Average Error: 0.4 → 0.5
Time: 10.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r165443 = 1.0;
        double r165444 = k;
        double r165445 = sqrt(r165444);
        double r165446 = r165443 / r165445;
        double r165447 = 2.0;
        double r165448 = atan2(1.0, 0.0);
        double r165449 = r165447 * r165448;
        double r165450 = n;
        double r165451 = r165449 * r165450;
        double r165452 = r165443 - r165444;
        double r165453 = r165452 / r165447;
        double r165454 = pow(r165451, r165453);
        double r165455 = r165446 * r165454;
        return r165455;
}
↓
double f(double k, double n) {
        double r165456 = 1.0;
        double r165457 = k;
        double r165458 = sqrt(r165457);
        double r165459 = r165456 / r165458;
        double r165460 = 2.0;
        double r165461 = atan2(1.0, 0.0);
        double r165462 = r165460 * r165461;
        double r165463 = n;
        double r165464 = r165462 * r165463;
        double r165465 = r165456 - r165457;
        double r165466 = r165465 / r165460;
        double r165467 = 2.0;
        double r165468 = r165466 / r165467;
        double r165469 = pow(r165464, r165468);
        double r165470 = r165459 * r165469;
        double r165471 = r165470 * r165469;
        return r165471;
}

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 sqr-pow0.5

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

  4. Applied associate-*r*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))