Average Error: 32.8 → 25.5
Time: 50.9s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t \cdot 2\right) + n \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{n \cdot \left(U* - U\right)}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.4044174827788283 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(\left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right) \cdot \sqrt[3]{U}\right)}\\ \end{array}\]
\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}\;\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{U \cdot \left(n \cdot \left(t \cdot 2\right) + n \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{n \cdot \left(U* - U\right)}{\frac{Om}{\ell}}\right)\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 1.4044174827788283 \cdot 10^{+153}:\\
\;\;\;\;\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{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(\left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right) \cdot \sqrt[3]{U}\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3078210 = 2.0;
        double r3078211 = n;
        double r3078212 = r3078210 * r3078211;
        double r3078213 = U;
        double r3078214 = r3078212 * r3078213;
        double r3078215 = t;
        double r3078216 = l;
        double r3078217 = r3078216 * r3078216;
        double r3078218 = Om;
        double r3078219 = r3078217 / r3078218;
        double r3078220 = r3078210 * r3078219;
        double r3078221 = r3078215 - r3078220;
        double r3078222 = r3078216 / r3078218;
        double r3078223 = pow(r3078222, r3078210);
        double r3078224 = r3078211 * r3078223;
        double r3078225 = U_;
        double r3078226 = r3078213 - r3078225;
        double r3078227 = r3078224 * r3078226;
        double r3078228 = r3078221 - r3078227;
        double r3078229 = r3078214 * r3078228;
        double r3078230 = sqrt(r3078229);
        return r3078230;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3078231 = 2.0;
        double r3078232 = n;
        double r3078233 = r3078231 * r3078232;
        double r3078234 = U;
        double r3078235 = r3078233 * r3078234;
        double r3078236 = t;
        double r3078237 = l;
        double r3078238 = r3078237 * r3078237;
        double r3078239 = Om;
        double r3078240 = r3078238 / r3078239;
        double r3078241 = r3078240 * r3078231;
        double r3078242 = r3078236 - r3078241;
        double r3078243 = r3078237 / r3078239;
        double r3078244 = pow(r3078243, r3078231);
        double r3078245 = r3078232 * r3078244;
        double r3078246 = U_;
        double r3078247 = r3078234 - r3078246;
        double r3078248 = r3078245 * r3078247;
        double r3078249 = r3078242 - r3078248;
        double r3078250 = r3078235 * r3078249;
        double r3078251 = sqrt(r3078250);
        double r3078252 = 0.0;
        bool r3078253 = r3078251 <= r3078252;
        double r3078254 = r3078236 * r3078231;
        double r3078255 = r3078232 * r3078254;
        double r3078256 = -2.0;
        double r3078257 = r3078231 * r3078237;
        double r3078258 = r3078246 - r3078234;
        double r3078259 = r3078232 * r3078258;
        double r3078260 = r3078239 / r3078237;
        double r3078261 = r3078259 / r3078260;
        double r3078262 = r3078257 - r3078261;
        double r3078263 = r3078243 * r3078262;
        double r3078264 = r3078256 * r3078263;
        double r3078265 = r3078232 * r3078264;
        double r3078266 = r3078255 + r3078265;
        double r3078267 = r3078234 * r3078266;
        double r3078268 = sqrt(r3078267);
        double r3078269 = 1.4044174827788283e+153;
        bool r3078270 = r3078251 <= r3078269;
        double r3078271 = cbrt(r3078234);
        double r3078272 = r3078271 * r3078271;
        double r3078273 = r3078232 / r3078260;
        double r3078274 = r3078258 * r3078273;
        double r3078275 = r3078257 - r3078274;
        double r3078276 = r3078243 * r3078275;
        double r3078277 = r3078236 - r3078276;
        double r3078278 = r3078231 * r3078277;
        double r3078279 = r3078232 * r3078278;
        double r3078280 = r3078279 * r3078271;
        double r3078281 = r3078272 * r3078280;
        double r3078282 = sqrt(r3078281);
        double r3078283 = r3078270 ? r3078251 : r3078282;
        double r3078284 = r3078253 ? r3078268 : r3078283;
        return r3078284;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 0.0

    1. Initial program 55.9

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

    2. Simplified55.4

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

    4. Applied associate-*l*38.0

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

    6. Applied sub-neg38.0

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

    7. Applied distribute-lft-in38.0

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

    8. Applied distribute-lft-in38.0

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

    9. Simplified38.2

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

    if 0.0 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.4044174827788283e+153

    1. Initial program 1.8

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

    if 1.4044174827788283e+153 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

    1. Initial program 60.6

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

    2. Simplified51.1

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

    4. Applied associate-*l*48.6

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

    6. Applied add-cube-cbrt48.7

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

    7. Applied associate-*l*48.7

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t \cdot 2\right) + n \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{n \cdot \left(U* - U\right)}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.4044174827788283 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(\left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right) \cdot \sqrt[3]{U}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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*))))))