Average Error: 0.4 → 0.5
Time: 8.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r140888 = 1.0;
        double r140889 = k;
        double r140890 = sqrt(r140889);
        double r140891 = r140888 / r140890;
        double r140892 = 2.0;
        double r140893 = atan2(1.0, 0.0);
        double r140894 = r140892 * r140893;
        double r140895 = n;
        double r140896 = r140894 * r140895;
        double r140897 = r140888 - r140889;
        double r140898 = r140897 / r140892;
        double r140899 = pow(r140896, r140898);
        double r140900 = r140891 * r140899;
        return r140900;
}
↓
double f(double k, double n) {
        double r140901 = 1.0;
        double r140902 = k;
        double r140903 = sqrt(r140902);
        double r140904 = r140901 / r140903;
        double r140905 = 2.0;
        double r140906 = atan2(1.0, 0.0);
        double r140907 = r140905 * r140906;
        double r140908 = n;
        double r140909 = r140907 * r140908;
        double r140910 = r140901 - r140902;
        double r140911 = r140910 / r140905;
        double r140912 = 2.0;
        double r140913 = r140911 / r140912;
        double r140914 = pow(r140909, r140913);
        double r140915 = r140904 * r140914;
        double r140916 = r140915 * r140914;
        return r140916;
}

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. Applied associate-*r*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019346 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))