Average Error: 1.6 → 1.6
Time: 26.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}\right)}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}\right)}\right)}
double f(double l, double Om, double kx, double ky) {
        double r42670 = 1.0;
        double r42671 = 2.0;
        double r42672 = r42670 / r42671;
        double r42673 = l;
        double r42674 = r42671 * r42673;
        double r42675 = Om;
        double r42676 = r42674 / r42675;
        double r42677 = pow(r42676, r42671);
        double r42678 = kx;
        double r42679 = sin(r42678);
        double r42680 = pow(r42679, r42671);
        double r42681 = ky;
        double r42682 = sin(r42681);
        double r42683 = pow(r42682, r42671);
        double r42684 = r42680 + r42683;
        double r42685 = r42677 * r42684;
        double r42686 = r42670 + r42685;
        double r42687 = sqrt(r42686);
        double r42688 = r42670 / r42687;
        double r42689 = r42670 + r42688;
        double r42690 = r42672 * r42689;
        double r42691 = sqrt(r42690);
        return r42691;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r42692 = 1.0;
        double r42693 = 2.0;
        double r42694 = r42692 / r42693;
        double r42695 = kx;
        double r42696 = sin(r42695);
        double r42697 = pow(r42696, r42693);
        double r42698 = ky;
        double r42699 = sin(r42698);
        double r42700 = pow(r42699, r42693);
        double r42701 = r42697 + r42700;
        double r42702 = l;
        double r42703 = r42693 * r42702;
        double r42704 = Om;
        double r42705 = r42703 / r42704;
        double r42706 = pow(r42705, r42693);
        double r42707 = fma(r42701, r42706, r42692);
        double r42708 = sqrt(r42707);
        double r42709 = cbrt(r42708);
        double r42710 = r42709 * r42709;
        double r42711 = sqrt(r42708);
        double r42712 = cbrt(r42711);
        double r42713 = r42712 * r42712;
        double r42714 = r42710 * r42713;
        double r42715 = r42692 / r42714;
        double r42716 = r42692 + r42715;
        double r42717 = r42694 * r42716;
        double r42718 = sqrt(r42717);
        return r42718;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.6

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  4. Simplified1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right)} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]

  5. Simplified1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \color{blue}{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}}\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}}}\right)}\]

  8. Applied sqrt-prod1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \sqrt[3]{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}}}\right)}\]

  9. Applied cbrt-prod1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}\right)}}\right)}\]

  10. Final simplification1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, 1\right)}}}\right)}\right)}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))