Average Error: 0.4 → 0.5
Time: 7.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r146941 = 1.0;
        double r146942 = k;
        double r146943 = sqrt(r146942);
        double r146944 = r146941 / r146943;
        double r146945 = 2.0;
        double r146946 = atan2(1.0, 0.0);
        double r146947 = r146945 * r146946;
        double r146948 = n;
        double r146949 = r146947 * r146948;
        double r146950 = r146941 - r146942;
        double r146951 = r146950 / r146945;
        double r146952 = pow(r146949, r146951);
        double r146953 = r146944 * r146952;
        return r146953;
}
↓
double f(double k, double n) {
        double r146954 = 1.0;
        double r146955 = k;
        double r146956 = sqrt(r146955);
        double r146957 = r146954 / r146956;
        double r146958 = sqrt(r146957);
        double r146959 = 2.0;
        double r146960 = atan2(1.0, 0.0);
        double r146961 = r146959 * r146960;
        double r146962 = n;
        double r146963 = r146961 * r146962;
        double r146964 = r146954 - r146955;
        double r146965 = r146964 / r146959;
        double r146966 = pow(r146963, r146965);
        double r146967 = r146958 * r146966;
        double r146968 = r146958 * r146967;
        return r146968;
}

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 add-sqr-sqrt0.5

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

  4. Applied associate-*l*0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))