Average Error: 0.4 → 0.4
Time: 4.1m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{1}{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\sqrt{k}}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{1}{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\sqrt{k}}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r20454350 = 1.0;
        double r20454351 = k;
        double r20454352 = sqrt(r20454351);
        double r20454353 = r20454350 / r20454352;
        double r20454354 = 2.0;
        double r20454355 = atan2(1.0, 0.0);
        double r20454356 = r20454354 * r20454355;
        double r20454357 = n;
        double r20454358 = r20454356 * r20454357;
        double r20454359 = r20454350 - r20454351;
        double r20454360 = r20454359 / r20454354;
        double r20454361 = pow(r20454358, r20454360);
        double r20454362 = r20454353 * r20454361;
        return r20454362;
}
↓
double f(double k, double n) {
        double r20454363 = n;
        double r20454364 = 0.5;
        double r20454365 = k;
        double r20454366 = 2.0;
        double r20454367 = r20454365 / r20454366;
        double r20454368 = r20454364 - r20454367;
        double r20454369 = pow(r20454363, r20454368);
        double r20454370 = 1.0;
        double r20454371 = pow(r20454366, r20454368);
        double r20454372 = r20454370 / r20454371;
        double r20454373 = sqrt(r20454365);
        double r20454374 = atan2(1.0, 0.0);
        double r20454375 = pow(r20454374, r20454368);
        double r20454376 = r20454373 / r20454375;
        double r20454377 = r20454372 * r20454376;
        double r20454378 = r20454369 / r20454377;
        return r20454378;
}

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. Simplified0.4

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

  4. Applied unpow-prod-down0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied unpow-prod-down0.5

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

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

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

  9. Applied times-frac0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))