Average Error: 0.4 → 0.4
Time: 8.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1 \cdot 1}{k}} \cdot \left(\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1 \cdot 1}{k}} \cdot \left(\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)
double f(double k, double n) {
        double r140729 = 1.0;
        double r140730 = k;
        double r140731 = sqrt(r140730);
        double r140732 = r140729 / r140731;
        double r140733 = 2.0;
        double r140734 = atan2(1.0, 0.0);
        double r140735 = r140733 * r140734;
        double r140736 = n;
        double r140737 = r140735 * r140736;
        double r140738 = r140729 - r140730;
        double r140739 = r140738 / r140733;
        double r140740 = pow(r140737, r140739);
        double r140741 = r140732 * r140740;
        return r140741;
}
↓
double f(double k, double n) {
        double r140742 = 1.0;
        double r140743 = r140742 * r140742;
        double r140744 = k;
        double r140745 = r140743 / r140744;
        double r140746 = sqrt(r140745);
        double r140747 = 2.0;
        double r140748 = r140742 - r140744;
        double r140749 = r140748 / r140747;
        double r140750 = pow(r140747, r140749);
        double r140751 = atan2(1.0, 0.0);
        double r140752 = pow(r140751, r140749);
        double r140753 = r140750 * r140752;
        double r140754 = n;
        double r140755 = pow(r140754, r140749);
        double r140756 = r140753 * r140755;
        double r140757 = r140746 * r140756;
        return r140757;
}

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 add-sqr-sqrt0.5

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

  5. Applied sqrt-unprod0.4

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

  6. Simplified0.4

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

  8. Applied unpow-prod-down0.5

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

  10. Applied unpow-prod-down0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))