Average Error: 0.4 → 0.5
Time: 28.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r112738 = 1.0;
        double r112739 = k;
        double r112740 = sqrt(r112739);
        double r112741 = r112738 / r112740;
        double r112742 = 2.0;
        double r112743 = atan2(1.0, 0.0);
        double r112744 = r112742 * r112743;
        double r112745 = n;
        double r112746 = r112744 * r112745;
        double r112747 = r112738 - r112739;
        double r112748 = r112747 / r112742;
        double r112749 = pow(r112746, r112748);
        double r112750 = r112741 * r112749;
        return r112750;
}
↓
double f(double k, double n) {
        double r112751 = 1.0;
        double r112752 = k;
        double r112753 = sqrt(r112752);
        double r112754 = sqrt(r112753);
        double r112755 = r112751 / r112754;
        double r112756 = r112755 / r112754;
        double r112757 = 2.0;
        double r112758 = atan2(1.0, 0.0);
        double r112759 = r112757 * r112758;
        double r112760 = n;
        double r112761 = r112759 * r112760;
        double r112762 = r112751 - r112752;
        double r112763 = r112762 / r112757;
        double r112764 = pow(r112761, r112763);
        double r112765 = r112756 * r112764;
        return r112765;
}

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-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))