Average Error: 33.2 → 26.6
Time: 3.4m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -9.183046660180342 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \le 1.0657230548376832 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}}\\ \end{array}\]
\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 -9.183046660180342 \cdot 10^{-109}:\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)}\\

\mathbf{elif}\;U \le 1.0657230548376832 \cdot 10^{+24}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot U\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r25640387 = 2.0;
        double r25640388 = n;
        double r25640389 = r25640387 * r25640388;
        double r25640390 = U;
        double r25640391 = r25640389 * r25640390;
        double r25640392 = t;
        double r25640393 = l;
        double r25640394 = r25640393 * r25640393;
        double r25640395 = Om;
        double r25640396 = r25640394 / r25640395;
        double r25640397 = r25640387 * r25640396;
        double r25640398 = r25640392 - r25640397;
        double r25640399 = r25640393 / r25640395;
        double r25640400 = pow(r25640399, r25640387);
        double r25640401 = r25640388 * r25640400;
        double r25640402 = U_;
        double r25640403 = r25640390 - r25640402;
        double r25640404 = r25640401 * r25640403;
        double r25640405 = r25640398 - r25640404;
        double r25640406 = r25640391 * r25640405;
        double r25640407 = sqrt(r25640406);
        return r25640407;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r25640408 = U;
        double r25640409 = -9.183046660180342e-109;
        bool r25640410 = r25640408 <= r25640409;
        double r25640411 = 2.0;
        double r25640412 = n;
        double r25640413 = r25640411 * r25640412;
        double r25640414 = r25640413 * r25640408;
        double r25640415 = t;
        double r25640416 = l;
        double r25640417 = Om;
        double r25640418 = r25640417 / r25640416;
        double r25640419 = r25640416 / r25640418;
        double r25640420 = r25640411 * r25640419;
        double r25640421 = r25640415 - r25640420;
        double r25640422 = 1.0;
        double r25640423 = r25640422 / r25640418;
        double r25640424 = U_;
        double r25640425 = r25640408 - r25640424;
        double r25640426 = r25640423 * r25640425;
        double r25640427 = r25640412 * r25640426;
        double r25640428 = r25640427 / r25640418;
        double r25640429 = r25640421 - r25640428;
        double r25640430 = r25640414 * r25640429;
        double r25640431 = sqrt(r25640430);
        double r25640432 = 1.0657230548376832e+24;
        bool r25640433 = r25640408 <= r25640432;
        double r25640434 = r25640412 / r25640418;
        double r25640435 = r25640425 * r25640434;
        double r25640436 = r25640435 / r25640418;
        double r25640437 = r25640421 - r25640436;
        double r25640438 = r25640437 * r25640408;
        double r25640439 = r25640413 * r25640438;
        double r25640440 = sqrt(r25640439);
        double r25640441 = r25640414 * r25640437;
        double r25640442 = sqrt(r25640441);
        double r25640443 = sqrt(r25640442);
        double r25640444 = r25640443 * r25640443;
        double r25640445 = r25640433 ? r25640440 : r25640444;
        double r25640446 = r25640410 ? r25640431 : r25640445;
        return r25640446;
}

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if U < -9.183046660180342e-109

    1. Initial program 28.3

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

    3. Applied associate-/l*24.9

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

    4. Taylor expanded around -inf 31.1

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

    5. Simplified24.1

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

    7. Applied associate-*l/23.6

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

    9. Applied div-inv23.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\color{blue}{\left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)}\]

    10. Applied associate-*l*24.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\color{blue}{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell}}\right)}\]

    if -9.183046660180342e-109 < U < 1.0657230548376832e+24

    1. Initial program 37.3

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

    3. Applied associate-/l*34.9

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

    4. Taylor expanded around -inf 41.1

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

    5. Simplified33.9

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

    7. Applied associate-*l/33.4

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

    9. Applied associate-*l*28.6

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

    if 1.0657230548376832e+24 < U

    1. Initial program 28.5

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

    3. Applied associate-/l*25.9

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

    4. Taylor expanded around -inf 30.9

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

    5. Simplified25.2

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

    7. Applied associate-*l/24.6

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

    9. Applied add-sqr-sqrt24.8

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

  4. Final simplification26.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -9.183046660180342 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \le 1.0657230548376832 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(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*))))))