Average Error: 0.4 → 0.3
Time: 4.7m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r33165456 = 1.0;
        double r33165457 = k;
        double r33165458 = sqrt(r33165457);
        double r33165459 = r33165456 / r33165458;
        double r33165460 = 2.0;
        double r33165461 = atan2(1.0, 0.0);
        double r33165462 = r33165460 * r33165461;
        double r33165463 = n;
        double r33165464 = r33165462 * r33165463;
        double r33165465 = r33165456 - r33165457;
        double r33165466 = r33165465 / r33165460;
        double r33165467 = pow(r33165464, r33165466);
        double r33165468 = r33165459 * r33165467;
        return r33165468;
}
↓
double f(double k, double n) {
        double r33165469 = atan2(1.0, 0.0);
        double r33165470 = n;
        double r33165471 = 2.0;
        double r33165472 = r33165470 * r33165471;
        double r33165473 = r33165469 * r33165472;
        double r33165474 = 1.0;
        double r33165475 = k;
        double r33165476 = r33165474 - r33165475;
        double r33165477 = r33165476 / r33165471;
        double r33165478 = pow(r33165473, r33165477);
        double r33165479 = sqrt(r33165475);
        double r33165480 = r33165478 / r33165479;
        return r33165480;
}

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 clear-num0.4

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

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

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

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

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

  8. Applied times-frac0.4

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

  9. Simplified0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))