Average Error: 0.4 → 0.5
Time: 7.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\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)}
double f(double k, double n) {
        double r153128 = 1.0;
        double r153129 = k;
        double r153130 = sqrt(r153129);
        double r153131 = r153128 / r153130;
        double r153132 = 2.0;
        double r153133 = atan2(1.0, 0.0);
        double r153134 = r153132 * r153133;
        double r153135 = n;
        double r153136 = r153134 * r153135;
        double r153137 = r153128 - r153129;
        double r153138 = r153137 / r153132;
        double r153139 = pow(r153136, r153138);
        double r153140 = r153131 * r153139;
        return r153140;
}
↓
double f(double k, double n) {
        double r153141 = 1.0;
        double r153142 = k;
        double r153143 = sqrt(r153142);
        double r153144 = sqrt(r153143);
        double r153145 = r153141 / r153144;
        double r153146 = r153145 / r153144;
        double r153147 = 2.0;
        double r153148 = atan2(1.0, 0.0);
        double r153149 = r153147 * r153148;
        double r153150 = n;
        double r153151 = r153149 * r153150;
        double r153152 = r153141 - r153142;
        double r153153 = r153152 / r153147;
        double r153154 = pow(r153151, r153153);
        double r153155 = r153146 * r153154;
        return r153155;
}

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))