Average Error: 1.8 → 1.8
Time: 21.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2}{Om} \cdot \ell\right)}^{2}, 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2}{Om} \cdot \ell\right)}^{2}, 1\right)}}} + 1\right) \cdot \frac{1}{2}}\]
\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{\left(\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2}{Om} \cdot \ell\right)}^{2}, 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, {\left(\frac{2}{Om} \cdot \ell\right)}^{2}, 1\right)}}} + 1\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r41253 = 1.0;
        double r41254 = 2.0;
        double r41255 = r41253 / r41254;
        double r41256 = l;
        double r41257 = r41254 * r41256;
        double r41258 = Om;
        double r41259 = r41257 / r41258;
        double r41260 = pow(r41259, r41254);
        double r41261 = kx;
        double r41262 = sin(r41261);
        double r41263 = pow(r41262, r41254);
        double r41264 = ky;
        double r41265 = sin(r41264);
        double r41266 = pow(r41265, r41254);
        double r41267 = r41263 + r41266;
        double r41268 = r41260 * r41267;
        double r41269 = r41253 + r41268;
        double r41270 = sqrt(r41269);
        double r41271 = r41253 / r41270;
        double r41272 = r41253 + r41271;
        double r41273 = r41255 * r41272;
        double r41274 = sqrt(r41273);
        return r41274;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r41275 = 1.0;
        double r41276 = kx;
        double r41277 = sin(r41276);
        double r41278 = 2.0;
        double r41279 = pow(r41277, r41278);
        double r41280 = ky;
        double r41281 = sin(r41280);
        double r41282 = pow(r41281, r41278);
        double r41283 = r41279 + r41282;
        double r41284 = Om;
        double r41285 = r41278 / r41284;
        double r41286 = l;
        double r41287 = r41285 * r41286;
        double r41288 = pow(r41287, r41278);
        double r41289 = fma(r41283, r41288, r41275);
        double r41290 = sqrt(r41289);
        double r41291 = sqrt(r41290);
        double r41292 = r41291 * r41291;
        double r41293 = r41275 / r41292;
        double r41294 = r41293 + r41275;
        double r41295 = r41275 / r41278;
        double r41296 = r41294 * r41295;
        double r41297 = sqrt(r41296);
        return r41297;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.8

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

  2. Simplified1.8

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

  4. Applied add-sqr-sqrt1.8

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

  5. Applied sqrt-prod1.8

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

  6. Simplified1.8

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

  7. Simplified1.8

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

  8. Final simplification1.8

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))