Average Error: 0.4 → 0.4
Time: 39.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}
double f(double k, double n) {
        double r2707951 = 1.0;
        double r2707952 = k;
        double r2707953 = sqrt(r2707952);
        double r2707954 = r2707951 / r2707953;
        double r2707955 = 2.0;
        double r2707956 = atan2(1.0, 0.0);
        double r2707957 = r2707955 * r2707956;
        double r2707958 = n;
        double r2707959 = r2707957 * r2707958;
        double r2707960 = r2707951 - r2707952;
        double r2707961 = r2707960 / r2707955;
        double r2707962 = pow(r2707959, r2707961);
        double r2707963 = r2707954 * r2707962;
        return r2707963;
}
↓
double f(double k, double n) {
        double r2707964 = 1.0;
        double r2707965 = k;
        double r2707966 = sqrt(r2707965);
        double r2707967 = n;
        double r2707968 = 2.0;
        double r2707969 = atan2(1.0, 0.0);
        double r2707970 = r2707968 * r2707969;
        double r2707971 = r2707967 * r2707970;
        double r2707972 = r2707964 - r2707965;
        double r2707973 = r2707972 / r2707968;
        double r2707974 = pow(r2707971, r2707973);
        double r2707975 = r2707966 / r2707974;
        double r2707976 = r2707964 / r2707975;
        return r2707976;
}

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 *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))