Average Error: 32.9 → 27.9
Time: 49.1s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.141827696680125 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{U \cdot n}\\ \end{array}\]
\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 1.141827696680125 \cdot 10^{+46}:\\
\;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3373027 = 2.0;
        double r3373028 = n;
        double r3373029 = r3373027 * r3373028;
        double r3373030 = U;
        double r3373031 = r3373029 * r3373030;
        double r3373032 = t;
        double r3373033 = l;
        double r3373034 = r3373033 * r3373033;
        double r3373035 = Om;
        double r3373036 = r3373034 / r3373035;
        double r3373037 = r3373027 * r3373036;
        double r3373038 = r3373032 - r3373037;
        double r3373039 = r3373033 / r3373035;
        double r3373040 = pow(r3373039, r3373027);
        double r3373041 = r3373028 * r3373040;
        double r3373042 = U_;
        double r3373043 = r3373030 - r3373042;
        double r3373044 = r3373041 * r3373043;
        double r3373045 = r3373038 - r3373044;
        double r3373046 = r3373031 * r3373045;
        double r3373047 = sqrt(r3373046);
        return r3373047;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3373048 = t;
        double r3373049 = 1.141827696680125e+46;
        bool r3373050 = r3373048 <= r3373049;
        double r3373051 = U;
        double r3373052 = n;
        double r3373053 = 2.0;
        double r3373054 = l;
        double r3373055 = r3373053 * r3373054;
        double r3373056 = Om;
        double r3373057 = r3373056 / r3373054;
        double r3373058 = r3373052 / r3373057;
        double r3373059 = U_;
        double r3373060 = r3373059 - r3373051;
        double r3373061 = r3373058 * r3373060;
        double r3373062 = r3373055 - r3373061;
        double r3373063 = r3373054 / r3373056;
        double r3373064 = r3373062 * r3373063;
        double r3373065 = r3373048 - r3373064;
        double r3373066 = r3373053 * r3373065;
        double r3373067 = r3373052 * r3373066;
        double r3373068 = r3373051 * r3373067;
        double r3373069 = sqrt(r3373068);
        double r3373070 = sqrt(r3373066);
        double r3373071 = r3373051 * r3373052;
        double r3373072 = sqrt(r3373071);
        double r3373073 = r3373070 * r3373072;
        double r3373074 = r3373050 ? r3373069 : r3373073;
        return r3373074;
}

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 < 1.141827696680125e+46

    1. Initial program 32.5

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

    2. Simplified28.6

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

    4. Applied associate-*l*28.8

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

    if 1.141827696680125e+46 < t

    1. Initial program 34.4

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

    2. Simplified30.7

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

    4. Applied sqrt-prod25.0

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

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.141827696680125 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{U \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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*))))))