Average Error: 0.4 → 0.4
Time: 3.8m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r4280635 = 1.0;
        double r4280636 = k;
        double r4280637 = sqrt(r4280636);
        double r4280638 = r4280635 / r4280637;
        double r4280639 = 2.0;
        double r4280640 = atan2(1.0, 0.0);
        double r4280641 = r4280639 * r4280640;
        double r4280642 = n;
        double r4280643 = r4280641 * r4280642;
        double r4280644 = r4280635 - r4280636;
        double r4280645 = r4280644 / r4280639;
        double r4280646 = pow(r4280643, r4280645);
        double r4280647 = r4280638 * r4280646;
        return r4280647;
}
↓
double f(double k, double n) {
        double r4280648 = 1.0;
        double r4280649 = k;
        double r4280650 = sqrt(r4280649);
        double r4280651 = atan2(1.0, 0.0);
        double r4280652 = n;
        double r4280653 = 2.0;
        double r4280654 = r4280652 * r4280653;
        double r4280655 = r4280651 * r4280654;
        double r4280656 = 0.5;
        double r4280657 = r4280649 / r4280653;
        double r4280658 = r4280656 - r4280657;
        double r4280659 = pow(r4280655, r4280658);
        double r4280660 = r4280650 / r4280659;
        double r4280661 = r4280648 / r4280660;
        return r4280661;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))