Average Error: 0.4 → 0.4
Time: 13.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r137145 = 1.0;
        double r137146 = k;
        double r137147 = sqrt(r137146);
        double r137148 = r137145 / r137147;
        double r137149 = 2.0;
        double r137150 = atan2(1.0, 0.0);
        double r137151 = r137149 * r137150;
        double r137152 = n;
        double r137153 = r137151 * r137152;
        double r137154 = r137145 - r137146;
        double r137155 = r137154 / r137149;
        double r137156 = pow(r137153, r137155);
        double r137157 = r137148 * r137156;
        return r137157;
}
↓
double f(double k, double n) {
        double r137158 = 1.0;
        double r137159 = 1.0;
        double r137160 = k;
        double r137161 = r137159 / r137160;
        double r137162 = 0.25;
        double r137163 = pow(r137161, r137162);
        double r137164 = r137158 * r137163;
        double r137165 = sqrt(r137160);
        double r137166 = sqrt(r137165);
        double r137167 = r137164 / r137166;
        double r137168 = 2.0;
        double r137169 = atan2(1.0, 0.0);
        double r137170 = r137168 * r137169;
        double r137171 = n;
        double r137172 = r137170 * r137171;
        double r137173 = r137158 - r137160;
        double r137174 = r137173 / r137168;
        double r137175 = pow(r137172, r137174);
        double r137176 = r137167 * r137175;
        return r137176;
}

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 add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  6. Taylor expanded around 0 0.4

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

  7. Final simplification0.4

    \[\leadsto \frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))