Average Error: 34.6 → 27.4
Time: 1.7m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right)}\\ \end{array}\]
\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}\;U \le -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\
\;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\

\mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2539235 = 2.0;
        double r2539236 = n;
        double r2539237 = r2539235 * r2539236;
        double r2539238 = U;
        double r2539239 = r2539237 * r2539238;
        double r2539240 = t;
        double r2539241 = l;
        double r2539242 = r2539241 * r2539241;
        double r2539243 = Om;
        double r2539244 = r2539242 / r2539243;
        double r2539245 = r2539235 * r2539244;
        double r2539246 = r2539240 - r2539245;
        double r2539247 = r2539241 / r2539243;
        double r2539248 = pow(r2539247, r2539235);
        double r2539249 = r2539236 * r2539248;
        double r2539250 = U_;
        double r2539251 = r2539238 - r2539250;
        double r2539252 = r2539249 * r2539251;
        double r2539253 = r2539246 - r2539252;
        double r2539254 = r2539239 * r2539253;
        double r2539255 = sqrt(r2539254);
        return r2539255;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2539256 = U;
        double r2539257 = -1.0845919563821882e+140;
        bool r2539258 = r2539256 <= r2539257;
        double r2539259 = t;
        double r2539260 = l;
        double r2539261 = Om;
        double r2539262 = r2539260 / r2539261;
        double r2539263 = 2.0;
        double r2539264 = r2539260 * r2539263;
        double r2539265 = -r2539264;
        double r2539266 = r2539262 * r2539265;
        double r2539267 = r2539259 + r2539266;
        double r2539268 = n;
        double r2539269 = r2539263 * r2539268;
        double r2539270 = r2539267 * r2539269;
        double r2539271 = r2539270 * r2539256;
        double r2539272 = sqrt(r2539271);
        double r2539273 = 4.990611334173369e-293;
        bool r2539274 = r2539256 <= r2539273;
        double r2539275 = 2.0;
        double r2539276 = r2539263 / r2539275;
        double r2539277 = pow(r2539262, r2539276);
        double r2539278 = r2539268 * r2539277;
        double r2539279 = r2539278 * r2539277;
        double r2539280 = U_;
        double r2539281 = r2539280 - r2539256;
        double r2539282 = r2539279 * r2539281;
        double r2539283 = r2539262 * r2539264;
        double r2539284 = r2539282 - r2539283;
        double r2539285 = r2539284 + r2539259;
        double r2539286 = r2539285 * r2539256;
        double r2539287 = r2539269 * r2539286;
        double r2539288 = sqrt(r2539287);
        double r2539289 = sqrt(r2539256);
        double r2539290 = r2539269 * r2539285;
        double r2539291 = sqrt(r2539290);
        double r2539292 = r2539289 * r2539291;
        double r2539293 = r2539274 ? r2539288 : r2539292;
        double r2539294 = r2539258 ? r2539272 : r2539293;
        return r2539294;
}

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 U < -1.0845919563821882e+140

    1. Initial program 31.8

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

    2. Simplified32.8

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

    3. Taylor expanded around 0 33.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \left(\color{blue}{0} - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\]

    if -1.0845919563821882e+140 < U < 4.990611334173369e-293

    1. Initial program 35.1

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

    2. Simplified32.1

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

    4. Applied sqr-pow32.1

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

    5. Applied associate-*r*30.9

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

    7. Applied associate-*l*29.4

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

    if 4.990611334173369e-293 < U

    1. Initial program 34.7

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

    2. Simplified32.3

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

    4. Applied sqr-pow32.3

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

    5. Applied associate-*r*31.2

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

    7. Applied sqrt-prod24.6

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

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))