Average Error: 33.7 → 23.6
Time: 41.6s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 9.396757433575657 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-\left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right) + t \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]
\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 9.396757433575657 \cdot 10^{-152}:\\
\;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1765102 = 2.0;
        double r1765103 = n;
        double r1765104 = r1765102 * r1765103;
        double r1765105 = U;
        double r1765106 = r1765104 * r1765105;
        double r1765107 = t;
        double r1765108 = l;
        double r1765109 = r1765108 * r1765108;
        double r1765110 = Om;
        double r1765111 = r1765109 / r1765110;
        double r1765112 = r1765102 * r1765111;
        double r1765113 = r1765107 - r1765112;
        double r1765114 = r1765108 / r1765110;
        double r1765115 = pow(r1765114, r1765102);
        double r1765116 = r1765103 * r1765115;
        double r1765117 = U_;
        double r1765118 = r1765105 - r1765117;
        double r1765119 = r1765116 * r1765118;
        double r1765120 = r1765113 - r1765119;
        double r1765121 = r1765106 * r1765120;
        double r1765122 = sqrt(r1765121);
        return r1765122;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1765123 = 2.0;
        double r1765124 = n;
        double r1765125 = r1765123 * r1765124;
        double r1765126 = U;
        double r1765127 = r1765125 * r1765126;
        double r1765128 = t;
        double r1765129 = l;
        double r1765130 = r1765129 * r1765129;
        double r1765131 = Om;
        double r1765132 = r1765130 / r1765131;
        double r1765133 = r1765132 * r1765123;
        double r1765134 = r1765128 - r1765133;
        double r1765135 = r1765129 / r1765131;
        double r1765136 = pow(r1765135, r1765123);
        double r1765137 = r1765124 * r1765136;
        double r1765138 = U_;
        double r1765139 = r1765126 - r1765138;
        double r1765140 = r1765137 * r1765139;
        double r1765141 = r1765134 - r1765140;
        double r1765142 = r1765127 * r1765141;
        double r1765143 = sqrt(r1765142);
        double r1765144 = 9.396757433575657e-152;
        bool r1765145 = r1765143 <= r1765144;
        double r1765146 = r1765123 * r1765129;
        double r1765147 = r1765138 - r1765126;
        double r1765148 = r1765124 * r1765147;
        double r1765149 = r1765135 * r1765148;
        double r1765150 = r1765146 - r1765149;
        double r1765151 = r1765150 * r1765135;
        double r1765152 = r1765128 - r1765151;
        double r1765153 = r1765124 * r1765152;
        double r1765154 = cbrt(r1765153);
        double r1765155 = r1765154 * r1765154;
        double r1765156 = r1765154 * r1765155;
        double r1765157 = r1765126 * r1765156;
        double r1765158 = r1765123 * r1765157;
        double r1765159 = sqrt(r1765158);
        double r1765160 = r1765124 * r1765135;
        double r1765161 = r1765126 * r1765160;
        double r1765162 = r1765160 * r1765147;
        double r1765163 = r1765146 - r1765162;
        double r1765164 = -r1765163;
        double r1765165 = r1765161 * r1765164;
        double r1765166 = r1765126 * r1765124;
        double r1765167 = r1765128 * r1765166;
        double r1765168 = r1765165 + r1765167;
        double r1765169 = r1765123 * r1765168;
        double r1765170 = sqrt(r1765169);
        double r1765171 = r1765145 ? r1765159 : r1765170;
        return r1765171;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 9.396757433575657e-152

    1. Initial program 55.0

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

    2. Simplified39.3

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

    4. Applied add-cube-cbrt39.5

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

    if 9.396757433575657e-152 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

    1. Initial program 29.5

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

    2. Simplified30.3

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

    4. Applied associate-*r*27.3

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

    5. Taylor expanded around 0 28.4

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

    6. Simplified25.9

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

    8. Applied sub-neg25.9

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

    9. Applied distribute-rgt-in25.9

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

    10. Simplified22.8

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

    12. Applied associate-*r*20.5

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 9.396757433575657 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-\left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right) + t \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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*))))))