Average Error: 0.4 → 0.4
Time: 13.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r147876 = 1.0;
        double r147877 = k;
        double r147878 = sqrt(r147877);
        double r147879 = r147876 / r147878;
        double r147880 = 2.0;
        double r147881 = atan2(1.0, 0.0);
        double r147882 = r147880 * r147881;
        double r147883 = n;
        double r147884 = r147882 * r147883;
        double r147885 = r147876 - r147877;
        double r147886 = r147885 / r147880;
        double r147887 = pow(r147884, r147886);
        double r147888 = r147879 * r147887;
        return r147888;
}
↓
double f(double k, double n) {
        double r147889 = 1.0;
        double r147890 = 1.0;
        double r147891 = k;
        double r147892 = r147890 / r147891;
        double r147893 = 0.25;
        double r147894 = pow(r147892, r147893);
        double r147895 = r147889 * r147894;
        double r147896 = sqrt(r147891);
        double r147897 = sqrt(r147896);
        double r147898 = r147895 / r147897;
        double r147899 = 2.0;
        double r147900 = atan2(1.0, 0.0);
        double r147901 = r147899 * r147900;
        double r147902 = n;
        double r147903 = r147901 * r147902;
        double r147904 = r147889 - r147891;
        double r147905 = r147904 / r147899;
        double r147906 = pow(r147903, r147905);
        double r147907 = r147898 * r147906;
        return r147907;
}

Error

Bits error versus k

Bits error versus n

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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 associate-/r*0.5

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

  6. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  7. Final simplification0.4

    \[\leadsto \frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))