Average Error: 0.4 → 0.4
Time: 26.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\right)\]
\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(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\right)
double f(double k, double n) {
        double r103977 = 1.0;
        double r103978 = k;
        double r103979 = sqrt(r103978);
        double r103980 = r103977 / r103979;
        double r103981 = 2.0;
        double r103982 = atan2(1.0, 0.0);
        double r103983 = r103981 * r103982;
        double r103984 = n;
        double r103985 = r103983 * r103984;
        double r103986 = r103977 - r103978;
        double r103987 = r103986 / r103981;
        double r103988 = pow(r103985, r103987);
        double r103989 = r103980 * r103988;
        return r103989;
}
↓
double f(double k, double n) {
        double r103990 = 1.0;
        double r103991 = k;
        double r103992 = sqrt(r103991);
        double r103993 = r103990 / r103992;
        double r103994 = 2.0;
        double r103995 = atan2(1.0, 0.0);
        double r103996 = r103994 * r103995;
        double r103997 = r103990 - r103991;
        double r103998 = r103997 / r103994;
        double r103999 = pow(r103996, r103998);
        double r104000 = sqrt(r103999);
        double r104001 = 1.0;
        double r104002 = n;
        double r104003 = r104001 / r104002;
        double r104004 = -0.5;
        double r104005 = r104004 * r103997;
        double r104006 = pow(r104003, r104005);
        double r104007 = r104000 * r104006;
        double r104008 = r104000 * r104007;
        double r104009 = r103993 * r104008;
        return r104009;
}

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. Using strategy rm

  5. Applied add-sqr-sqrt0.5

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around inf 17.3

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \color{blue}{e^{-0.5 \cdot \left(\left(1 - k\right) \cdot \log \left(\frac{1}{n}\right)\right)}}\right)\right)\]

  8. Simplified0.4

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \color{blue}{{\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}}\right)\right)\]

  9. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \left(\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))