Average Error: 0.5 → 0.5
Time: 13.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double f(double k, double n) {
        double r146206 = 1.0;
        double r146207 = k;
        double r146208 = sqrt(r146207);
        double r146209 = r146206 / r146208;
        double r146210 = 2.0;
        double r146211 = atan2(1.0, 0.0);
        double r146212 = r146210 * r146211;
        double r146213 = n;
        double r146214 = r146212 * r146213;
        double r146215 = r146206 - r146207;
        double r146216 = r146215 / r146210;
        double r146217 = pow(r146214, r146216);
        double r146218 = r146209 * r146217;
        return r146218;
}
↓
double f(double k, double n) {
        double r146219 = 1.0;
        double r146220 = 2.0;
        double r146221 = atan2(1.0, 0.0);
        double r146222 = r146220 * r146221;
        double r146223 = n;
        double r146224 = r146222 * r146223;
        double r146225 = r146219 / r146220;
        double r146226 = pow(r146224, r146225);
        double r146227 = r146219 * r146226;
        double r146228 = 1.0;
        double r146229 = k;
        double r146230 = sqrt(r146229);
        double r146231 = r146229 / r146220;
        double r146232 = pow(r146224, r146231);
        double r146233 = r146230 * r146232;
        double r146234 = r146228 / r146233;
        double r146235 = r146227 * r146234;
        return r146235;
}

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 0.5

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

  3. Applied div-sub0.5

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

  4. Applied pow-sub0.5

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

  5. Applied frac-times0.4

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

  7. Applied div-inv0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))