Average Error: 0.4 → 0.9
Time: 2.4m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left({\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)} \cdot {k}^{\frac{-1}{2}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left({\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)} \cdot {k}^{\frac{-1}{2}}
double f(double k, double n) {
        double r15720444 = 1.0;
        double r15720445 = k;
        double r15720446 = sqrt(r15720445);
        double r15720447 = r15720444 / r15720446;
        double r15720448 = 2.0;
        double r15720449 = atan2(1.0, 0.0);
        double r15720450 = r15720448 * r15720449;
        double r15720451 = n;
        double r15720452 = r15720450 * r15720451;
        double r15720453 = r15720444 - r15720445;
        double r15720454 = r15720453 / r15720448;
        double r15720455 = pow(r15720452, r15720454);
        double r15720456 = r15720447 * r15720455;
        return r15720456;
}
↓
double f(double k, double n) {
        double r15720457 = atan2(1.0, 0.0);
        double r15720458 = 2.0;
        double r15720459 = r15720457 * r15720458;
        double r15720460 = n;
        double r15720461 = r15720459 * r15720460;
        double r15720462 = 1.0;
        double r15720463 = k;
        double r15720464 = sqrt(r15720463);
        double r15720465 = r15720462 + r15720464;
        double r15720466 = pow(r15720461, r15720465);
        double r15720467 = r15720462 - r15720464;
        double r15720468 = r15720467 / r15720458;
        double r15720469 = pow(r15720466, r15720468);
        double r15720470 = -0.5;
        double r15720471 = pow(r15720463, r15720470);
        double r15720472 = r15720469 * r15720471;
        return r15720472;
}

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

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

  5. Applied add-sqr-sqrt0.4

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

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

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

  7. Applied difference-of-squares0.5

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

  8. Applied times-frac0.5

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

  9. Applied pow-unpow0.9

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

  11. Applied div-inv1.0

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

  13. Applied pow1/21.0

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

  14. Applied pow-flip0.9

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

  15. Simplified0.9

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

  16. Final simplification0.9

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))