Average Error: 0.4 → 0.4
Time: 23.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(2 \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \left(\sqrt{{\pi}^{\left(\frac{1 - k}{2}\right)}} \cdot \frac{1 \cdot \sqrt{{\pi}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(2 \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \left(\sqrt{{\pi}^{\left(\frac{1 - k}{2}\right)}} \cdot \frac{1 \cdot \sqrt{{\pi}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}}\right)
double f(double k, double n) {
        double r132312 = 1.0;
        double r132313 = k;
        double r132314 = sqrt(r132313);
        double r132315 = r132312 / r132314;
        double r132316 = 2.0;
        double r132317 = atan2(1.0, 0.0);
        double r132318 = r132316 * r132317;
        double r132319 = n;
        double r132320 = r132318 * r132319;
        double r132321 = r132312 - r132313;
        double r132322 = r132321 / r132316;
        double r132323 = pow(r132320, r132322);
        double r132324 = r132315 * r132323;
        return r132324;
}
↓
double f(double k, double n) {
        double r132325 = 2.0;
        double r132326 = n;
        double r132327 = r132325 * r132326;
        double r132328 = 1.0;
        double r132329 = k;
        double r132330 = r132328 - r132329;
        double r132331 = r132330 / r132325;
        double r132332 = pow(r132327, r132331);
        double r132333 = atan2(1.0, 0.0);
        double r132334 = pow(r132333, r132331);
        double r132335 = sqrt(r132334);
        double r132336 = r132328 * r132335;
        double r132337 = sqrt(r132329);
        double r132338 = r132336 / r132337;
        double r132339 = r132335 * r132338;
        double r132340 = r132332 * r132339;
        return r132340;
}

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

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

  4. Applied unpow-prod-down0.5

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

  5. Applied associate-*r*0.5

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied associate-*r*0.5

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))