Average Error: 0.4 → 0.4
Time: 27.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\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{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\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 r4463120 = 1.0;
        double r4463121 = k;
        double r4463122 = sqrt(r4463121);
        double r4463123 = r4463120 / r4463122;
        double r4463124 = 2.0;
        double r4463125 = atan2(1.0, 0.0);
        double r4463126 = r4463124 * r4463125;
        double r4463127 = n;
        double r4463128 = r4463126 * r4463127;
        double r4463129 = r4463120 - r4463121;
        double r4463130 = r4463129 / r4463124;
        double r4463131 = pow(r4463128, r4463130);
        double r4463132 = r4463123 * r4463131;
        return r4463132;
}
↓
double f(double k, double n) {
        double r4463133 = 1.0;
        double r4463134 = k;
        double r4463135 = sqrt(r4463134);
        double r4463136 = r4463133 / r4463135;
        double r4463137 = sqrt(r4463136);
        double r4463138 = sqrt(r4463135);
        double r4463139 = r4463133 / r4463138;
        double r4463140 = r4463139 / r4463138;
        double r4463141 = sqrt(r4463140);
        double r4463142 = 2.0;
        double r4463143 = atan2(1.0, 0.0);
        double r4463144 = r4463142 * r4463143;
        double r4463145 = n;
        double r4463146 = r4463144 * r4463145;
        double r4463147 = r4463133 - r4463134;
        double r4463148 = r4463147 / r4463142;
        double r4463149 = pow(r4463146, r4463148);
        double r4463150 = r4463141 * r4463149;
        double r4463151 = r4463137 * r4463150;
        return r4463151;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt0.5

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

  7. Applied sqrt-prod0.5

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))