Average Error: 0.4 → 0.5
Time: 4.3m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{{\pi}^{\left(\frac{k}{2}\right)}} \cdot \frac{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\frac{1}{2}}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{{\pi}^{\left(\frac{k}{2}\right)}} \cdot \frac{{n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\frac{1}{2}}}}
double f(double k, double n) {
        double r28919130 = 1.0;
        double r28919131 = k;
        double r28919132 = sqrt(r28919131);
        double r28919133 = r28919130 / r28919132;
        double r28919134 = 2.0;
        double r28919135 = atan2(1.0, 0.0);
        double r28919136 = r28919134 * r28919135;
        double r28919137 = n;
        double r28919138 = r28919136 * r28919137;
        double r28919139 = r28919130 - r28919131;
        double r28919140 = r28919139 / r28919134;
        double r28919141 = pow(r28919138, r28919140);
        double r28919142 = r28919133 * r28919141;
        return r28919142;
}
↓
double f(double k, double n) {
        double r28919143 = 2.0;
        double r28919144 = 0.5;
        double r28919145 = k;
        double r28919146 = r28919145 / r28919143;
        double r28919147 = r28919144 - r28919146;
        double r28919148 = pow(r28919143, r28919147);
        double r28919149 = atan2(1.0, 0.0);
        double r28919150 = pow(r28919149, r28919146);
        double r28919151 = r28919148 / r28919150;
        double r28919152 = n;
        double r28919153 = pow(r28919152, r28919147);
        double r28919154 = sqrt(r28919145);
        double r28919155 = pow(r28919149, r28919144);
        double r28919156 = r28919154 / r28919155;
        double r28919157 = r28919153 / r28919156;
        double r28919158 = r28919151 * r28919157;
        return r28919158;
}

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. Simplified0.3

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

  4. Applied unpow-prod-down0.4

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

  5. Applied associate-/l*0.4

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

  7. Applied pow-sub0.4

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

  8. Applied associate-/r/0.4

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

  9. Applied unpow-prod-down0.5

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

  10. Applied times-frac0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))