Average Error: 0.4 → 0.5
Time: 29.7s
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 r3771707 = 1.0;
        double r3771708 = k;
        double r3771709 = sqrt(r3771708);
        double r3771710 = r3771707 / r3771709;
        double r3771711 = 2.0;
        double r3771712 = atan2(1.0, 0.0);
        double r3771713 = r3771711 * r3771712;
        double r3771714 = n;
        double r3771715 = r3771713 * r3771714;
        double r3771716 = r3771707 - r3771708;
        double r3771717 = r3771716 / r3771711;
        double r3771718 = pow(r3771715, r3771717);
        double r3771719 = r3771710 * r3771718;
        return r3771719;
}
↓
double f(double k, double n) {
        double r3771720 = 1.0;
        double r3771721 = k;
        double r3771722 = sqrt(r3771721);
        double r3771723 = r3771720 / r3771722;
        double r3771724 = sqrt(r3771723);
        double r3771725 = 2.0;
        double r3771726 = atan2(1.0, 0.0);
        double r3771727 = r3771725 * r3771726;
        double r3771728 = n;
        double r3771729 = r3771727 * r3771728;
        double r3771730 = r3771720 - r3771721;
        double r3771731 = r3771730 / r3771725;
        double r3771732 = pow(r3771729, r3771731);
        double r3771733 = r3771724 * r3771732;
        double r3771734 = r3771724 * r3771733;
        return r3771734;
}

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 2019169 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))