Average Error: 0.4 → 0.5
Time: 8.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\frac{1}{\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(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r133822 = 1.0;
        double r133823 = k;
        double r133824 = sqrt(r133823);
        double r133825 = r133822 / r133824;
        double r133826 = 2.0;
        double r133827 = atan2(1.0, 0.0);
        double r133828 = r133826 * r133827;
        double r133829 = n;
        double r133830 = r133828 * r133829;
        double r133831 = r133822 - r133823;
        double r133832 = r133831 / r133826;
        double r133833 = pow(r133830, r133832);
        double r133834 = r133825 * r133833;
        return r133834;
}
↓
double f(double k, double n) {
        double r133835 = 1.0;
        double r133836 = k;
        double r133837 = sqrt(r133836);
        double r133838 = r133835 / r133837;
        double r133839 = 2.0;
        double r133840 = atan2(1.0, 0.0);
        double r133841 = r133839 * r133840;
        double r133842 = n;
        double r133843 = r133841 * r133842;
        double r133844 = r133835 - r133836;
        double r133845 = r133844 / r133839;
        double r133846 = 2.0;
        double r133847 = r133845 / r133846;
        double r133848 = pow(r133843, r133847);
        double r133849 = r133848 * r133848;
        double r133850 = r133838 * r133849;
        return r133850;
}

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 sqr-pow0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))