Average Error: 1.8 → 0.7
Time: 39.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}\]
\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{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1711593 = 1.0;
        double r1711594 = 2.0;
        double r1711595 = r1711593 / r1711594;
        double r1711596 = l;
        double r1711597 = r1711594 * r1711596;
        double r1711598 = Om;
        double r1711599 = r1711597 / r1711598;
        double r1711600 = pow(r1711599, r1711594);
        double r1711601 = kx;
        double r1711602 = sin(r1711601);
        double r1711603 = pow(r1711602, r1711594);
        double r1711604 = ky;
        double r1711605 = sin(r1711604);
        double r1711606 = pow(r1711605, r1711594);
        double r1711607 = r1711603 + r1711606;
        double r1711608 = r1711600 * r1711607;
        double r1711609 = r1711593 + r1711608;
        double r1711610 = sqrt(r1711609);
        double r1711611 = r1711593 / r1711610;
        double r1711612 = r1711593 + r1711611;
        double r1711613 = r1711595 * r1711612;
        double r1711614 = sqrt(r1711613);
        return r1711614;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1711615 = 0.5;
        double r1711616 = 4.0;
        double r1711617 = l;
        double r1711618 = kx;
        double r1711619 = sin(r1711618);
        double r1711620 = r1711617 * r1711619;
        double r1711621 = Om;
        double r1711622 = r1711620 / r1711621;
        double r1711623 = r1711622 * r1711622;
        double r1711624 = ky;
        double r1711625 = sin(r1711624);
        double r1711626 = r1711625 * r1711617;
        double r1711627 = r1711626 / r1711621;
        double r1711628 = r1711627 * r1711627;
        double r1711629 = r1711623 + r1711628;
        double r1711630 = 1.0;
        double r1711631 = fma(r1711616, r1711629, r1711630);
        double r1711632 = sqrt(r1711631);
        double r1711633 = r1711615 / r1711632;
        double r1711634 = r1711633 + r1711615;
        double r1711635 = sqrt(r1711634);
        return r1711635;
}

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

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

  4. Applied insert-posit162.6

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

  5. Taylor expanded around inf 16.2

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

  6. Simplified0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019144 +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))))))))))