Average Error: 0.4 → 0.3
Time: 4.5m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r38152792 = 1.0;
        double r38152793 = k;
        double r38152794 = sqrt(r38152793);
        double r38152795 = r38152792 / r38152794;
        double r38152796 = 2.0;
        double r38152797 = atan2(1.0, 0.0);
        double r38152798 = r38152796 * r38152797;
        double r38152799 = n;
        double r38152800 = r38152798 * r38152799;
        double r38152801 = r38152792 - r38152793;
        double r38152802 = r38152801 / r38152796;
        double r38152803 = pow(r38152800, r38152802);
        double r38152804 = r38152795 * r38152803;
        return r38152804;
}
↓
double f(double k, double n) {
        double r38152805 = atan2(1.0, 0.0);
        double r38152806 = n;
        double r38152807 = 2.0;
        double r38152808 = r38152806 * r38152807;
        double r38152809 = r38152805 * r38152808;
        double r38152810 = 1.0;
        double r38152811 = k;
        double r38152812 = r38152810 - r38152811;
        double r38152813 = r38152812 / r38152807;
        double r38152814 = pow(r38152809, r38152813);
        double r38152815 = sqrt(r38152811);
        double r38152816 = r38152814 / r38152815;
        return r38152816;
}

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 clear-num0.4

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

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

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

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

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

  8. Applied sqrt-prod0.4

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

  9. Applied times-frac0.4

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

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

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

  11. Applied times-frac0.4

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

  12. Simplified0.4

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

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