Average Error: 0.4 → 0.4
Time: 4.8m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\left(\frac{1 - k}{2}\right)}}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{\pi}^{\left(\frac{1 - k}{2}\right)}}} \cdot {2}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r41699564 = 1.0;
        double r41699565 = k;
        double r41699566 = sqrt(r41699565);
        double r41699567 = r41699564 / r41699566;
        double r41699568 = 2.0;
        double r41699569 = atan2(1.0, 0.0);
        double r41699570 = r41699568 * r41699569;
        double r41699571 = n;
        double r41699572 = r41699570 * r41699571;
        double r41699573 = r41699564 - r41699565;
        double r41699574 = r41699573 / r41699568;
        double r41699575 = pow(r41699572, r41699574);
        double r41699576 = r41699567 * r41699575;
        return r41699576;
}
↓
double f(double k, double n) {
        double r41699577 = n;
        double r41699578 = 1.0;
        double r41699579 = k;
        double r41699580 = r41699578 - r41699579;
        double r41699581 = 2.0;
        double r41699582 = r41699580 / r41699581;
        double r41699583 = pow(r41699577, r41699582);
        double r41699584 = sqrt(r41699579);
        double r41699585 = atan2(1.0, 0.0);
        double r41699586 = pow(r41699585, r41699582);
        double r41699587 = r41699584 / r41699586;
        double r41699588 = r41699583 / r41699587;
        double r41699589 = pow(r41699581, r41699582);
        double r41699590 = r41699588 * r41699589;
        return r41699590;
}

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 unpow-prod-down0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied unpow-prod-down0.5

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

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

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

  9. Applied sqrt-prod0.5

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

  10. Applied times-frac0.4

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

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

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))