Average Error: 0.4 → 0.4
Time: 33.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(2 \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot e^{\log \pi \cdot \frac{1 - k}{2}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(2 \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot e^{\log \pi \cdot \frac{1 - k}{2}}
double f(double k, double n) {
        double r2285961 = 1.0;
        double r2285962 = k;
        double r2285963 = sqrt(r2285962);
        double r2285964 = r2285961 / r2285963;
        double r2285965 = 2.0;
        double r2285966 = atan2(1.0, 0.0);
        double r2285967 = r2285965 * r2285966;
        double r2285968 = n;
        double r2285969 = r2285967 * r2285968;
        double r2285970 = r2285961 - r2285962;
        double r2285971 = r2285970 / r2285965;
        double r2285972 = pow(r2285969, r2285971);
        double r2285973 = r2285964 * r2285972;
        return r2285973;
}
↓
double f(double k, double n) {
        double r2285974 = 2.0;
        double r2285975 = n;
        double r2285976 = r2285974 * r2285975;
        double r2285977 = 1.0;
        double r2285978 = k;
        double r2285979 = r2285977 - r2285978;
        double r2285980 = r2285979 / r2285974;
        double r2285981 = pow(r2285976, r2285980);
        double r2285982 = sqrt(r2285978);
        double r2285983 = r2285981 / r2285982;
        double r2285984 = atan2(1.0, 0.0);
        double r2285985 = log(r2285984);
        double r2285986 = r2285985 * r2285980;
        double r2285987 = exp(r2285986);
        double r2285988 = r2285983 * r2285987;
        return r2285988;
}

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

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied unpow-prod-down0.4

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

  6. Applied times-frac0.4

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

  7. Simplified0.4

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

  9. Applied add-exp-log0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))