Average Error: 0.4 → 0.5
Time: 8.8s
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 r141092 = 1.0;
        double r141093 = k;
        double r141094 = sqrt(r141093);
        double r141095 = r141092 / r141094;
        double r141096 = 2.0;
        double r141097 = atan2(1.0, 0.0);
        double r141098 = r141096 * r141097;
        double r141099 = n;
        double r141100 = r141098 * r141099;
        double r141101 = r141092 - r141093;
        double r141102 = r141101 / r141096;
        double r141103 = pow(r141100, r141102);
        double r141104 = r141095 * r141103;
        return r141104;
}
↓
double f(double k, double n) {
        double r141105 = 1.0;
        double r141106 = k;
        double r141107 = sqrt(r141106);
        double r141108 = r141105 / r141107;
        double r141109 = sqrt(r141108);
        double r141110 = 2.0;
        double r141111 = atan2(1.0, 0.0);
        double r141112 = r141110 * r141111;
        double r141113 = n;
        double r141114 = r141112 * r141113;
        double r141115 = r141105 - r141106;
        double r141116 = r141115 / r141110;
        double r141117 = pow(r141114, r141116);
        double r141118 = r141109 * r141117;
        double r141119 = r141109 * r141118;
        return r141119;
}

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 2019362 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))