Average Error: 0.4 → 0.5
Time: 10.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r135550 = 1.0;
        double r135551 = k;
        double r135552 = sqrt(r135551);
        double r135553 = r135550 / r135552;
        double r135554 = 2.0;
        double r135555 = atan2(1.0, 0.0);
        double r135556 = r135554 * r135555;
        double r135557 = n;
        double r135558 = r135556 * r135557;
        double r135559 = r135550 - r135551;
        double r135560 = r135559 / r135554;
        double r135561 = pow(r135558, r135560);
        double r135562 = r135553 * r135561;
        return r135562;
}
↓
double f(double k, double n) {
        double r135563 = 1.0;
        double r135564 = k;
        double r135565 = sqrt(r135564);
        double r135566 = r135563 / r135565;
        double r135567 = 2.0;
        double r135568 = atan2(1.0, 0.0);
        double r135569 = r135567 * r135568;
        double r135570 = n;
        double r135571 = r135569 * r135570;
        double r135572 = r135563 - r135564;
        double r135573 = r135572 / r135567;
        double r135574 = 2.0;
        double r135575 = r135573 / r135574;
        double r135576 = r135575 / r135574;
        double r135577 = pow(r135571, r135576);
        double r135578 = r135577 * r135577;
        double r135579 = r135566 * r135578;
        double r135580 = pow(r135571, r135575);
        double r135581 = r135579 * r135580;
        return r135581;
}

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 sqr-pow0.4

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

  4. Applied associate-*r*0.5

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

  6. Applied sqr-pow0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))