Average Error: 0.4 → 0.5
Time: 54.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}
double f(double k, double n) {
        double r4179866 = 1.0;
        double r4179867 = k;
        double r4179868 = sqrt(r4179867);
        double r4179869 = r4179866 / r4179868;
        double r4179870 = 2.0;
        double r4179871 = atan2(1.0, 0.0);
        double r4179872 = r4179870 * r4179871;
        double r4179873 = n;
        double r4179874 = r4179872 * r4179873;
        double r4179875 = r4179866 - r4179867;
        double r4179876 = r4179875 / r4179870;
        double r4179877 = pow(r4179874, r4179876);
        double r4179878 = r4179869 * r4179877;
        return r4179878;
}
↓
double f(double k, double n) {
        double r4179879 = atan2(1.0, 0.0);
        double r4179880 = 2.0;
        double r4179881 = r4179879 * r4179880;
        double r4179882 = n;
        double r4179883 = r4179881 * r4179882;
        double r4179884 = 1.0;
        double r4179885 = k;
        double r4179886 = r4179884 - r4179885;
        double r4179887 = r4179886 / r4179880;
        double r4179888 = r4179887 / r4179880;
        double r4179889 = r4179888 / r4179880;
        double r4179890 = pow(r4179883, r4179889);
        double r4179891 = r4179890 * r4179890;
        double r4179892 = sqrt(r4179885);
        double r4179893 = pow(r4179883, r4179888);
        double r4179894 = r4179892 / r4179893;
        double r4179895 = r4179891 / r4179894;
        return r4179895;
}

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. Simplified0.3

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

  4. Applied sqr-pow0.4

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

  5. Applied associate-/l*0.4

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

  7. Applied sqr-pow0.5

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

  9. Applied *-commutative0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))