Average Error: 0.4 → 0.4
Time: 1.0m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{k}}}{\frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{k}}}{\frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r3887969 = 1.0;
        double r3887970 = k;
        double r3887971 = sqrt(r3887970);
        double r3887972 = r3887969 / r3887971;
        double r3887973 = 2.0;
        double r3887974 = atan2(1.0, 0.0);
        double r3887975 = r3887973 * r3887974;
        double r3887976 = n;
        double r3887977 = r3887975 * r3887976;
        double r3887978 = r3887969 - r3887970;
        double r3887979 = r3887978 / r3887973;
        double r3887980 = pow(r3887977, r3887979);
        double r3887981 = r3887972 * r3887980;
        return r3887981;
}
↓
double f(double k, double n) {
        double r3887982 = 1.0;
        double r3887983 = k;
        double r3887984 = sqrt(r3887983);
        double r3887985 = r3887982 / r3887984;
        double r3887986 = 2.0;
        double r3887987 = atan2(1.0, 0.0);
        double r3887988 = r3887986 * r3887987;
        double r3887989 = n;
        double r3887990 = r3887988 * r3887989;
        double r3887991 = 0.5;
        double r3887992 = r3887983 / r3887986;
        double r3887993 = r3887991 - r3887992;
        double r3887994 = pow(r3887990, r3887993);
        double r3887995 = r3887982 / r3887994;
        double r3887996 = r3887985 / r3887995;
        return r3887996;
}

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. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied associate-/l*0.4

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

  7. Applied div-inv0.4

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))