Average Error: 34.4 → 32.0
Time: 43.3s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\ \mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\ \;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right)}\\ \end{array}\]
\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 r100333 = 2.0;
        double r100334 = n;
        double r100335 = r100333 * r100334;
        double r100336 = U;
        double r100337 = r100335 * r100336;
        double r100338 = t;
        double r100339 = l;
        double r100340 = r100339 * r100339;
        double r100341 = Om;
        double r100342 = r100340 / r100341;
        double r100343 = r100333 * r100342;
        double r100344 = r100338 - r100343;
        double r100345 = r100339 / r100341;
        double r100346 = pow(r100345, r100333);
        double r100347 = r100334 * r100346;
        double r100348 = U_;
        double r100349 = r100336 - r100348;
        double r100350 = r100347 * r100349;
        double r100351 = r100344 - r100350;
        double r100352 = r100337 * r100351;
        double r100353 = sqrt(r100352);
        return r100353;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r100354 = U;
        double r100355 = -7.011733841081501e+54;
        bool r100356 = r100354 <= r100355;
        double r100357 = U_;
        double r100358 = r100357 - r100354;
        double r100359 = n;
        double r100360 = r100358 * r100359;
        double r100361 = l;
        double r100362 = Om;
        double r100363 = r100361 / r100362;
        double r100364 = 2.0;
        double r100365 = pow(r100363, r100364);
        double r100366 = -r100364;
        double r100367 = r100366 * r100361;
        double r100368 = t;
        double r100369 = fma(r100363, r100367, r100368);
        double r100370 = fma(r100360, r100365, r100369);
        double r100371 = r100359 * r100354;
        double r100372 = r100364 * r100371;
        double r100373 = r100370 * r100372;
        double r100374 = sqrt(r100373);
        double r100375 = sqrt(r100374);
        double r100376 = r100375 * r100375;
        double r100377 = 1.7143508691560957e+62;
        bool r100378 = r100354 <= r100377;
        double r100379 = cbrt(r100363);
        double r100380 = pow(r100379, r100364);
        double r100381 = r100354 * r100364;
        double r100382 = r100359 * r100381;
        double r100383 = r100380 * r100382;
        double r100384 = r100379 * r100379;
        double r100385 = pow(r100384, r100364);
        double r100386 = r100383 * r100385;
        double r100387 = r100386 * r100360;
        double r100388 = r100369 * r100354;
        double r100389 = r100359 * r100364;
        double r100390 = r100388 * r100389;
        double r100391 = r100387 + r100390;
        double r100392 = sqrt(r100391);
        double r100393 = sqrt(r100372);
        double r100394 = sqrt(r100370);
        double r100395 = r100393 * r100394;
        double r100396 = r100378 ? r100392 : r100395;
        double r100397 = r100356 ? r100376 : r100396;
        return r100397;
}

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*))))))