Average Error: 0.4 → 0.5
Time: 10.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\frac{\left|\sqrt[3]{k}\right|}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\frac{\left|\sqrt[3]{k}\right|}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r125650 = 1.0;
        double r125651 = k;
        double r125652 = sqrt(r125651);
        double r125653 = r125650 / r125652;
        double r125654 = 2.0;
        double r125655 = atan2(1.0, 0.0);
        double r125656 = r125654 * r125655;
        double r125657 = n;
        double r125658 = r125656 * r125657;
        double r125659 = r125650 - r125651;
        double r125660 = r125659 / r125654;
        double r125661 = pow(r125658, r125660);
        double r125662 = r125653 * r125661;
        return r125662;
}
↓
double f(double k, double n) {
        double r125663 = 1.0;
        double r125664 = k;
        double r125665 = sqrt(r125664);
        double r125666 = r125663 / r125665;
        double r125667 = sqrt(r125666);
        double r125668 = cbrt(r125663);
        double r125669 = cbrt(r125664);
        double r125670 = fabs(r125669);
        double r125671 = r125670 / r125668;
        double r125672 = r125668 / r125671;
        double r125673 = sqrt(r125669);
        double r125674 = r125668 / r125673;
        double r125675 = r125672 * r125674;
        double r125676 = sqrt(r125675);
        double r125677 = 2.0;
        double r125678 = atan2(1.0, 0.0);
        double r125679 = r125677 * r125678;
        double r125680 = n;
        double r125681 = r125679 * r125680;
        double r125682 = r125663 - r125664;
        double r125683 = r125682 / r125677;
        double r125684 = pow(r125681, r125683);
        double r125685 = r125676 * r125684;
        double r125686 = r125667 * r125685;
        return r125686;
}

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-sqrt0.5

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

  4. Applied associate-*l*0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied sqrt-prod0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\frac{\left|\sqrt[3]{k}\right|}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020089 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))