Average Error: 0.4 → 0.5
Time: 25.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}\right) \cdot \left({n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}\right) \cdot \left({n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r89000 = 1.0;
        double r89001 = k;
        double r89002 = sqrt(r89001);
        double r89003 = r89000 / r89002;
        double r89004 = 2.0;
        double r89005 = atan2(1.0, 0.0);
        double r89006 = r89004 * r89005;
        double r89007 = n;
        double r89008 = r89006 * r89007;
        double r89009 = r89000 - r89001;
        double r89010 = r89009 / r89004;
        double r89011 = pow(r89008, r89010);
        double r89012 = r89003 * r89011;
        return r89012;
}
↓
double f(double k, double n) {
        double r89013 = atan2(1.0, 0.0);
        double r89014 = 1.0;
        double r89015 = k;
        double r89016 = r89014 - r89015;
        double r89017 = 2.0;
        double r89018 = r89016 / r89017;
        double r89019 = pow(r89013, r89018);
        double r89020 = pow(r89017, r89018);
        double r89021 = r89019 * r89020;
        double r89022 = sqrt(r89015);
        double r89023 = r89014 / r89022;
        double r89024 = r89021 * r89023;
        double r89025 = n;
        double r89026 = 2.0;
        double r89027 = r89018 / r89026;
        double r89028 = pow(r89025, r89027);
        double r89029 = r89028 * r89028;
        double r89030 = r89024 * r89029;
        return r89030;
}

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. Simplified0.5

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

  7. Applied unpow-prod-down0.4

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

  9. Applied sqr-pow0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019174 +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))))