Average Error: 33.3 → 26.3
Time: 44.8s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 1.0337592973629363 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]
\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 1.0337592973629363 \cdot 10^{-288}:\\
\;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1314145 = 2.0;
        double r1314146 = n;
        double r1314147 = r1314145 * r1314146;
        double r1314148 = U;
        double r1314149 = r1314147 * r1314148;
        double r1314150 = t;
        double r1314151 = l;
        double r1314152 = r1314151 * r1314151;
        double r1314153 = Om;
        double r1314154 = r1314152 / r1314153;
        double r1314155 = r1314145 * r1314154;
        double r1314156 = r1314150 - r1314155;
        double r1314157 = r1314151 / r1314153;
        double r1314158 = pow(r1314157, r1314145);
        double r1314159 = r1314146 * r1314158;
        double r1314160 = U_;
        double r1314161 = r1314148 - r1314160;
        double r1314162 = r1314159 * r1314161;
        double r1314163 = r1314156 - r1314162;
        double r1314164 = r1314149 * r1314163;
        double r1314165 = sqrt(r1314164);
        return r1314165;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1314166 = U;
        double r1314167 = 1.0337592973629363e-288;
        bool r1314168 = r1314166 <= r1314167;
        double r1314169 = 2.0;
        double r1314170 = r1314169 * r1314166;
        double r1314171 = n;
        double r1314172 = t;
        double r1314173 = l;
        double r1314174 = Om;
        double r1314175 = r1314173 / r1314174;
        double r1314176 = r1314169 * r1314175;
        double r1314177 = r1314176 * r1314173;
        double r1314178 = r1314175 * r1314171;
        double r1314179 = U_;
        double r1314180 = r1314166 - r1314179;
        double r1314181 = r1314175 * r1314180;
        double r1314182 = r1314178 * r1314181;
        double r1314183 = r1314177 + r1314182;
        double r1314184 = r1314172 - r1314183;
        double r1314185 = r1314171 * r1314184;
        double r1314186 = r1314170 * r1314185;
        double r1314187 = sqrt(r1314186);
        double r1314188 = sqrt(r1314185);
        double r1314189 = sqrt(r1314170);
        double r1314190 = r1314188 * r1314189;
        double r1314191 = r1314168 ? r1314187 : r1314190;
        return r1314191;
}

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 U < 1.0337592973629363e-288

    1. Initial program 33.5

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

    2. Simplified30.0

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

    4. Applied associate-*l*30.3

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

    if 1.0337592973629363e-288 < U

    1. Initial program 33.1

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

    2. Simplified29.2

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

    4. Applied associate-*l*29.3

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

    6. Applied sqrt-prod22.0

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 1.0337592973629363 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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*))))))