Average Error: 0.4 → 0.4
Time: 1.6m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{k}^{\frac{-1}{2}}}{\frac{1}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{1}{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{k}^{\frac{-1}{2}}}{\frac{1}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{1}{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r7800575 = 1.0;
        double r7800576 = k;
        double r7800577 = sqrt(r7800576);
        double r7800578 = r7800575 / r7800577;
        double r7800579 = 2.0;
        double r7800580 = atan2(1.0, 0.0);
        double r7800581 = r7800579 * r7800580;
        double r7800582 = n;
        double r7800583 = r7800581 * r7800582;
        double r7800584 = r7800575 - r7800576;
        double r7800585 = r7800584 / r7800579;
        double r7800586 = pow(r7800583, r7800585);
        double r7800587 = r7800578 * r7800586;
        return r7800587;
}
↓
double f(double k, double n) {
        double r7800588 = k;
        double r7800589 = -0.5;
        double r7800590 = pow(r7800588, r7800589);
        double r7800591 = 1.0;
        double r7800592 = atan2(1.0, 0.0);
        double r7800593 = 0.5;
        double r7800594 = 2.0;
        double r7800595 = r7800588 / r7800594;
        double r7800596 = r7800593 - r7800595;
        double r7800597 = pow(r7800592, r7800596);
        double r7800598 = r7800591 / r7800597;
        double r7800599 = n;
        double r7800600 = r7800599 * r7800594;
        double r7800601 = pow(r7800600, r7800596);
        double r7800602 = r7800591 / r7800601;
        double r7800603 = r7800598 * r7800602;
        double r7800604 = r7800590 / r7800603;
        return r7800604;
}

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 clear-num0.4

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

  6. Applied div-inv0.4

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

  7. Applied associate-/r*0.4

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

  9. Applied pow1/20.4

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

  10. Applied pow-flip0.4

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

  12. Applied unpow-prod-down0.5

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

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

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

  14. Applied times-frac0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))