Average Error: 33.2 → 25.2
Time: 1.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 9.680598668789347 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\\ \end{array}\]
\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 9.680598668789347 \cdot 10^{+126}:\\
\;\;\;\;\sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r7779003 = 2.0;
        double r7779004 = n;
        double r7779005 = r7779003 * r7779004;
        double r7779006 = U;
        double r7779007 = r7779005 * r7779006;
        double r7779008 = t;
        double r7779009 = l;
        double r7779010 = r7779009 * r7779009;
        double r7779011 = Om;
        double r7779012 = r7779010 / r7779011;
        double r7779013 = r7779003 * r7779012;
        double r7779014 = r7779008 - r7779013;
        double r7779015 = r7779009 / r7779011;
        double r7779016 = pow(r7779015, r7779003);
        double r7779017 = r7779004 * r7779016;
        double r7779018 = U_;
        double r7779019 = r7779006 - r7779018;
        double r7779020 = r7779017 * r7779019;
        double r7779021 = r7779014 - r7779020;
        double r7779022 = r7779007 * r7779021;
        double r7779023 = sqrt(r7779022);
        return r7779023;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r7779024 = t;
        double r7779025 = 9.680598668789347e+126;
        bool r7779026 = r7779024 <= r7779025;
        double r7779027 = 2.0;
        double r7779028 = n;
        double r7779029 = l;
        double r7779030 = Om;
        double r7779031 = r7779029 / r7779030;
        double r7779032 = r7779028 * r7779031;
        double r7779033 = U;
        double r7779034 = r7779032 * r7779033;
        double r7779035 = -2.0;
        double r7779036 = r7779029 * r7779035;
        double r7779037 = U_;
        double r7779038 = r7779033 - r7779037;
        double r7779039 = r7779032 * r7779038;
        double r7779040 = r7779036 - r7779039;
        double r7779041 = r7779034 * r7779040;
        double r7779042 = r7779027 * r7779041;
        double r7779043 = r7779027 * r7779028;
        double r7779044 = r7779043 * r7779033;
        double r7779045 = r7779044 * r7779024;
        double r7779046 = r7779042 + r7779045;
        double r7779047 = sqrt(r7779046);
        double r7779048 = sqrt(r7779047);
        double r7779049 = r7779048 * r7779048;
        double r7779050 = sqrt(r7779044);
        double r7779051 = r7779029 * r7779029;
        double r7779052 = r7779051 / r7779030;
        double r7779053 = r7779052 * r7779027;
        double r7779054 = r7779024 - r7779053;
        double r7779055 = pow(r7779031, r7779027);
        double r7779056 = r7779055 * r7779028;
        double r7779057 = r7779038 * r7779056;
        double r7779058 = r7779054 - r7779057;
        double r7779059 = sqrt(r7779058);
        double r7779060 = r7779050 * r7779059;
        double r7779061 = r7779026 ? r7779049 : r7779060;
        return r7779061;
}

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 < 9.680598668789347e+126

    1. Initial program 32.8

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

    3. Applied *-un-lft-identity32.8

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

    4. Applied associate-*r*32.8

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

    5. Simplified28.6

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

    7. Applied sub-neg28.6

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

    8. Applied distribute-rgt-in28.6

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

    9. Simplified24.9

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

    11. Applied add-sqr-sqrt25.1

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

    if 9.680598668789347e+126 < t

    1. Initial program 35.6

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

    3. Applied sqrt-prod26.0

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

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 9.680598668789347 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(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*))))))