Average Error: 0.4 → 1.0
Time: 35.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left({\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(\sqrt{1} + \sqrt{k}\right)}\right)}^{\left(\frac{\sqrt{1} - \sqrt{k}}{2}\right)}}{\sqrt{k}}\]
\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(2 \cdot \left(n \cdot \pi\right)\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 r118329 = 1.0;
        double r118330 = k;
        double r118331 = sqrt(r118330);
        double r118332 = r118329 / r118331;
        double r118333 = 2.0;
        double r118334 = atan2(1.0, 0.0);
        double r118335 = r118333 * r118334;
        double r118336 = n;
        double r118337 = r118335 * r118336;
        double r118338 = r118329 - r118330;
        double r118339 = r118338 / r118333;
        double r118340 = pow(r118337, r118339);
        double r118341 = r118332 * r118340;
        return r118341;
}
↓
double f(double k, double n) {
        double r118342 = 1.0;
        double r118343 = 2.0;
        double r118344 = n;
        double r118345 = atan2(1.0, 0.0);
        double r118346 = r118344 * r118345;
        double r118347 = r118343 * r118346;
        double r118348 = sqrt(r118342);
        double r118349 = k;
        double r118350 = sqrt(r118349);
        double r118351 = r118348 + r118350;
        double r118352 = pow(r118347, r118351);
        double r118353 = r118348 - r118350;
        double r118354 = r118353 / r118343;
        double r118355 = pow(r118352, r118354);
        double r118356 = r118342 * r118355;
        double r118357 = r118356 / r118350;
        return r118357;
}

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

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2019199 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))