Average Error: 0.4 → 0.4
Time: 3.5m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{\sqrt{2}}{\sqrt{k}} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{{2}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{\sqrt{2}}{\sqrt{k}} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{{2}^{\left(\frac{k}{2}\right)}}
double f(double k, double n) {
        double r27720307 = 1.0;
        double r27720308 = k;
        double r27720309 = sqrt(r27720308);
        double r27720310 = r27720307 / r27720309;
        double r27720311 = 2.0;
        double r27720312 = atan2(1.0, 0.0);
        double r27720313 = r27720311 * r27720312;
        double r27720314 = n;
        double r27720315 = r27720313 * r27720314;
        double r27720316 = r27720307 - r27720308;
        double r27720317 = r27720316 / r27720311;
        double r27720318 = pow(r27720315, r27720317);
        double r27720319 = r27720310 * r27720318;
        return r27720319;
}
↓
double f(double k, double n) {
        double r27720320 = 2.0;
        double r27720321 = sqrt(r27720320);
        double r27720322 = k;
        double r27720323 = sqrt(r27720322);
        double r27720324 = r27720321 / r27720323;
        double r27720325 = atan2(1.0, 0.0);
        double r27720326 = 1.0;
        double r27720327 = r27720326 - r27720322;
        double r27720328 = r27720327 / r27720320;
        double r27720329 = pow(r27720325, r27720328);
        double r27720330 = r27720324 * r27720329;
        double r27720331 = n;
        double r27720332 = pow(r27720331, r27720328);
        double r27720333 = r27720322 / r27720320;
        double r27720334 = pow(r27720320, r27720333);
        double r27720335 = r27720332 / r27720334;
        double r27720336 = r27720330 * r27720335;
        return r27720336;
}

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. Simplified0.3

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

  4. Applied unpow-prod-down0.5

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

  6. Applied unpow-prod-down0.5

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

  8. Applied associate-/l*0.5

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

  10. Applied div-sub0.5

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

  11. Applied pow-sub0.5

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

  12. Applied associate-*l/0.5

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

  13. Applied associate-/r/0.5

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

  14. Applied *-un-lft-identity0.5

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

  15. Applied times-frac0.5

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))