Average Error: 0.4 → 0.4
Time: 46.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(n \cdot \pi\right)}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)} \cdot {2}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(n \cdot \pi\right)}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)} \cdot {2}^{\left(\frac{1}{2} - k \cdot \frac{1}{2}\right)}}{\sqrt{k}}
double f(double k, double n) {
        double r4511535 = 1.0;
        double r4511536 = k;
        double r4511537 = sqrt(r4511536);
        double r4511538 = r4511535 / r4511537;
        double r4511539 = 2.0;
        double r4511540 = atan2(1.0, 0.0);
        double r4511541 = r4511539 * r4511540;
        double r4511542 = n;
        double r4511543 = r4511541 * r4511542;
        double r4511544 = r4511535 - r4511536;
        double r4511545 = r4511544 / r4511539;
        double r4511546 = pow(r4511543, r4511545);
        double r4511547 = r4511538 * r4511546;
        return r4511547;
}
↓
double f(double k, double n) {
        double r4511548 = n;
        double r4511549 = atan2(1.0, 0.0);
        double r4511550 = r4511548 * r4511549;
        double r4511551 = 0.5;
        double r4511552 = k;
        double r4511553 = r4511552 * r4511551;
        double r4511554 = r4511551 - r4511553;
        double r4511555 = pow(r4511550, r4511554);
        double r4511556 = 2.0;
        double r4511557 = pow(r4511556, r4511554);
        double r4511558 = r4511555 * r4511557;
        double r4511559 = sqrt(r4511552);
        double r4511560 = r4511558 / r4511559;
        return r4511560;
}

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

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

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

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

  5. Applied sqrt-prod0.3

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

  6. Applied associate-/r*0.3

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

  7. Simplified0.3

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

  9. Applied unpow-prod-down0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))