Average Error: 0.4 → 0.4
Time: 8.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\frac{1 \cdot 1}{k}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\frac{1 \cdot 1}{k}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r133546 = 1.0;
        double r133547 = k;
        double r133548 = sqrt(r133547);
        double r133549 = r133546 / r133548;
        double r133550 = 2.0;
        double r133551 = atan2(1.0, 0.0);
        double r133552 = r133550 * r133551;
        double r133553 = n;
        double r133554 = r133552 * r133553;
        double r133555 = r133546 - r133547;
        double r133556 = r133555 / r133550;
        double r133557 = pow(r133554, r133556);
        double r133558 = r133549 * r133557;
        return r133558;
}
↓
double f(double k, double n) {
        double r133559 = 1.0;
        double r133560 = r133559 * r133559;
        double r133561 = k;
        double r133562 = r133560 / r133561;
        double r133563 = 1.0;
        double r133564 = 2.0;
        double r133565 = r133563 / r133564;
        double r133566 = pow(r133562, r133565);
        double r133567 = 2.0;
        double r133568 = atan2(1.0, 0.0);
        double r133569 = r133567 * r133568;
        double r133570 = n;
        double r133571 = r133569 * r133570;
        double r133572 = r133559 - r133561;
        double r133573 = r133572 / r133567;
        double r133574 = pow(r133571, r133573);
        double r133575 = r133566 * r133574;
        return r133575;
}

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. Using strategy rm

  5. Applied pow10.5

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

  6. Applied sqrt-pow10.5

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

  7. Applied pow10.5

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

  8. Applied sqrt-pow10.5

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

  9. Applied pow-prod-down0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020027 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))