Average Error: 0.3 → 0.4
Time: 7.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)\]
\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(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)
double f(double k, double n) {
        double r131045 = 1.0;
        double r131046 = k;
        double r131047 = sqrt(r131046);
        double r131048 = r131045 / r131047;
        double r131049 = 2.0;
        double r131050 = atan2(1.0, 0.0);
        double r131051 = r131049 * r131050;
        double r131052 = n;
        double r131053 = r131051 * r131052;
        double r131054 = r131045 - r131046;
        double r131055 = r131054 / r131049;
        double r131056 = pow(r131053, r131055);
        double r131057 = r131048 * r131056;
        return r131057;
}
↓
double f(double k, double n) {
        double r131058 = 1.0;
        double r131059 = k;
        double r131060 = sqrt(r131059);
        double r131061 = r131058 / r131060;
        double r131062 = 2.0;
        double r131063 = atan2(1.0, 0.0);
        double r131064 = r131062 * r131063;
        double r131065 = n;
        double r131066 = r131064 * r131065;
        double r131067 = r131058 - r131059;
        double r131068 = r131067 / r131062;
        double r131069 = pow(r131066, r131068);
        double r131070 = sqrt(r131069);
        double r131071 = r131070 * r131070;
        double r131072 = r131061 * r131071;
        return r131072;
}

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 0.3

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

  3. Applied add-sqr-sqrt0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))