Average Error: 0.4 → 0.5
Time: 28.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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 r123361 = 1.0;
        double r123362 = k;
        double r123363 = sqrt(r123362);
        double r123364 = r123361 / r123363;
        double r123365 = 2.0;
        double r123366 = atan2(1.0, 0.0);
        double r123367 = r123365 * r123366;
        double r123368 = n;
        double r123369 = r123367 * r123368;
        double r123370 = r123361 - r123362;
        double r123371 = r123370 / r123365;
        double r123372 = pow(r123369, r123371);
        double r123373 = r123364 * r123372;
        return r123373;
}
↓
double f(double k, double n) {
        double r123374 = 1.0;
        double r123375 = k;
        double r123376 = sqrt(r123375);
        double r123377 = r123374 / r123376;
        double r123378 = 2.0;
        double r123379 = atan2(1.0, 0.0);
        double r123380 = r123378 * r123379;
        double r123381 = n;
        double r123382 = r123380 * r123381;
        double r123383 = r123374 - r123375;
        double r123384 = r123383 / r123378;
        double r123385 = 2.0;
        double r123386 = r123384 / r123385;
        double r123387 = pow(r123382, r123386);
        double r123388 = r123377 * r123387;
        double r123389 = r123388 * r123387;
        return r123389;
}

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 2019198 +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))))