Average Error: 0.5 → 0.4
Time: 11.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{\frac{1}{2}}{2}\right)}}{\frac{\sqrt{k}}{1}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{\frac{1}{2}}{2}\right)}}{\frac{\sqrt{k}}{1}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double f(double k, double n) {
        double r142676 = 1.0;
        double r142677 = k;
        double r142678 = sqrt(r142677);
        double r142679 = r142676 / r142678;
        double r142680 = 2.0;
        double r142681 = atan2(1.0, 0.0);
        double r142682 = r142680 * r142681;
        double r142683 = n;
        double r142684 = r142682 * r142683;
        double r142685 = r142676 - r142677;
        double r142686 = r142685 / r142680;
        double r142687 = pow(r142684, r142686);
        double r142688 = r142679 * r142687;
        return r142688;
}
↓
double f(double k, double n) {
        double r142689 = 2.0;
        double r142690 = atan2(1.0, 0.0);
        double r142691 = r142689 * r142690;
        double r142692 = n;
        double r142693 = r142691 * r142692;
        double r142694 = 2.0;
        double r142695 = 1.0;
        double r142696 = r142695 / r142689;
        double r142697 = r142696 / r142694;
        double r142698 = r142694 * r142697;
        double r142699 = pow(r142693, r142698);
        double r142700 = k;
        double r142701 = sqrt(r142700);
        double r142702 = r142701 / r142695;
        double r142703 = r142699 / r142702;
        double r142704 = r142700 / r142689;
        double r142705 = pow(r142693, r142704);
        double r142706 = r142703 / r142705;
        return r142706;
}

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 0.5

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

  3. Applied div-sub0.5

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

  4. Applied pow-sub0.4

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

  5. Applied frac-times0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied associate-*r*0.4

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

  10. Applied associate-/r*0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))