Average Error: 1.7 → 0.6
Time: 1.5m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}\]
\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, \left(\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}\right), 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r9321336 = 1.0;
        double r9321337 = 2.0;
        double r9321338 = r9321336 / r9321337;
        double r9321339 = l;
        double r9321340 = r9321337 * r9321339;
        double r9321341 = Om;
        double r9321342 = r9321340 / r9321341;
        double r9321343 = pow(r9321342, r9321337);
        double r9321344 = kx;
        double r9321345 = sin(r9321344);
        double r9321346 = pow(r9321345, r9321337);
        double r9321347 = ky;
        double r9321348 = sin(r9321347);
        double r9321349 = pow(r9321348, r9321337);
        double r9321350 = r9321346 + r9321349;
        double r9321351 = r9321343 * r9321350;
        double r9321352 = r9321336 + r9321351;
        double r9321353 = sqrt(r9321352);
        double r9321354 = r9321336 / r9321353;
        double r9321355 = r9321336 + r9321354;
        double r9321356 = r9321338 * r9321355;
        double r9321357 = sqrt(r9321356);
        return r9321357;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r9321358 = 0.5;
        double r9321359 = 4.0;
        double r9321360 = l;
        double r9321361 = kx;
        double r9321362 = sin(r9321361);
        double r9321363 = r9321360 * r9321362;
        double r9321364 = Om;
        double r9321365 = r9321363 / r9321364;
        double r9321366 = r9321365 * r9321365;
        double r9321367 = ky;
        double r9321368 = sin(r9321367);
        double r9321369 = r9321368 * r9321360;
        double r9321370 = r9321369 / r9321364;
        double r9321371 = r9321370 * r9321370;
        double r9321372 = r9321366 + r9321371;
        double r9321373 = 1.0;
        double r9321374 = fma(r9321359, r9321372, r9321373);
        double r9321375 = sqrt(r9321374);
        double r9321376 = r9321358 / r9321375;
        double r9321377 = r9321376 + r9321358;
        double r9321378 = sqrt(r9321377);
        return r9321378;
}

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

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

  3. Taylor expanded around inf 16.4

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019120 +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))))))))))