Average Error: 0.4 → 0.5
Time: 28.6s
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 r4410133 = 1.0;
        double r4410134 = k;
        double r4410135 = sqrt(r4410134);
        double r4410136 = r4410133 / r4410135;
        double r4410137 = 2.0;
        double r4410138 = atan2(1.0, 0.0);
        double r4410139 = r4410137 * r4410138;
        double r4410140 = n;
        double r4410141 = r4410139 * r4410140;
        double r4410142 = r4410133 - r4410134;
        double r4410143 = r4410142 / r4410137;
        double r4410144 = pow(r4410141, r4410143);
        double r4410145 = r4410136 * r4410144;
        return r4410145;
}
↓
double f(double k, double n) {
        double r4410146 = 1.0;
        double r4410147 = k;
        double r4410148 = sqrt(r4410147);
        double r4410149 = r4410146 / r4410148;
        double r4410150 = 2.0;
        double r4410151 = r4410146 - r4410147;
        double r4410152 = r4410151 / r4410150;
        double r4410153 = pow(r4410150, r4410152);
        double r4410154 = r4410149 * r4410153;
        double r4410155 = atan2(1.0, 0.0);
        double r4410156 = pow(r4410155, r4410152);
        double r4410157 = r4410154 * r4410156;
        double r4410158 = n;
        double r4410159 = pow(r4410158, r4410152);
        double r4410160 = r4410157 * r4410159;
        return r4410160;
}

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-down0.6

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

  4. Applied associate-*r*0.6

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}}\]
  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 2019169 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))