Average Error: 33.7 → 24.0
Time: 34.8s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.108446529047814 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0427927580921139 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \end{array}\]
\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 6.108446529047814 \cdot 10^{-226}:\\
\;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\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 1.0427927580921139 \cdot 10^{+304}:\\
\;\;\;\;\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}:\\
\;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1813153 = 2.0;
        double r1813154 = n;
        double r1813155 = r1813153 * r1813154;
        double r1813156 = U;
        double r1813157 = r1813155 * r1813156;
        double r1813158 = t;
        double r1813159 = l;
        double r1813160 = r1813159 * r1813159;
        double r1813161 = Om;
        double r1813162 = r1813160 / r1813161;
        double r1813163 = r1813153 * r1813162;
        double r1813164 = r1813158 - r1813163;
        double r1813165 = r1813159 / r1813161;
        double r1813166 = pow(r1813165, r1813153);
        double r1813167 = r1813154 * r1813166;
        double r1813168 = U_;
        double r1813169 = r1813156 - r1813168;
        double r1813170 = r1813167 * r1813169;
        double r1813171 = r1813164 - r1813170;
        double r1813172 = r1813157 * r1813171;
        double r1813173 = sqrt(r1813172);
        return r1813173;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1813174 = 2.0;
        double r1813175 = n;
        double r1813176 = r1813174 * r1813175;
        double r1813177 = U;
        double r1813178 = r1813176 * r1813177;
        double r1813179 = t;
        double r1813180 = l;
        double r1813181 = r1813180 * r1813180;
        double r1813182 = Om;
        double r1813183 = r1813181 / r1813182;
        double r1813184 = r1813183 * r1813174;
        double r1813185 = r1813179 - r1813184;
        double r1813186 = r1813180 / r1813182;
        double r1813187 = pow(r1813186, r1813174);
        double r1813188 = r1813175 * r1813187;
        double r1813189 = U_;
        double r1813190 = r1813177 - r1813189;
        double r1813191 = r1813188 * r1813190;
        double r1813192 = r1813185 - r1813191;
        double r1813193 = r1813178 * r1813192;
        double r1813194 = 6.108446529047814e-226;
        bool r1813195 = r1813193 <= r1813194;
        double r1813196 = r1813179 * r1813175;
        double r1813197 = r1813174 * r1813180;
        double r1813198 = -r1813175;
        double r1813199 = r1813198 * r1813180;
        double r1813200 = r1813199 / r1813182;
        double r1813201 = r1813190 * r1813200;
        double r1813202 = r1813197 - r1813201;
        double r1813203 = r1813202 * r1813200;
        double r1813204 = r1813196 + r1813203;
        double r1813205 = r1813177 * r1813174;
        double r1813206 = r1813204 * r1813205;
        double r1813207 = 0.5;
        double r1813208 = pow(r1813206, r1813207);
        double r1813209 = 1.0427927580921139e+304;
        bool r1813210 = r1813193 <= r1813209;
        double r1813211 = sqrt(r1813193);
        double r1813212 = -r1813186;
        double r1813213 = r1813175 * r1813212;
        double r1813214 = r1813213 * r1813190;
        double r1813215 = r1813197 - r1813214;
        double r1813216 = r1813175 * r1813215;
        double r1813217 = r1813216 * r1813212;
        double r1813218 = r1813196 + r1813217;
        double r1813219 = r1813218 * r1813205;
        double r1813220 = pow(r1813219, r1813207);
        double r1813221 = r1813210 ? r1813211 : r1813220;
        double r1813222 = r1813195 ? r1813208 : r1813221;
        return r1813222;
}

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*)))) < 6.108446529047814e-226

    1. Initial program 46.6

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

    2. Simplified33.5

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

    4. Applied sub-neg33.5

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

    5. Applied distribute-rgt-in33.5

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

    6. Simplified31.9

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

    8. Applied associate-*r*33.0

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

    10. Applied pow1/233.0

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

    12. Applied *-un-lft-identity33.0

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

    13. Applied pow-unpow33.0

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

    14. Simplified32.6

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

    if 6.108446529047814e-226 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.0427927580921139e+304

    1. Initial program 1.4

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

    if 1.0427927580921139e+304 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 60.4

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

    2. Simplified52.2

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

    4. Applied sub-neg52.2

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

    5. Applied distribute-rgt-in52.2

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

    6. Simplified51.0

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

    8. Applied associate-*r*43.1

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

    10. Applied pow1/243.1

      \[\leadsto \color{blue}{{\left(\left(U \cdot 2\right) \cdot \left(t \cdot n + \left(\left(-n\right) \cdot \left(\ell \cdot 2 - \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)}^{\frac{1}{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.108446529047814 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0427927580921139 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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*))))))