Average Error: 1.7 → 1.7
Time: 31.7s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{\frac{2}{\frac{1}{\sqrt{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} + 1}}}\]
\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}{\frac{2}{\frac{1}{\sqrt{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} + 1}}}
double f(double l, double Om, double kx, double ky) {
        double r52479 = 1.0;
        double r52480 = 2.0;
        double r52481 = r52479 / r52480;
        double r52482 = l;
        double r52483 = r52480 * r52482;
        double r52484 = Om;
        double r52485 = r52483 / r52484;
        double r52486 = pow(r52485, r52480);
        double r52487 = kx;
        double r52488 = sin(r52487);
        double r52489 = pow(r52488, r52480);
        double r52490 = ky;
        double r52491 = sin(r52490);
        double r52492 = pow(r52491, r52480);
        double r52493 = r52489 + r52492;
        double r52494 = r52486 * r52493;
        double r52495 = r52479 + r52494;
        double r52496 = sqrt(r52495);
        double r52497 = r52479 / r52496;
        double r52498 = r52479 + r52497;
        double r52499 = r52481 * r52498;
        double r52500 = sqrt(r52499);
        return r52500;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r52501 = 1.0;
        double r52502 = 2.0;
        double r52503 = l;
        double r52504 = r52502 * r52503;
        double r52505 = Om;
        double r52506 = r52504 / r52505;
        double r52507 = pow(r52506, r52502);
        double r52508 = kx;
        double r52509 = sin(r52508);
        double r52510 = pow(r52509, r52502);
        double r52511 = ky;
        double r52512 = sin(r52511);
        double r52513 = pow(r52512, r52502);
        double r52514 = r52510 + r52513;
        double r52515 = fma(r52507, r52514, r52501);
        double r52516 = cbrt(r52515);
        double r52517 = r52516 * r52516;
        double r52518 = r52517 * r52516;
        double r52519 = sqrt(r52518);
        double r52520 = r52501 / r52519;
        double r52521 = r52520 + r52501;
        double r52522 = r52502 / r52521;
        double r52523 = r52501 / r52522;
        double r52524 = sqrt(r52523);
        return r52524;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.7

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

  2. Simplified1.7

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

  4. Applied add-cube-cbrt1.7

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

  5. Final simplification1.7

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

Reproduce

herbie shell --seed 2019325 +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))))))))))