Average Error: 0.4 → 0.5
Time: 29.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(\pi \cdot n\right) \cdot 2\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\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(\pi \cdot n\right) \cdot 2\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r92157 = 1.0;
        double r92158 = k;
        double r92159 = sqrt(r92158);
        double r92160 = r92157 / r92159;
        double r92161 = 2.0;
        double r92162 = atan2(1.0, 0.0);
        double r92163 = r92161 * r92162;
        double r92164 = n;
        double r92165 = r92163 * r92164;
        double r92166 = r92157 - r92158;
        double r92167 = r92166 / r92161;
        double r92168 = pow(r92165, r92167);
        double r92169 = r92160 * r92168;
        return r92169;
}
↓
double f(double k, double n) {
        double r92170 = 1.0;
        double r92171 = k;
        double r92172 = sqrt(r92171);
        double r92173 = r92170 / r92172;
        double r92174 = sqrt(r92173);
        double r92175 = atan2(1.0, 0.0);
        double r92176 = n;
        double r92177 = r92175 * r92176;
        double r92178 = 2.0;
        double r92179 = r92177 * r92178;
        double r92180 = r92170 - r92171;
        double r92181 = r92180 / r92178;
        double r92182 = pow(r92179, r92181);
        double r92183 = r92174 * r92182;
        double r92184 = r92174 * r92183;
        return r92184;
}

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. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))