Average Error: 0.4 → 0.5
Time: 1.0m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{\sqrt{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \cdot \sqrt{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{\sqrt{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \cdot \sqrt{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}
double f(double k, double n) {
        double r12867838 = 1.0;
        double r12867839 = k;
        double r12867840 = sqrt(r12867839);
        double r12867841 = r12867838 / r12867840;
        double r12867842 = 2.0;
        double r12867843 = atan2(1.0, 0.0);
        double r12867844 = r12867842 * r12867843;
        double r12867845 = n;
        double r12867846 = r12867844 * r12867845;
        double r12867847 = r12867838 - r12867839;
        double r12867848 = r12867847 / r12867842;
        double r12867849 = pow(r12867846, r12867848);
        double r12867850 = r12867841 * r12867849;
        return r12867850;
}
↓
double f(double k, double n) {
        double r12867851 = n;
        double r12867852 = 2.0;
        double r12867853 = r12867851 * r12867852;
        double r12867854 = 0.5;
        double r12867855 = k;
        double r12867856 = r12867855 / r12867852;
        double r12867857 = r12867854 - r12867856;
        double r12867858 = pow(r12867853, r12867857);
        double r12867859 = sqrt(r12867858);
        double r12867860 = sqrt(r12867855);
        double r12867861 = r12867859 / r12867860;
        double r12867862 = atan2(1.0, 0.0);
        double r12867863 = r12867853 * r12867862;
        double r12867864 = pow(r12867863, r12867857);
        double r12867865 = sqrt(r12867864);
        double r12867866 = r12867861 * r12867865;
        double r12867867 = pow(r12867862, r12867857);
        double r12867868 = sqrt(r12867867);
        double r12867869 = r12867866 * r12867868;
        return r12867869;
}

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

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

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

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied times-frac0.5

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

  8. Applied unpow-prod-down0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied associate-/l*0.5

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

  12. Applied associate-/r/0.5

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

  13. Applied associate-*r*0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019162 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))