Average Error: 0.4 → 0.4
Time: 42.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r4082431 = 1.0;
        double r4082432 = k;
        double r4082433 = sqrt(r4082432);
        double r4082434 = r4082431 / r4082433;
        double r4082435 = 2.0;
        double r4082436 = atan2(1.0, 0.0);
        double r4082437 = r4082435 * r4082436;
        double r4082438 = n;
        double r4082439 = r4082437 * r4082438;
        double r4082440 = r4082431 - r4082432;
        double r4082441 = r4082440 / r4082435;
        double r4082442 = pow(r4082439, r4082441);
        double r4082443 = r4082434 * r4082442;
        return r4082443;
}
↓
double f(double k, double n) {
        double r4082444 = atan2(1.0, 0.0);
        double r4082445 = 1.0;
        double r4082446 = k;
        double r4082447 = r4082445 - r4082446;
        double r4082448 = 2.0;
        double r4082449 = r4082447 / r4082448;
        double r4082450 = pow(r4082444, r4082449);
        double r4082451 = sqrt(r4082446);
        double r4082452 = r4082450 / r4082451;
        double r4082453 = pow(r4082448, r4082449);
        double r4082454 = r4082452 * r4082453;
        double r4082455 = n;
        double r4082456 = pow(r4082455, r4082449);
        double r4082457 = r4082454 * r4082456;
        return r4082457;
}

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. Simplified0.4

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

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

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

  5. Applied unpow-prod-down0.5

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

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

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

  10. Applied sqrt-prod0.5

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

  11. Applied unpow-prod-down0.5

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))