Average Error: 0.4 → 0.5
Time: 17.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r128949 = 1.0;
        double r128950 = k;
        double r128951 = sqrt(r128950);
        double r128952 = r128949 / r128951;
        double r128953 = 2.0;
        double r128954 = atan2(1.0, 0.0);
        double r128955 = r128953 * r128954;
        double r128956 = n;
        double r128957 = r128955 * r128956;
        double r128958 = r128949 - r128950;
        double r128959 = r128958 / r128953;
        double r128960 = pow(r128957, r128959);
        double r128961 = r128952 * r128960;
        return r128961;
}
↓
double f(double k, double n) {
        double r128962 = 1.0;
        double r128963 = k;
        double r128964 = sqrt(r128963);
        double r128965 = sqrt(r128964);
        double r128966 = r128962 / r128965;
        double r128967 = 1.0;
        double r128968 = r128967 / r128965;
        double r128969 = r128966 * r128968;
        double r128970 = 2.0;
        double r128971 = atan2(1.0, 0.0);
        double r128972 = r128970 * r128971;
        double r128973 = n;
        double r128974 = r128972 * r128973;
        double r128975 = r128967 - r128963;
        double r128976 = r128975 / r128970;
        double r128977 = pow(r128974, r128976);
        double r128978 = r128969 * r128977;
        return r128978;
}

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-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied *-un-lft-identity0.5

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

  6. Applied times-frac0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))