Average Error: 0.4 → 0.4
Time: 13.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\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 r142909 = 1.0;
        double r142910 = k;
        double r142911 = sqrt(r142910);
        double r142912 = r142909 / r142911;
        double r142913 = 2.0;
        double r142914 = atan2(1.0, 0.0);
        double r142915 = r142913 * r142914;
        double r142916 = n;
        double r142917 = r142915 * r142916;
        double r142918 = r142909 - r142910;
        double r142919 = r142918 / r142913;
        double r142920 = pow(r142917, r142919);
        double r142921 = r142912 * r142920;
        return r142921;
}
↓
double f(double k, double n) {
        double r142922 = 1.0;
        double r142923 = 1.0;
        double r142924 = k;
        double r142925 = r142923 / r142924;
        double r142926 = 0.25;
        double r142927 = pow(r142925, r142926);
        double r142928 = r142922 * r142927;
        double r142929 = sqrt(r142924);
        double r142930 = sqrt(r142929);
        double r142931 = r142928 / r142930;
        double r142932 = 2.0;
        double r142933 = atan2(1.0, 0.0);
        double r142934 = r142932 * r142933;
        double r142935 = n;
        double r142936 = r142934 * r142935;
        double r142937 = r142922 - r142924;
        double r142938 = r142937 / r142932;
        double r142939 = pow(r142936, r142938);
        double r142940 = r142931 * r142939;
        return r142940;
}

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. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  7. Final simplification0.4

    \[\leadsto \frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))