Average Error: 0.4 → 0.4
Time: 3.3m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{1}{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{1}{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}}
double f(double k, double n) {
        double r44145519 = 1.0;
        double r44145520 = k;
        double r44145521 = sqrt(r44145520);
        double r44145522 = r44145519 / r44145521;
        double r44145523 = 2.0;
        double r44145524 = atan2(1.0, 0.0);
        double r44145525 = r44145523 * r44145524;
        double r44145526 = n;
        double r44145527 = r44145525 * r44145526;
        double r44145528 = r44145519 - r44145520;
        double r44145529 = r44145528 / r44145523;
        double r44145530 = pow(r44145527, r44145529);
        double r44145531 = r44145522 * r44145530;
        return r44145531;
}
↓
double f(double k, double n) {
        double r44145532 = 1.0;
        double r44145533 = n;
        double r44145534 = 2.0;
        double r44145535 = atan2(1.0, 0.0);
        double r44145536 = r44145534 * r44145535;
        double r44145537 = r44145533 * r44145536;
        double r44145538 = k;
        double r44145539 = r44145532 - r44145538;
        double r44145540 = r44145539 / r44145534;
        double r44145541 = pow(r44145537, r44145540);
        double r44145542 = sqrt(r44145538);
        double r44145543 = r44145541 / r44145542;
        double r44145544 = r44145532 / r44145543;
        double r44145545 = r44145532 / r44145544;
        return r44145545;
}

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. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied associate-/l*0.4

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

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

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

  8. Applied sqrt-prod0.4

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

  9. Applied associate-/l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))