Average Error: 33.7 → 27.6
Time: 47.3s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1063631272675845 \cdot 10^{-282}:\\ \;\;\;\;{\left(\left(\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.51509684742799 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2}\\ \end{array}\]
\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}\;\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 1.1063631272675845 \cdot 10^{-282}:\\
\;\;\;\;{\left(\left(\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(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\

\mathbf{elif}\;\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 6.51509684742799 \cdot 10^{+287}:\\
\;\;\;\;\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)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\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 2}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1633155 = 2.0;
        double r1633156 = n;
        double r1633157 = r1633155 * r1633156;
        double r1633158 = U;
        double r1633159 = r1633157 * r1633158;
        double r1633160 = t;
        double r1633161 = l;
        double r1633162 = r1633161 * r1633161;
        double r1633163 = Om;
        double r1633164 = r1633162 / r1633163;
        double r1633165 = r1633155 * r1633164;
        double r1633166 = r1633160 - r1633165;
        double r1633167 = r1633161 / r1633163;
        double r1633168 = pow(r1633167, r1633155);
        double r1633169 = r1633156 * r1633168;
        double r1633170 = U_;
        double r1633171 = r1633158 - r1633170;
        double r1633172 = r1633169 * r1633171;
        double r1633173 = r1633166 - r1633172;
        double r1633174 = r1633159 * r1633173;
        double r1633175 = sqrt(r1633174);
        return r1633175;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1633176 = 2.0;
        double r1633177 = n;
        double r1633178 = r1633176 * r1633177;
        double r1633179 = U;
        double r1633180 = r1633178 * r1633179;
        double r1633181 = t;
        double r1633182 = l;
        double r1633183 = r1633182 * r1633182;
        double r1633184 = Om;
        double r1633185 = r1633183 / r1633184;
        double r1633186 = r1633185 * r1633176;
        double r1633187 = r1633181 - r1633186;
        double r1633188 = r1633182 / r1633184;
        double r1633189 = pow(r1633188, r1633176);
        double r1633190 = r1633177 * r1633189;
        double r1633191 = U_;
        double r1633192 = r1633179 - r1633191;
        double r1633193 = r1633190 * r1633192;
        double r1633194 = r1633187 - r1633193;
        double r1633195 = r1633180 * r1633194;
        double r1633196 = 1.1063631272675845e-282;
        bool r1633197 = r1633195 <= r1633196;
        double r1633198 = r1633176 * r1633182;
        double r1633199 = r1633191 - r1633179;
        double r1633200 = r1633177 * r1633199;
        double r1633201 = r1633188 * r1633200;
        double r1633202 = r1633198 - r1633201;
        double r1633203 = r1633202 * r1633188;
        double r1633204 = r1633181 - r1633203;
        double r1633205 = r1633204 * r1633178;
        double r1633206 = r1633205 * r1633179;
        double r1633207 = 0.5;
        double r1633208 = pow(r1633206, r1633207);
        double r1633209 = 6.51509684742799e+287;
        bool r1633210 = r1633195 <= r1633209;
        double r1633211 = sqrt(r1633195);
        double r1633212 = r1633179 * r1633177;
        double r1633213 = sqrt(r1633212);
        double r1633214 = r1633204 * r1633176;
        double r1633215 = sqrt(r1633214);
        double r1633216 = r1633213 * r1633215;
        double r1633217 = r1633210 ? r1633211 : r1633216;
        double r1633218 = r1633197 ? r1633208 : r1633217;
        return r1633218;
}

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 3 regimes

  2. if (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.1063631272675845e-282

    1. Initial program 53.5

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

    2. Simplified48.1

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

    4. Applied pow1/248.1

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

    6. Applied associate-*l*35.9

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

    7. Simplified36.8

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

    if 1.1063631272675845e-282 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 6.51509684742799e+287

    1. Initial program 1.6

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

    if 6.51509684742799e+287 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 58.9

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

    2. Simplified51.9

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

    4. Applied pow1/251.9

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

    6. Applied unpow-prod-down51.0

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

    7. Simplified51.0

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

    8. Simplified51.4

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

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1063631272675845 \cdot 10^{-282}:\\ \;\;\;\;{\left(\left(\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.51509684742799 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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*))))))