Average Error: 0.4 → 0.5
Time: 9.6m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}
double f(double k, double n) {
        double r70516002 = 1.0;
        double r70516003 = k;
        double r70516004 = sqrt(r70516003);
        double r70516005 = r70516002 / r70516004;
        double r70516006 = 2.0;
        double r70516007 = atan2(1.0, 0.0);
        double r70516008 = r70516006 * r70516007;
        double r70516009 = n;
        double r70516010 = r70516008 * r70516009;
        double r70516011 = r70516002 - r70516003;
        double r70516012 = r70516011 / r70516006;
        double r70516013 = pow(r70516010, r70516012);
        double r70516014 = r70516005 * r70516013;
        return r70516014;
}
↓
double f(double k, double n) {
        double r70516015 = atan2(1.0, 0.0);
        double r70516016 = 2.0;
        double r70516017 = r70516015 * r70516016;
        double r70516018 = n;
        double r70516019 = r70516017 * r70516018;
        double r70516020 = 1.0;
        double r70516021 = k;
        double r70516022 = r70516020 - r70516021;
        double r70516023 = r70516022 / r70516016;
        double r70516024 = pow(r70516019, r70516023);
        double r70516025 = sqrt(r70516021);
        double r70516026 = sqrt(r70516025);
        double r70516027 = r70516024 / r70516026;
        double r70516028 = r70516027 / r70516026;
        return r70516028;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019125 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))