Average Error: 0.4 → 0.4
Time: 14.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r125782 = 1.0;
        double r125783 = k;
        double r125784 = sqrt(r125783);
        double r125785 = r125782 / r125784;
        double r125786 = 2.0;
        double r125787 = atan2(1.0, 0.0);
        double r125788 = r125786 * r125787;
        double r125789 = n;
        double r125790 = r125788 * r125789;
        double r125791 = r125782 - r125783;
        double r125792 = r125791 / r125786;
        double r125793 = pow(r125790, r125792);
        double r125794 = r125785 * r125793;
        return r125794;
}
↓
double f(double k, double n) {
        double r125795 = 1.0;
        double r125796 = 1.0;
        double r125797 = k;
        double r125798 = r125796 / r125797;
        double r125799 = 0.25;
        double r125800 = pow(r125798, r125799);
        double r125801 = r125795 * r125800;
        double r125802 = sqrt(r125797);
        double r125803 = sqrt(r125802);
        double r125804 = r125801 / r125803;
        double r125805 = 2.0;
        double r125806 = atan2(1.0, 0.0);
        double r125807 = r125805 * r125806;
        double r125808 = n;
        double r125809 = r125807 * r125808;
        double r125810 = r125795 - r125797;
        double r125811 = r125810 / r125805;
        double r125812 = pow(r125809, r125811);
        double r125813 = r125804 * r125812;
        return r125813;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied sqrt-prod0.6

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

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

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

  11. Applied times-frac0.6

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

  12. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  13. Final simplification0.4

    \[\leadsto \frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))