Average Error: 0.5 → 0.6
Time: 10.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\right)
double f(double k, double n) {
        double r150269 = 1.0;
        double r150270 = k;
        double r150271 = sqrt(r150270);
        double r150272 = r150269 / r150271;
        double r150273 = 2.0;
        double r150274 = atan2(1.0, 0.0);
        double r150275 = r150273 * r150274;
        double r150276 = n;
        double r150277 = r150275 * r150276;
        double r150278 = r150269 - r150270;
        double r150279 = r150278 / r150273;
        double r150280 = pow(r150277, r150279);
        double r150281 = r150272 * r150280;
        return r150281;
}
↓
double f(double k, double n) {
        double r150282 = 1.0;
        double r150283 = k;
        double r150284 = sqrt(r150283);
        double r150285 = r150282 / r150284;
        double r150286 = 2.0;
        double r150287 = r150282 - r150283;
        double r150288 = r150287 / r150286;
        double r150289 = pow(r150286, r150288);
        double r150290 = atan2(1.0, 0.0);
        double r150291 = pow(r150290, r150288);
        double r150292 = n;
        double r150293 = pow(r150292, r150288);
        double r150294 = r150291 * r150293;
        double r150295 = r150289 * r150294;
        double r150296 = r150285 * r150295;
        return r150296;
}

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 0.5

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

  3. Applied unpow-prod-down0.7

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

  5. Applied unpow-prod-down0.6

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

  6. Applied associate-*l*0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020065 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))