Average Error: 0.4 → 0.5
Time: 9.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\left(\frac{1}{\sqrt{k}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\left(\frac{1}{\sqrt{k}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r119132 = 1.0;
        double r119133 = k;
        double r119134 = sqrt(r119133);
        double r119135 = r119132 / r119134;
        double r119136 = 2.0;
        double r119137 = atan2(1.0, 0.0);
        double r119138 = r119136 * r119137;
        double r119139 = n;
        double r119140 = r119138 * r119139;
        double r119141 = r119132 - r119133;
        double r119142 = r119141 / r119136;
        double r119143 = pow(r119140, r119142);
        double r119144 = r119135 * r119143;
        return r119144;
}
↓
double f(double k, double n) {
        double r119145 = 1.0;
        double r119146 = k;
        double r119147 = sqrt(r119146);
        double r119148 = r119145 / r119147;
        double r119149 = 2.0;
        double r119150 = r119145 - r119146;
        double r119151 = r119150 / r119149;
        double r119152 = pow(r119149, r119151);
        double r119153 = r119148 * r119152;
        double r119154 = atan2(1.0, 0.0);
        double r119155 = pow(r119154, r119151);
        double r119156 = r119153 * r119155;
        double r119157 = n;
        double r119158 = pow(r119157, r119151);
        double r119159 = r119156 * r119158;
        return r119159;
}

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 unpow-prod-down0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.5

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

  7. Applied associate-*r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))