Average Error: 0.4 → 0.5
Time: 13.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r102421 = 1.0;
        double r102422 = k;
        double r102423 = sqrt(r102422);
        double r102424 = r102421 / r102423;
        double r102425 = 2.0;
        double r102426 = atan2(1.0, 0.0);
        double r102427 = r102425 * r102426;
        double r102428 = n;
        double r102429 = r102427 * r102428;
        double r102430 = r102421 - r102422;
        double r102431 = r102430 / r102425;
        double r102432 = pow(r102429, r102431);
        double r102433 = r102424 * r102432;
        return r102433;
}
↓
double f(double k, double n) {
        double r102434 = 1.0;
        double r102435 = k;
        double r102436 = sqrt(r102435);
        double r102437 = r102434 / r102436;
        double r102438 = 2.0;
        double r102439 = atan2(1.0, 0.0);
        double r102440 = r102438 * r102439;
        double r102441 = n;
        double r102442 = r102440 * r102441;
        double r102443 = r102434 - r102435;
        double r102444 = r102443 / r102438;
        double r102445 = 2.0;
        double r102446 = r102444 / r102445;
        double r102447 = pow(r102442, r102446);
        double r102448 = r102437 * r102447;
        double r102449 = cbrt(r102441);
        double r102450 = r102449 * r102449;
        double r102451 = r102440 * r102450;
        double r102452 = r102451 * r102449;
        double r102453 = pow(r102452, r102446);
        double r102454 = r102448 * r102453;
        return r102454;
}

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 sqr-pow0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied associate-*r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))