Average Error: 34.4 → 32.0
Time: 43.9s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\ \mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\ \;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right)}\\ \end{array}\]
\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}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\
\;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\

\mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\
\;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r100289 = 2.0;
        double r100290 = n;
        double r100291 = r100289 * r100290;
        double r100292 = U;
        double r100293 = r100291 * r100292;
        double r100294 = t;
        double r100295 = l;
        double r100296 = r100295 * r100295;
        double r100297 = Om;
        double r100298 = r100296 / r100297;
        double r100299 = r100289 * r100298;
        double r100300 = r100294 - r100299;
        double r100301 = r100295 / r100297;
        double r100302 = pow(r100301, r100289);
        double r100303 = r100290 * r100302;
        double r100304 = U_;
        double r100305 = r100292 - r100304;
        double r100306 = r100303 * r100305;
        double r100307 = r100300 - r100306;
        double r100308 = r100293 * r100307;
        double r100309 = sqrt(r100308);
        return r100309;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r100310 = U;
        double r100311 = -7.011733841081501e+54;
        bool r100312 = r100310 <= r100311;
        double r100313 = U_;
        double r100314 = r100313 - r100310;
        double r100315 = n;
        double r100316 = r100314 * r100315;
        double r100317 = l;
        double r100318 = Om;
        double r100319 = r100317 / r100318;
        double r100320 = 2.0;
        double r100321 = pow(r100319, r100320);
        double r100322 = -r100320;
        double r100323 = r100322 * r100317;
        double r100324 = t;
        double r100325 = fma(r100319, r100323, r100324);
        double r100326 = fma(r100316, r100321, r100325);
        double r100327 = r100315 * r100310;
        double r100328 = r100320 * r100327;
        double r100329 = r100326 * r100328;
        double r100330 = sqrt(r100329);
        double r100331 = sqrt(r100330);
        double r100332 = r100331 * r100331;
        double r100333 = 1.7143508691560957e+62;
        bool r100334 = r100310 <= r100333;
        double r100335 = cbrt(r100319);
        double r100336 = pow(r100335, r100320);
        double r100337 = r100310 * r100320;
        double r100338 = r100315 * r100337;
        double r100339 = r100336 * r100338;
        double r100340 = r100335 * r100335;
        double r100341 = pow(r100340, r100320);
        double r100342 = r100339 * r100341;
        double r100343 = r100342 * r100316;
        double r100344 = r100325 * r100310;
        double r100345 = r100315 * r100320;
        double r100346 = r100344 * r100345;
        double r100347 = r100343 + r100346;
        double r100348 = sqrt(r100347);
        double r100349 = sqrt(r100328);
        double r100350 = sqrt(r100326);
        double r100351 = r100349 * r100350;
        double r100352 = r100334 ? r100348 : r100351;
        double r100353 = r100312 ? r100332 : r100352;
        return r100353;
}

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*

Derivation

  1. Split input into 3 regimes

  2. if U < -7.011733841081501e+54

    1. Initial program 29.1

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

    2. Simplified26.1

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

    4. Applied add-sqr-sqrt26.3

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

    5. Simplified27.2

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

    6. Simplified27.1

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

    if -7.011733841081501e+54 < U < 1.7143508691560957e+62

    1. Initial program 36.1

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

    2. Simplified33.2

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

    4. Applied fma-udef33.2

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

    5. Applied distribute-lft-in33.2

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

    6. Simplified34.9

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

    7. Simplified32.1

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

    9. Applied add-cube-cbrt32.2

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

    10. Applied unpow-prod-down32.1

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

    11. Applied associate-*l*31.0

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

    12. Simplified31.0

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

    if 1.7143508691560957e+62 < U

    1. Initial program 31.3

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

    2. Simplified28.0

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

    4. Applied sqrt-prod42.1

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

    5. Simplified42.1

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

    6. Simplified42.6

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

  4. Final simplification32.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\ \mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\ \;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(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*))))))