Average Error: 0.4 → 0.4
Time: 29.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\pi \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot {2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\pi \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot {2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r3808139 = 1.0;
        double r3808140 = k;
        double r3808141 = sqrt(r3808140);
        double r3808142 = r3808139 / r3808141;
        double r3808143 = 2.0;
        double r3808144 = atan2(1.0, 0.0);
        double r3808145 = r3808143 * r3808144;
        double r3808146 = n;
        double r3808147 = r3808145 * r3808146;
        double r3808148 = r3808139 - r3808140;
        double r3808149 = r3808148 / r3808143;
        double r3808150 = pow(r3808147, r3808149);
        double r3808151 = r3808142 * r3808150;
        return r3808151;
}
↓
double f(double k, double n) {
        double r3808152 = atan2(1.0, 0.0);
        double r3808153 = n;
        double r3808154 = r3808152 * r3808153;
        double r3808155 = 0.5;
        double r3808156 = k;
        double r3808157 = 2.0;
        double r3808158 = r3808156 / r3808157;
        double r3808159 = r3808155 - r3808158;
        double r3808160 = pow(r3808154, r3808159);
        double r3808161 = pow(r3808157, r3808159);
        double r3808162 = r3808160 * r3808161;
        double r3808163 = sqrt(r3808156);
        double r3808164 = r3808162 / r3808163;
        return r3808164;
}

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. Taylor expanded around 0 0.3

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

  5. Applied unpow-prod-down0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))