Average Error: 0.4 → 0.5
Time: 59.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\sqrt{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{\sqrt{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}
double f(double k, double n) {
        double r3066840 = 1.0;
        double r3066841 = k;
        double r3066842 = sqrt(r3066841);
        double r3066843 = r3066840 / r3066842;
        double r3066844 = 2.0;
        double r3066845 = atan2(1.0, 0.0);
        double r3066846 = r3066844 * r3066845;
        double r3066847 = n;
        double r3066848 = r3066846 * r3066847;
        double r3066849 = r3066840 - r3066841;
        double r3066850 = r3066849 / r3066844;
        double r3066851 = pow(r3066848, r3066850);
        double r3066852 = r3066843 * r3066851;
        return r3066852;
}
↓
double f(double k, double n) {
        double r3066853 = atan2(1.0, 0.0);
        double r3066854 = 2.0;
        double r3066855 = r3066853 * r3066854;
        double r3066856 = n;
        double r3066857 = r3066855 * r3066856;
        double r3066858 = 0.5;
        double r3066859 = k;
        double r3066860 = r3066859 / r3066854;
        double r3066861 = r3066858 - r3066860;
        double r3066862 = pow(r3066857, r3066861);
        double r3066863 = sqrt(r3066862);
        double r3066864 = sqrt(r3066859);
        double r3066865 = r3066863 / r3066864;
        double r3066866 = r3066863 * r3066865;
        return r3066866;
}

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

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

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

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied times-frac0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))