Toniolo and Linder, Equation (13)

Average Error: 33.4 → 25.3

Time: 44.7s

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0865908327746399 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2031302 = 2.0;
        double r2031303 = n;
        double r2031304 = r2031302 * r2031303;
        double r2031305 = U;
        double r2031306 = r2031304 * r2031305;
        double r2031307 = t;
        double r2031308 = l;
        double r2031309 = r2031308 * r2031308;
        double r2031310 = Om;
        double r2031311 = r2031309 / r2031310;
        double r2031312 = r2031302 * r2031311;
        double r2031313 = r2031307 - r2031312;
        double r2031314 = r2031308 / r2031310;
        double r2031315 = pow(r2031314, r2031302);
        double r2031316 = r2031303 * r2031315;
        double r2031317 = U_;
        double r2031318 = r2031305 - r2031317;
        double r2031319 = r2031316 * r2031318;
        double r2031320 = r2031313 - r2031319;
        double r2031321 = r2031306 * r2031320;
        double r2031322 = sqrt(r2031321);
        return r2031322;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2031323 = 2.0;
        double r2031324 = n;
        double r2031325 = r2031323 * r2031324;
        double r2031326 = U;
        double r2031327 = r2031325 * r2031326;
        double r2031328 = t;
        double r2031329 = l;
        double r2031330 = r2031329 * r2031329;
        double r2031331 = Om;
        double r2031332 = r2031330 / r2031331;
        double r2031333 = r2031332 * r2031323;
        double r2031334 = r2031328 - r2031333;
        double r2031335 = r2031329 / r2031331;
        double r2031336 = pow(r2031335, r2031323);
        double r2031337 = r2031324 * r2031336;
        double r2031338 = U_;
        double r2031339 = r2031326 - r2031338;
        double r2031340 = r2031337 * r2031339;
        double r2031341 = r2031334 - r2031340;
        double r2031342 = r2031327 * r2031341;
        double r2031343 = sqrt(r2031342);
        double r2031344 = 0.0;
        bool r2031345 = r2031343 <= r2031344;
        double r2031346 = r2031328 * r2031324;
        double r2031347 = r2031323 * r2031329;
        double r2031348 = r2031338 - r2031326;
        double r2031349 = r2031324 * r2031348;
        double r2031350 = r2031335 * r2031349;
        double r2031351 = r2031347 - r2031350;
        double r2031352 = -r2031351;
        double r2031353 = r2031324 * r2031335;
        double r2031354 = r2031352 * r2031353;
        double r2031355 = r2031346 + r2031354;
        double r2031356 = r2031326 * r2031355;
        double r2031357 = r2031323 * r2031356;
        double r2031358 = sqrt(r2031357);
        double r2031359 = 1.0865908327746399e+112;
        bool r2031360 = r2031343 <= r2031359;
        double r2031361 = r2031360 ? r2031343 : r2031358;
        double r2031362 = r2031345 ? r2031358 : r2031361;
        return r2031362;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 2 regimes

2. if (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 0.0 or 1.0865908327746399e+112 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 54.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 46.1

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 46.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)} \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}\]

   5. Applied associate-*l* 46.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}\right)}\]
   6. Using strategy rm

   7. Applied sub-neg 46.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \color{blue}{\left(t + \left(-\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\right)\right)\right)}\]

   8. Applied distribute-lft-in 46.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \color{blue}{\left(\sqrt[3]{n} \cdot t + \sqrt[3]{n} \cdot \left(-\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\right)\right)}\]

   9. Applied distribute-lft-in 46.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot t\right) + \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(-\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}\right)}\]

   10. Simplified 46.1

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{t \cdot n} + \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(-\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)\right)}\]

   11. Simplified 41.3

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \color{blue}{\left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot \ell - \left(\left(U* - U\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)}\right)\right)}\]

   if 0.0 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0865908327746399e+112

   1. Initial program 1.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0865908327746399 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))