Average Error: 0.4 → 0.5
Time: 34.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({k}^{\frac{-1}{2}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({k}^{\frac{-1}{2}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}
double f(double k, double n) {
        double r2784430 = 1.0;
        double r2784431 = k;
        double r2784432 = sqrt(r2784431);
        double r2784433 = r2784430 / r2784432;
        double r2784434 = 2.0;
        double r2784435 = atan2(1.0, 0.0);
        double r2784436 = r2784434 * r2784435;
        double r2784437 = n;
        double r2784438 = r2784436 * r2784437;
        double r2784439 = r2784430 - r2784431;
        double r2784440 = r2784439 / r2784434;
        double r2784441 = pow(r2784438, r2784440);
        double r2784442 = r2784433 * r2784441;
        return r2784442;
}
↓
double f(double k, double n) {
        double r2784443 = k;
        double r2784444 = -0.5;
        double r2784445 = pow(r2784443, r2784444);
        double r2784446 = 2.0;
        double r2784447 = atan2(1.0, 0.0);
        double r2784448 = r2784446 * r2784447;
        double r2784449 = n;
        double r2784450 = r2784448 * r2784449;
        double r2784451 = 1.0;
        double r2784452 = r2784451 - r2784443;
        double r2784453 = r2784452 / r2784446;
        double r2784454 = pow(r2784450, r2784453);
        double r2784455 = sqrt(r2784454);
        double r2784456 = r2784445 * r2784455;
        double r2784457 = r2784456 * r2784455;
        return r2784457;
}

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. Using strategy rm

  3. Applied pow10.4

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

  4. Applied sqrt-pow10.4

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

  5. Applied pow-flip0.3

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

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-*r*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))