Average Error: 0.4 → 0.5
Time: 13.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}
double f(double k, double n) {
        double r129397 = 1.0;
        double r129398 = k;
        double r129399 = sqrt(r129398);
        double r129400 = r129397 / r129399;
        double r129401 = 2.0;
        double r129402 = atan2(1.0, 0.0);
        double r129403 = r129401 * r129402;
        double r129404 = n;
        double r129405 = r129403 * r129404;
        double r129406 = r129397 - r129398;
        double r129407 = r129406 / r129401;
        double r129408 = pow(r129405, r129407);
        double r129409 = r129400 * r129408;
        return r129409;
}
↓
double f(double k, double n) {
        double r129410 = 1.0;
        double r129411 = k;
        double r129412 = sqrt(r129411);
        double r129413 = r129410 / r129412;
        double r129414 = 2.0;
        double r129415 = atan2(1.0, 0.0);
        double r129416 = r129414 * r129415;
        double r129417 = n;
        double r129418 = r129416 * r129417;
        double r129419 = r129410 - r129411;
        double r129420 = r129419 / r129414;
        double r129421 = pow(r129418, r129420);
        double r129422 = r129413 * r129421;
        double r129423 = sqrt(r129422);
        double r129424 = r129423 * r129423;
        return r129424;
}

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 add-sqr-sqrt0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020045 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))