Average Error: 33.9 → 29.2
Time: 42.0s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 8.2293020532058349 \cdot 10^{-305} \lor \neg \left(t \le 3.01844021419666986 \cdot 10^{-183} \lor \neg \left(t \le 2.8623757609740864 \cdot 10^{193}\right)\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]
\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}\;t \le 8.2293020532058349 \cdot 10^{-305} \lor \neg \left(t \le 3.01844021419666986 \cdot 10^{-183} \lor \neg \left(t \le 2.8623757609740864 \cdot 10^{193}\right)\right):\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r171203 = 2.0;
        double r171204 = n;
        double r171205 = r171203 * r171204;
        double r171206 = U;
        double r171207 = r171205 * r171206;
        double r171208 = t;
        double r171209 = l;
        double r171210 = r171209 * r171209;
        double r171211 = Om;
        double r171212 = r171210 / r171211;
        double r171213 = r171203 * r171212;
        double r171214 = r171208 - r171213;
        double r171215 = r171209 / r171211;
        double r171216 = pow(r171215, r171203);
        double r171217 = r171204 * r171216;
        double r171218 = U_;
        double r171219 = r171206 - r171218;
        double r171220 = r171217 * r171219;
        double r171221 = r171214 - r171220;
        double r171222 = r171207 * r171221;
        double r171223 = sqrt(r171222);
        return r171223;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r171224 = t;
        double r171225 = 8.229302053205835e-305;
        bool r171226 = r171224 <= r171225;
        double r171227 = 3.01844021419667e-183;
        bool r171228 = r171224 <= r171227;
        double r171229 = 2.8623757609740864e+193;
        bool r171230 = r171224 <= r171229;
        double r171231 = !r171230;
        bool r171232 = r171228 || r171231;
        double r171233 = !r171232;
        bool r171234 = r171226 || r171233;
        double r171235 = 2.0;
        double r171236 = n;
        double r171237 = r171235 * r171236;
        double r171238 = U;
        double r171239 = r171237 * r171238;
        double r171240 = l;
        double r171241 = Om;
        double r171242 = r171241 / r171240;
        double r171243 = r171240 / r171242;
        double r171244 = r171235 * r171243;
        double r171245 = r171224 - r171244;
        double r171246 = r171240 / r171241;
        double r171247 = 2.0;
        double r171248 = r171235 / r171247;
        double r171249 = pow(r171246, r171248);
        double r171250 = cbrt(r171249);
        double r171251 = r171250 * r171250;
        double r171252 = r171236 * r171251;
        double r171253 = r171252 * r171250;
        double r171254 = U_;
        double r171255 = r171238 - r171254;
        double r171256 = r171249 * r171255;
        double r171257 = r171253 * r171256;
        double r171258 = r171245 - r171257;
        double r171259 = r171239 * r171258;
        double r171260 = sqrt(r171259);
        double r171261 = sqrt(r171239);
        double r171262 = pow(r171246, r171235);
        double r171263 = r171236 * r171262;
        double r171264 = r171263 * r171255;
        double r171265 = r171245 - r171264;
        double r171266 = sqrt(r171265);
        double r171267 = r171261 * r171266;
        double r171268 = r171234 ? r171260 : r171267;
        return r171268;
}

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 2 regimes

  2. if t < 8.229302053205835e-305 or 3.01844021419667e-183 < t < 2.8623757609740864e+193

    1. Initial program 33.0

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

    3. Applied associate-/l*30.1

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

    5. Applied sqr-pow30.1

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

    6. Applied associate-*r*29.5

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

    8. Applied associate-*l*29.4

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

    10. Applied add-cube-cbrt29.4

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

    11. Applied associate-*r*29.4

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

    if 8.229302053205835e-305 < t < 3.01844021419667e-183 or 2.8623757609740864e+193 < t

    1. Initial program 37.8

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

    3. Applied associate-/l*35.2

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

    5. Applied sqrt-prod28.1

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

  4. Final simplification29.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.2293020532058349 \cdot 10^{-305} \lor \neg \left(t \le 3.01844021419666986 \cdot 10^{-183} \lor \neg \left(t \le 2.8623757609740864 \cdot 10^{193}\right)\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))