Average Error: 0.4 → 1.0
Time: 36.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\sqrt{1} + \sqrt{k}\right)}\right)}^{\left(\frac{\sqrt{1} - \sqrt{k}}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\sqrt{1} + \sqrt{k}\right)}\right)}^{\left(\frac{\sqrt{1} - \sqrt{k}}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r129217 = 1.0;
        double r129218 = k;
        double r129219 = sqrt(r129218);
        double r129220 = r129217 / r129219;
        double r129221 = 2.0;
        double r129222 = atan2(1.0, 0.0);
        double r129223 = r129221 * r129222;
        double r129224 = n;
        double r129225 = r129223 * r129224;
        double r129226 = r129217 - r129218;
        double r129227 = r129226 / r129221;
        double r129228 = pow(r129225, r129227);
        double r129229 = r129220 * r129228;
        return r129229;
}
↓
double f(double k, double n) {
        double r129230 = 1.0;
        double r129231 = 2.0;
        double r129232 = atan2(1.0, 0.0);
        double r129233 = r129231 * r129232;
        double r129234 = n;
        double r129235 = r129233 * r129234;
        double r129236 = sqrt(r129230);
        double r129237 = k;
        double r129238 = sqrt(r129237);
        double r129239 = r129236 + r129238;
        double r129240 = pow(r129235, r129239);
        double r129241 = r129236 - r129238;
        double r129242 = r129241 / r129231;
        double r129243 = pow(r129240, r129242);
        double r129244 = r129230 * r129243;
        double r129245 = r129244 / r129238;
        return r129245;
}

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 associate-*l/0.4

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

  5. Applied *-un-lft-identity0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied difference-of-squares0.5

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

  9. Applied times-frac0.5

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

  10. Applied pow-unpow1.0

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

  11. Simplified1.0

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2019199 +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))))