Average Error: 0.4 → 0.3
Time: 37.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}
double f(double k, double n) {
        double r2614671 = 1.0;
        double r2614672 = k;
        double r2614673 = sqrt(r2614672);
        double r2614674 = r2614671 / r2614673;
        double r2614675 = 2.0;
        double r2614676 = atan2(1.0, 0.0);
        double r2614677 = r2614675 * r2614676;
        double r2614678 = n;
        double r2614679 = r2614677 * r2614678;
        double r2614680 = r2614671 - r2614672;
        double r2614681 = r2614680 / r2614675;
        double r2614682 = pow(r2614679, r2614681);
        double r2614683 = r2614674 * r2614682;
        return r2614683;
}
↓
double f(double k, double n) {
        double r2614684 = 2.0;
        double r2614685 = atan2(1.0, 0.0);
        double r2614686 = r2614684 * r2614685;
        double r2614687 = n;
        double r2614688 = r2614686 * r2614687;
        double r2614689 = 1.0;
        double r2614690 = k;
        double r2614691 = r2614689 - r2614690;
        double r2614692 = r2614691 / r2614684;
        double r2614693 = pow(r2614688, r2614692);
        double r2614694 = -0.5;
        double r2614695 = pow(r2614690, r2614694);
        double r2614696 = r2614693 * r2614695;
        return r2614696;
}

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 pow1/20.4

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

  4. Applied pow-flip0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))