 * r148265;
        double r148267 = r148257 * r148263;
        double r148268 = pow(r148267, r148261);
        double r148269 = r148266 * r148268;
        return r148269;
}

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. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))