Average Error: 0.3 → 0.4
Time: 7.9s
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 r131033 = 1.0;
        double r131034 = k;
        double r131035 = sqrt(r131034);
        double r131036 = r131033 / r131035;
        double r131037 = 2.0;
        double r131038 = atan2(1.0, 0.0);
        double r131039 = r131037 * r131038;
        double r131040 = n;
        double r131041 = r131039 * r131040;
        double r131042 = r131033 - r131034;
        double r131043 = r131042 / r131037;
        double r131044 = pow(r131041, r131043);
        double r131045 = r131036 * r131044;
        return r131045;
}
↓
double f(double k, double n) {
        double r131046 = 1.0;
        double r131047 = k;
        double r131048 = sqrt(r131047);
        double r131049 = r131046 / r131048;
        double r131050 = 2.0;
        double r131051 = atan2(1.0, 0.0);
        double r131052 = r131050 * r131051;
        double r131053 = n;
        double r131054 = r131052 * r131053;
        double r131055 = r131046 - r131047;
        double r131056 = r131055 / r131050;
        double r131057 = pow(r131054, r131056);
        double r131058 = sqrt(r131057);
        double r131059 = r131058 * r131058;
        double r131060 = r131049 * r131059;
        return r131060;
}

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))))