Average Error: 1.6 → 0.6
Time: 30.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}\]
\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{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r660457 = 1.0;
        double r660458 = 2.0;
        double r660459 = r660457 / r660458;
        double r660460 = l;
        double r660461 = r660458 * r660460;
        double r660462 = Om;
        double r660463 = r660461 / r660462;
        double r660464 = pow(r660463, r660458);
        double r660465 = kx;
        double r660466 = sin(r660465);
        double r660467 = pow(r660466, r660458);
        double r660468 = ky;
        double r660469 = sin(r660468);
        double r660470 = pow(r660469, r660458);
        double r660471 = r660467 + r660470;
        double r660472 = r660464 * r660471;
        double r660473 = r660457 + r660472;
        double r660474 = sqrt(r660473);
        double r660475 = r660457 / r660474;
        double r660476 = r660457 + r660475;
        double r660477 = r660459 * r660476;
        double r660478 = sqrt(r660477);
        return r660478;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r660479 = 0.5;
        double r660480 = 4.0;
        double r660481 = kx;
        double r660482 = sin(r660481);
        double r660483 = Om;
        double r660484 = l;
        double r660485 = r660483 / r660484;
        double r660486 = r660482 / r660485;
        double r660487 = r660486 * r660486;
        double r660488 = ky;
        double r660489 = sin(r660488);
        double r660490 = r660483 / r660489;
        double r660491 = r660484 / r660490;
        double r660492 = r660491 * r660491;
        double r660493 = r660487 + r660492;
        double r660494 = 1.0;
        double r660495 = fma(r660480, r660493, r660494);
        double r660496 = sqrt(r660495);
        double r660497 = r660479 / r660496;
        double r660498 = exp(r660497);
        double r660499 = log(r660498);
        double r660500 = r660499 + r660479;
        double r660501 = sqrt(r660500);
        return r660501;
}

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. Simplified1.6

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

  4. Applied add-log-exp1.6

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

  5. Taylor expanded around inf 16.7

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\left(\sin ky\right)}^{2}}{{Om}^{2}} + \left(4 \cdot \frac{{\left(\sin kx\right)}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 1\right)}}}}\right) + \frac{1}{2}}\]

  6. Simplified0.6

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}} + \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}}, 1\right)}}}}\right) + \frac{1}{2}}\]

  7. Final simplification0.6

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))