Average Error: 0.4 → 0.4
Time: 38.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left({\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right)}^{\frac{1}{2}}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1}{2} - \frac{k}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left({\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right)}^{\frac{1}{2}}}}
double f(double k, double n) {
        double r5284117 = 1.0;
        double r5284118 = k;
        double r5284119 = sqrt(r5284118);
        double r5284120 = r5284117 / r5284119;
        double r5284121 = 2.0;
        double r5284122 = atan2(1.0, 0.0);
        double r5284123 = r5284121 * r5284122;
        double r5284124 = n;
        double r5284125 = r5284123 * r5284124;
        double r5284126 = r5284117 - r5284118;
        double r5284127 = r5284126 / r5284121;
        double r5284128 = pow(r5284125, r5284127);
        double r5284129 = r5284120 * r5284128;
        return r5284129;
}
↓
double f(double k, double n) {
        double r5284130 = atan2(1.0, 0.0);
        double r5284131 = 2.0;
        double r5284132 = r5284130 * r5284131;
        double r5284133 = n;
        double r5284134 = r5284132 * r5284133;
        double r5284135 = 0.5;
        double r5284136 = k;
        double r5284137 = r5284136 / r5284131;
        double r5284138 = r5284135 - r5284137;
        double r5284139 = r5284138 / r5284131;
        double r5284140 = pow(r5284134, r5284139);
        double r5284141 = sqrt(r5284136);
        double r5284142 = pow(r5284134, r5284138);
        double r5284143 = pow(r5284142, r5284135);
        double r5284144 = r5284141 / r5284143;
        double r5284145 = r5284140 / r5284144;
        return r5284145;
}

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

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

  4. Applied sqr-pow0.4

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

  5. Applied associate-/l*0.4

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

  7. Applied div-inv0.4

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

  8. Applied pow-unpow0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019138 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))