Average Error: 0.4 → 0.5
Time: 31.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r4446350 = 1.0;
        double r4446351 = k;
        double r4446352 = sqrt(r4446351);
        double r4446353 = r4446350 / r4446352;
        double r4446354 = 2.0;
        double r4446355 = atan2(1.0, 0.0);
        double r4446356 = r4446354 * r4446355;
        double r4446357 = n;
        double r4446358 = r4446356 * r4446357;
        double r4446359 = r4446350 - r4446351;
        double r4446360 = r4446359 / r4446354;
        double r4446361 = pow(r4446358, r4446360);
        double r4446362 = r4446353 * r4446361;
        return r4446362;
}
↓
double f(double k, double n) {
        double r4446363 = 1.0;
        double r4446364 = k;
        double r4446365 = sqrt(r4446364);
        double r4446366 = r4446363 / r4446365;
        double r4446367 = n;
        double r4446368 = 2.0;
        double r4446369 = atan2(1.0, 0.0);
        double r4446370 = r4446368 * r4446369;
        double r4446371 = r4446367 * r4446370;
        double r4446372 = r4446363 - r4446364;
        double r4446373 = r4446372 / r4446368;
        double r4446374 = 2.0;
        double r4446375 = r4446373 / r4446374;
        double r4446376 = pow(r4446371, r4446375);
        double r4446377 = r4446366 * r4446376;
        double r4446378 = r4446377 * r4446376;
        return r4446378;
}

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

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

Reproduce

herbie shell --seed 2019170 +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))))