Average Error: 0.4 → 0.4
Time: 8.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\frac{1 \cdot 1}{k}\right)}^{\frac{1}{2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\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)}^{\frac{1}{2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r123520 = 1.0;
        double r123521 = k;
        double r123522 = sqrt(r123521);
        double r123523 = r123520 / r123522;
        double r123524 = 2.0;
        double r123525 = atan2(1.0, 0.0);
        double r123526 = r123524 * r123525;
        double r123527 = n;
        double r123528 = r123526 * r123527;
        double r123529 = r123520 - r123521;
        double r123530 = r123529 / r123524;
        double r123531 = pow(r123528, r123530);
        double r123532 = r123523 * r123531;
        return r123532;
}
↓
double f(double k, double n) {
        double r123533 = 1.0;
        double r123534 = r123533 * r123533;
        double r123535 = k;
        double r123536 = r123534 / r123535;
        double r123537 = 0.5;
        double r123538 = pow(r123536, r123537);
        double r123539 = 2.0;
        double r123540 = atan2(1.0, 0.0);
        double r123541 = r123539 * r123540;
        double r123542 = n;
        double r123543 = r123541 * r123542;
        double r123544 = r123533 - r123535;
        double r123545 = r123544 / r123539;
        double r123546 = pow(r123543, r123545);
        double r123547 = r123538 * r123546;
        return r123547;
}

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 pow1/20.5

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

  6. Applied pow1/20.5

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

  7. Applied pow-prod-down0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

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