Average Error: 1.6 → 0.7
Time: 1.2m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}}} + \frac{1}{2}}\]
\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{\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)}} \cdot \sqrt{\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 r6283265 = 1.0;
        double r6283266 = 2.0;
        double r6283267 = r6283265 / r6283266;
        double r6283268 = l;
        double r6283269 = r6283266 * r6283268;
        double r6283270 = Om;
        double r6283271 = r6283269 / r6283270;
        double r6283272 = pow(r6283271, r6283266);
        double r6283273 = kx;
        double r6283274 = sin(r6283273);
        double r6283275 = pow(r6283274, r6283266);
        double r6283276 = ky;
        double r6283277 = sin(r6283276);
        double r6283278 = pow(r6283277, r6283266);
        double r6283279 = r6283275 + r6283278;
        double r6283280 = r6283272 * r6283279;
        double r6283281 = r6283265 + r6283280;
        double r6283282 = sqrt(r6283281);
        double r6283283 = r6283265 / r6283282;
        double r6283284 = r6283265 + r6283283;
        double r6283285 = r6283267 * r6283284;
        double r6283286 = sqrt(r6283285);
        return r6283286;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r6283287 = 0.5;
        double r6283288 = 4.0;
        double r6283289 = l;
        double r6283290 = kx;
        double r6283291 = sin(r6283290);
        double r6283292 = r6283289 * r6283291;
        double r6283293 = Om;
        double r6283294 = r6283292 / r6283293;
        double r6283295 = r6283294 * r6283294;
        double r6283296 = ky;
        double r6283297 = sin(r6283296);
        double r6283298 = r6283297 * r6283289;
        double r6283299 = r6283298 / r6283293;
        double r6283300 = r6283299 * r6283299;
        double r6283301 = r6283295 + r6283300;
        double r6283302 = 1.0;
        double r6283303 = fma(r6283288, r6283301, r6283302);
        double r6283304 = sqrt(r6283303);
        double r6283305 = sqrt(r6283304);
        double r6283306 = r6283305 * r6283305;
        double r6283307 = r6283287 / r6283306;
        double r6283308 = r6283307 + r6283287;
        double r6283309 = sqrt(r6283308);
        return r6283309;
}

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

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

  3. Taylor expanded around inf 16.5

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

  4. Simplified0.7

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

  6. Applied add-sqr-sqrt0.7

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

  7. Applied sqrt-prod0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019121 +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))))))))))