Toniolo and Linder, Equation (13)

Average Error: 33.1 → 26.7

Time: 46.0s

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}\;U \le -1.4421678277993846 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}\\ \mathbf{elif}\;U \le 9.442227136838079 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1941397 = 2.0;
        double r1941398 = n;
        double r1941399 = r1941397 * r1941398;
        double r1941400 = U;
        double r1941401 = r1941399 * r1941400;
        double r1941402 = t;
        double r1941403 = l;
        double r1941404 = r1941403 * r1941403;
        double r1941405 = Om;
        double r1941406 = r1941404 / r1941405;
        double r1941407 = r1941397 * r1941406;
        double r1941408 = r1941402 - r1941407;
        double r1941409 = r1941403 / r1941405;
        double r1941410 = pow(r1941409, r1941397);
        double r1941411 = r1941398 * r1941410;
        double r1941412 = U_;
        double r1941413 = r1941400 - r1941412;
        double r1941414 = r1941411 * r1941413;
        double r1941415 = r1941408 - r1941414;
        double r1941416 = r1941401 * r1941415;
        double r1941417 = sqrt(r1941416);
        return r1941417;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1941418 = U;
        double r1941419 = -1.4421678277993846e+75;
        bool r1941420 = r1941418 <= r1941419;
        double r1941421 = 2.0;
        double r1941422 = n;
        double r1941423 = r1941421 * r1941422;
        double r1941424 = r1941423 * r1941418;
        double r1941425 = l;
        double r1941426 = Om;
        double r1941427 = r1941425 / r1941426;
        double r1941428 = -2.0;
        double r1941429 = r1941428 * r1941425;
        double r1941430 = r1941422 * r1941427;
        double r1941431 = U_;
        double r1941432 = r1941418 - r1941431;
        double r1941433 = r1941430 * r1941432;
        double r1941434 = r1941429 - r1941433;
        double r1941435 = r1941427 * r1941434;
        double r1941436 = t;
        double r1941437 = r1941435 + r1941436;
        double r1941438 = r1941424 * r1941437;
        double r1941439 = cbrt(r1941438);
        double r1941440 = r1941439 * r1941439;
        double r1941441 = r1941440 * r1941439;
        double r1941442 = sqrt(r1941441);
        double r1941443 = sqrt(r1941442);
        double r1941444 = sqrt(r1941438);
        double r1941445 = sqrt(r1941444);
        double r1941446 = r1941443 * r1941445;
        double r1941447 = 9.442227136838079e+52;
        bool r1941448 = r1941418 <= r1941447;
        double r1941449 = r1941418 * r1941437;
        double r1941450 = r1941449 * r1941423;
        double r1941451 = sqrt(r1941450);
        double r1941452 = r1941448 ? r1941451 : r1941446;
        double r1941453 = r1941420 ? r1941446 : r1941452;
        return r1941453;
}

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 U < -1.4421678277993846e+75 or 9.442227136838079e+52 < U

   1. Initial program 28.7

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

   2. Simplified 25.5

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

   4. Applied add-sqr-sqrt 25.7

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

   6. Applied add-cube-cbrt 25.8

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

   if -1.4421678277993846e+75 < U < 9.442227136838079e+52

   1. Initial program 34.8

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

   2. Simplified 30.8

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

   4. Applied associate-*l* 27.1

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

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.4421678277993846 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}\\ \mathbf{elif}\;U \le 9.442227136838079 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) + t\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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*))))))