Average Error: 0.4 → 0.4
Time: 26.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(n \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}\]
\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) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(n \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}
double f(double k, double n) {
        double r62231 = 1.0;
        double r62232 = k;
        double r62233 = sqrt(r62232);
        double r62234 = r62231 / r62233;
        double r62235 = 2.0;
        double r62236 = atan2(1.0, 0.0);
        double r62237 = r62235 * r62236;
        double r62238 = n;
        double r62239 = r62237 * r62238;
        double r62240 = r62231 - r62232;
        double r62241 = r62240 / r62235;
        double r62242 = pow(r62239, r62241);
        double r62243 = r62234 * r62242;
        return r62243;
}
↓
double f(double k, double n) {
        double r62244 = 1.0;
        double r62245 = k;
        double r62246 = sqrt(r62245);
        double r62247 = r62244 / r62246;
        double r62248 = 2.0;
        double r62249 = atan2(1.0, 0.0);
        double r62250 = r62248 * r62249;
        double r62251 = n;
        double r62252 = r62250 * r62251;
        double r62253 = r62244 - r62245;
        double r62254 = r62253 / r62248;
        double r62255 = 2.0;
        double r62256 = r62254 / r62255;
        double r62257 = pow(r62252, r62256);
        double r62258 = r62247 * r62257;
        double r62259 = pow(r62248, r62254);
        double r62260 = r62251 * r62249;
        double r62261 = pow(r62260, r62254);
        double r62262 = r62259 * r62261;
        double r62263 = 0.5;
        double r62264 = pow(r62262, r62263);
        double r62265 = r62258 * r62264;
        return r62265;
}

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 div-inv0.5

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

  7. Applied pow-unpow0.5

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

  8. Simplified0.5

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

  10. Applied unpow-prod-down0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))