Average Error: 0.4 → 0.4
Time: 41.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}\]
\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({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}^{\frac{1}{2}}
double f(double k, double n) {
        double r62092 = 1.0;
        double r62093 = k;
        double r62094 = sqrt(r62093);
        double r62095 = r62092 / r62094;
        double r62096 = 2.0;
        double r62097 = atan2(1.0, 0.0);
        double r62098 = r62096 * r62097;
        double r62099 = n;
        double r62100 = r62098 * r62099;
        double r62101 = r62092 - r62093;
        double r62102 = r62101 / r62096;
        double r62103 = pow(r62100, r62102);
        double r62104 = r62095 * r62103;
        return r62104;
}
↓
double f(double k, double n) {
        double r62105 = 1.0;
        double r62106 = k;
        double r62107 = sqrt(r62106);
        double r62108 = r62105 / r62107;
        double r62109 = 2.0;
        double r62110 = atan2(1.0, 0.0);
        double r62111 = r62109 * r62110;
        double r62112 = n;
        double r62113 = r62111 * r62112;
        double r62114 = r62105 - r62106;
        double r62115 = r62114 / r62109;
        double r62116 = 2.0;
        double r62117 = r62115 / r62116;
        double r62118 = pow(r62113, r62117);
        double r62119 = r62108 * r62118;
        double r62120 = pow(r62109, r62115);
        double r62121 = r62110 * r62112;
        double r62122 = pow(r62121, r62115);
        double r62123 = r62120 * r62122;
        double r62124 = 0.5;
        double r62125 = pow(r62123, r62124);
        double r62126 = r62119 * r62125;
        return r62126;
}

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 div-inv0.5

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

  7. Applied pow-unpow0.5

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

  8. Simplified0.5

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*l*0.5

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

  13. Applied unpow-prod-down0.5

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))