Average Error: 0.4 → 0.5
Time: 9.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\frac{\sqrt{k}}{\sqrt[3]{1}}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\frac{\sqrt{k}}{\sqrt[3]{1}}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r159917 = 1.0;
        double r159918 = k;
        double r159919 = sqrt(r159918);
        double r159920 = r159917 / r159919;
        double r159921 = 2.0;
        double r159922 = atan2(1.0, 0.0);
        double r159923 = r159921 * r159922;
        double r159924 = n;
        double r159925 = r159923 * r159924;
        double r159926 = r159917 - r159918;
        double r159927 = r159926 / r159921;
        double r159928 = pow(r159925, r159927);
        double r159929 = r159920 * r159928;
        return r159929;
}
↓
double f(double k, double n) {
        double r159930 = 1.0;
        double r159931 = cbrt(r159930);
        double r159932 = r159931 * r159931;
        double r159933 = 1.0;
        double r159934 = r159932 / r159933;
        double r159935 = k;
        double r159936 = sqrt(r159935);
        double r159937 = r159936 / r159931;
        double r159938 = r159934 / r159937;
        double r159939 = 2.0;
        double r159940 = atan2(1.0, 0.0);
        double r159941 = r159939 * r159940;
        double r159942 = r159930 - r159935;
        double r159943 = r159942 / r159939;
        double r159944 = pow(r159941, r159943);
        double r159945 = n;
        double r159946 = pow(r159945, r159943);
        double r159947 = r159944 * r159946;
        double r159948 = r159938 * r159947;
        return r159948;
}

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 unpow-prod-down0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.6

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

  7. Applied associate-/r*0.6

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

  9. Applied *-un-lft-identity0.6

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

  10. Applied add-cube-cbrt0.6

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

  11. Applied times-frac0.6

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

  12. Applied associate-/l*0.6

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))