Average Error: 0.4 → 0.6
Time: 27.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right)\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right)\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r137679 = 1.0;
        double r137680 = k;
        double r137681 = sqrt(r137680);
        double r137682 = r137679 / r137681;
        double r137683 = 2.0;
        double r137684 = atan2(1.0, 0.0);
        double r137685 = r137683 * r137684;
        double r137686 = n;
        double r137687 = r137685 * r137686;
        double r137688 = r137679 - r137680;
        double r137689 = r137688 / r137683;
        double r137690 = pow(r137687, r137689);
        double r137691 = r137682 * r137690;
        return r137691;
}
↓
double f(double k, double n) {
        double r137692 = 1.0;
        double r137693 = k;
        double r137694 = sqrt(r137693);
        double r137695 = r137692 / r137694;
        double r137696 = 2.0;
        double r137697 = r137692 - r137693;
        double r137698 = r137697 / r137696;
        double r137699 = pow(r137696, r137698);
        double r137700 = atan2(1.0, 0.0);
        double r137701 = pow(r137700, r137698);
        double r137702 = r137699 * r137701;
        double r137703 = r137695 * r137702;
        double r137704 = n;
        double r137705 = pow(r137704, r137698);
        double r137706 = r137703 * r137705;
        return r137706;
}

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 unpow-prod-down0.6

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

  4. Applied associate-*r*0.6

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

  6. Applied unpow-prod-down0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))