Average Error: 0.4 → 0.4
Time: 34.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{2}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\frac{\sqrt{k}}{{\left(\pi \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{2}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}{\frac{\sqrt{k}}{{\left(\pi \cdot n\right)}^{\left(\left(1 - k\right) \cdot \frac{1}{2}\right)}}}
double f(double k, double n) {
        double r2323291 = 1.0;
        double r2323292 = k;
        double r2323293 = sqrt(r2323292);
        double r2323294 = r2323291 / r2323293;
        double r2323295 = 2.0;
        double r2323296 = atan2(1.0, 0.0);
        double r2323297 = r2323295 * r2323296;
        double r2323298 = n;
        double r2323299 = r2323297 * r2323298;
        double r2323300 = r2323291 - r2323292;
        double r2323301 = r2323300 / r2323295;
        double r2323302 = pow(r2323299, r2323301);
        double r2323303 = r2323294 * r2323302;
        return r2323303;
}
↓
double f(double k, double n) {
        double r2323304 = 2.0;
        double r2323305 = 1.0;
        double r2323306 = k;
        double r2323307 = r2323305 - r2323306;
        double r2323308 = 0.5;
        double r2323309 = r2323307 * r2323308;
        double r2323310 = pow(r2323304, r2323309);
        double r2323311 = sqrt(r2323306);
        double r2323312 = atan2(1.0, 0.0);
        double r2323313 = n;
        double r2323314 = r2323312 * r2323313;
        double r2323315 = pow(r2323314, r2323309);
        double r2323316 = r2323311 / r2323315;
        double r2323317 = r2323310 / r2323316;
        return r2323317;
}

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

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

  3. Taylor expanded around 0 0.3

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

  5. Applied unpow-prod-down0.4

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

  6. Applied associate-/l*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))