Average Error: 0.4 → 0.5
Time: 26.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\left(\frac{1}{\sqrt{k}} \cdot \sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\left(\frac{1}{\sqrt{k}} \cdot \sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r91347 = 1.0;
        double r91348 = k;
        double r91349 = sqrt(r91348);
        double r91350 = r91347 / r91349;
        double r91351 = 2.0;
        double r91352 = atan2(1.0, 0.0);
        double r91353 = r91351 * r91352;
        double r91354 = n;
        double r91355 = r91353 * r91354;
        double r91356 = r91347 - r91348;
        double r91357 = r91356 / r91351;
        double r91358 = pow(r91355, r91357);
        double r91359 = r91350 * r91358;
        return r91359;
}
↓
double f(double k, double n) {
        double r91360 = 1.0;
        double r91361 = k;
        double r91362 = sqrt(r91361);
        double r91363 = r91360 / r91362;
        double r91364 = 2.0;
        double r91365 = atan2(1.0, 0.0);
        double r91366 = r91364 * r91365;
        double r91367 = r91360 - r91361;
        double r91368 = r91367 / r91364;
        double r91369 = pow(r91366, r91368);
        double r91370 = sqrt(r91369);
        double r91371 = r91363 * r91370;
        double r91372 = r91371 * r91370;
        double r91373 = n;
        double r91374 = pow(r91373, r91368);
        double r91375 = r91372 * r91374;
        return r91375;
}

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 unpow-prod-down0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-*r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))