Average Error: 0.4 → 0.5
Time: 28.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r83695 = 1.0;
        double r83696 = k;
        double r83697 = sqrt(r83696);
        double r83698 = r83695 / r83697;
        double r83699 = 2.0;
        double r83700 = atan2(1.0, 0.0);
        double r83701 = r83699 * r83700;
        double r83702 = n;
        double r83703 = r83701 * r83702;
        double r83704 = r83695 - r83696;
        double r83705 = r83704 / r83699;
        double r83706 = pow(r83703, r83705);
        double r83707 = r83698 * r83706;
        return r83707;
}
↓
double f(double k, double n) {
        double r83708 = 1.0;
        double r83709 = k;
        double r83710 = sqrt(r83709);
        double r83711 = r83708 / r83710;
        double r83712 = sqrt(r83711);
        double r83713 = 2.0;
        double r83714 = atan2(1.0, 0.0);
        double r83715 = r83713 * r83714;
        double r83716 = n;
        double r83717 = r83715 * r83716;
        double r83718 = r83708 - r83709;
        double r83719 = r83718 / r83713;
        double r83720 = pow(r83717, r83719);
        double r83721 = r83712 * r83720;
        double r83722 = r83712 * r83721;
        return r83722;
}

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. Applied associate-*l*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))