Average Error: 0.4 → 0.5
Time: 8.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[1 \cdot \frac{{\left(\left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
1 \cdot \frac{{\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{\sqrt{k}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}
double f(double k, double n) {
        double r139009 = 1.0;
        double r139010 = k;
        double r139011 = sqrt(r139010);
        double r139012 = r139009 / r139011;
        double r139013 = 2.0;
        double r139014 = atan2(1.0, 0.0);
        double r139015 = r139013 * r139014;
        double r139016 = n;
        double r139017 = r139015 * r139016;
        double r139018 = r139009 - r139010;
        double r139019 = r139018 / r139013;
        double r139020 = pow(r139017, r139019);
        double r139021 = r139012 * r139020;
        return r139021;
}
↓
double f(double k, double n) {
        double r139022 = 1.0;
        double r139023 = 2.0;
        double r139024 = atan2(1.0, 0.0);
        double r139025 = r139023 * r139024;
        double r139026 = n;
        double r139027 = cbrt(r139026);
        double r139028 = r139027 * r139027;
        double r139029 = r139025 * r139028;
        double r139030 = r139029 * r139027;
        double r139031 = k;
        double r139032 = r139022 - r139031;
        double r139033 = r139032 / r139023;
        double r139034 = 2.0;
        double r139035 = r139033 / r139034;
        double r139036 = pow(r139030, r139035);
        double r139037 = sqrt(r139031);
        double r139038 = r139025 * r139026;
        double r139039 = pow(r139038, r139035);
        double r139040 = r139037 / r139039;
        double r139041 = r139036 / r139040;
        double r139042 = r139022 * r139041;
        return r139042;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  7. Applied sqr-pow0.5

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

  8. Applied associate-/l*0.5

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))