Average Error: 1.7 → 0.7
Time: 28.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\ell \cdot \sin ky}{Om} \cdot \frac{\ell \cdot \sin ky}{Om} + \frac{\sin kx \cdot \ell}{Om} \cdot \frac{\sin kx \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}\]
\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 ky}{Om} \cdot \frac{\ell \cdot \sin ky}{Om} + \frac{\sin kx \cdot \ell}{Om} \cdot \frac{\sin kx \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r800386 = 1.0;
        double r800387 = 2.0;
        double r800388 = r800386 / r800387;
        double r800389 = l;
        double r800390 = r800387 * r800389;
        double r800391 = Om;
        double r800392 = r800390 / r800391;
        double r800393 = pow(r800392, r800387);
        double r800394 = kx;
        double r800395 = sin(r800394);
        double r800396 = pow(r800395, r800387);
        double r800397 = ky;
        double r800398 = sin(r800397);
        double r800399 = pow(r800398, r800387);
        double r800400 = r800396 + r800399;
        double r800401 = r800393 * r800400;
        double r800402 = r800386 + r800401;
        double r800403 = sqrt(r800402);
        double r800404 = r800386 / r800403;
        double r800405 = r800386 + r800404;
        double r800406 = r800388 * r800405;
        double r800407 = sqrt(r800406);
        return r800407;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r800408 = 0.5;
        double r800409 = 4.0;
        double r800410 = l;
        double r800411 = ky;
        double r800412 = sin(r800411);
        double r800413 = r800410 * r800412;
        double r800414 = Om;
        double r800415 = r800413 / r800414;
        double r800416 = r800415 * r800415;
        double r800417 = kx;
        double r800418 = sin(r800417);
        double r800419 = r800418 * r800410;
        double r800420 = r800419 / r800414;
        double r800421 = r800420 * r800420;
        double r800422 = r800416 + r800421;
        double r800423 = 1.0;
        double r800424 = fma(r800409, r800422, r800423);
        double r800425 = sqrt(r800424);
        double r800426 = r800408 / r800425;
        double r800427 = r800426 + r800408;
        double r800428 = sqrt(r800427);
        return r800428;
}

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

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

  3. Taylor expanded around inf 16.8

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019141 +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))))))))))