Average Error: 0.4 → 0.5
Time: 39.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{\sqrt{k}} \cdot \frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{\sqrt{k}} \cdot \frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r4529119 = 1.0;
        double r4529120 = k;
        double r4529121 = sqrt(r4529120);
        double r4529122 = r4529119 / r4529121;
        double r4529123 = 2.0;
        double r4529124 = atan2(1.0, 0.0);
        double r4529125 = r4529123 * r4529124;
        double r4529126 = n;
        double r4529127 = r4529125 * r4529126;
        double r4529128 = r4529119 - r4529120;
        double r4529129 = r4529128 / r4529123;
        double r4529130 = pow(r4529127, r4529129);
        double r4529131 = r4529122 * r4529130;
        return r4529131;
}
↓
double f(double k, double n) {
        double r4529132 = 1.0;
        double r4529133 = k;
        double r4529134 = sqrt(r4529133);
        double r4529135 = sqrt(r4529134);
        double r4529136 = 2.0;
        double r4529137 = atan2(1.0, 0.0);
        double r4529138 = r4529136 * r4529137;
        double r4529139 = n;
        double r4529140 = r4529138 * r4529139;
        double r4529141 = 0.5;
        double r4529142 = r4529133 / r4529136;
        double r4529143 = r4529141 - r4529142;
        double r4529144 = pow(r4529140, r4529143);
        double r4529145 = r4529135 / r4529144;
        double r4529146 = r4529135 * r4529145;
        double r4529147 = r4529132 / r4529146;
        return r4529147;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019139 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))