Average Error: 0.4 → 0.4
Time: 39.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{1}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}{\frac{\sqrt{k}}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{1}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}{\frac{\sqrt{k}}{{\left(\sqrt{\pi}\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r3399612 = 1.0;
        double r3399613 = k;
        double r3399614 = sqrt(r3399613);
        double r3399615 = r3399612 / r3399614;
        double r3399616 = 2.0;
        double r3399617 = atan2(1.0, 0.0);
        double r3399618 = r3399616 * r3399617;
        double r3399619 = n;
        double r3399620 = r3399618 * r3399619;
        double r3399621 = r3399612 - r3399613;
        double r3399622 = r3399621 / r3399616;
        double r3399623 = pow(r3399620, r3399622);
        double r3399624 = r3399615 * r3399623;
        return r3399624;
}
↓
double f(double k, double n) {
        double r3399625 = n;
        double r3399626 = 2.0;
        double r3399627 = r3399625 * r3399626;
        double r3399628 = 0.5;
        double r3399629 = k;
        double r3399630 = r3399629 / r3399626;
        double r3399631 = r3399628 - r3399630;
        double r3399632 = pow(r3399627, r3399631);
        double r3399633 = 1.0;
        double r3399634 = atan2(1.0, 0.0);
        double r3399635 = sqrt(r3399634);
        double r3399636 = pow(r3399635, r3399631);
        double r3399637 = r3399633 / r3399636;
        double r3399638 = r3399632 / r3399637;
        double r3399639 = sqrt(r3399629);
        double r3399640 = r3399639 / r3399636;
        double r3399641 = r3399638 / r3399640;
        return r3399641;
}

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 unpow-prod-down0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied unpow-prod-down0.4

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

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

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

  10. Applied times-frac0.4

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

  11. Applied associate-/r*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))