Average Error: 0.5 → 0.6
Time: 10.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left({2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)\right)
double f(double k, double n) {
        double r146379 = 1.0;
        double r146380 = k;
        double r146381 = sqrt(r146380);
        double r146382 = r146379 / r146381;
        double r146383 = 2.0;
        double r146384 = atan2(1.0, 0.0);
        double r146385 = r146383 * r146384;
        double r146386 = n;
        double r146387 = r146385 * r146386;
        double r146388 = r146379 - r146380;
        double r146389 = r146388 / r146383;
        double r146390 = pow(r146387, r146389);
        double r146391 = r146382 * r146390;
        return r146391;
}
↓
double f(double k, double n) {
        double r146392 = 1.0;
        double r146393 = k;
        double r146394 = sqrt(r146393);
        double r146395 = r146392 / r146394;
        double r146396 = 2.0;
        double r146397 = r146392 - r146393;
        double r146398 = r146397 / r146396;
        double r146399 = pow(r146396, r146398);
        double r146400 = atan2(1.0, 0.0);
        double r146401 = pow(r146400, r146398);
        double r146402 = n;
        double r146403 = pow(r146402, r146398);
        double r146404 = r146401 * r146403;
        double r146405 = r146399 * r146404;
        double r146406 = r146395 * r146405;
        return r146406;
}

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 unpow-prod-down0.7

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

  5. Applied unpow-prod-down0.6

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

  6. Applied associate-*l*0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))