Average Error: 34.6 → 28.4
Time: 57.4s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.776184263616495583834663238899089510483 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \le 7.786798218553425520162161300387875418626 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(2 \cdot t\right) \cdot \left(U \cdot n\right) - \frac{U \cdot 4}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(U* - U, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]
\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}\;n \le -2.776184263616495583834663238899089510483 \cdot 10^{-228}:\\
\;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\

\mathbf{elif}\;n \le 7.786798218553425520162161300387875418626 \cdot 10^{-276}:\\
\;\;\;\;\sqrt{\left(2 \cdot t\right) \cdot \left(U \cdot n\right) - \frac{U \cdot 4}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2187627 = 2.0;
        double r2187628 = n;
        double r2187629 = r2187627 * r2187628;
        double r2187630 = U;
        double r2187631 = r2187629 * r2187630;
        double r2187632 = t;
        double r2187633 = l;
        double r2187634 = r2187633 * r2187633;
        double r2187635 = Om;
        double r2187636 = r2187634 / r2187635;
        double r2187637 = r2187627 * r2187636;
        double r2187638 = r2187632 - r2187637;
        double r2187639 = r2187633 / r2187635;
        double r2187640 = pow(r2187639, r2187627);
        double r2187641 = r2187628 * r2187640;
        double r2187642 = U_;
        double r2187643 = r2187630 - r2187642;
        double r2187644 = r2187641 * r2187643;
        double r2187645 = r2187638 - r2187644;
        double r2187646 = r2187631 * r2187645;
        double r2187647 = sqrt(r2187646);
        return r2187647;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2187648 = n;
        double r2187649 = -2.7761842636164956e-228;
        bool r2187650 = r2187648 <= r2187649;
        double r2187651 = U;
        double r2187652 = U_;
        double r2187653 = r2187652 - r2187651;
        double r2187654 = l;
        double r2187655 = Om;
        double r2187656 = r2187654 / r2187655;
        double r2187657 = 2.0;
        double r2187658 = 2.0;
        double r2187659 = r2187657 / r2187658;
        double r2187660 = pow(r2187656, r2187659);
        double r2187661 = r2187660 * r2187648;
        double r2187662 = r2187661 * r2187660;
        double r2187663 = t;
        double r2187664 = r2187654 * r2187656;
        double r2187665 = r2187657 * r2187664;
        double r2187666 = r2187663 - r2187665;
        double r2187667 = fma(r2187653, r2187662, r2187666);
        double r2187668 = r2187651 * r2187667;
        double r2187669 = r2187657 * r2187648;
        double r2187670 = r2187668 * r2187669;
        double r2187671 = sqrt(r2187670);
        double r2187672 = 7.786798218553426e-276;
        bool r2187673 = r2187648 <= r2187672;
        double r2187674 = r2187657 * r2187663;
        double r2187675 = r2187651 * r2187648;
        double r2187676 = r2187674 * r2187675;
        double r2187677 = 4.0;
        double r2187678 = r2187651 * r2187677;
        double r2187679 = r2187654 * r2187654;
        double r2187680 = r2187679 * r2187648;
        double r2187681 = r2187655 / r2187680;
        double r2187682 = r2187678 / r2187681;
        double r2187683 = r2187676 - r2187682;
        double r2187684 = sqrt(r2187683);
        double r2187685 = sqrt(r2187668);
        double r2187686 = sqrt(r2187669);
        double r2187687 = r2187685 * r2187686;
        double r2187688 = r2187673 ? r2187684 : r2187687;
        double r2187689 = r2187650 ? r2187671 : r2187688;
        return r2187689;
}

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 n < -2.7761842636164956e-228

    1. Initial program 33.0

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

    2. Simplified30.0

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

    4. Applied sqr-pow30.0

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

    5. Applied associate-*r*29.2

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

    7. Applied associate-*l*30.1

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

    if -2.7761842636164956e-228 < n < 7.786798218553426e-276

    1. Initial program 41.2

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

    2. Simplified38.9

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

    4. Applied sqr-pow38.9

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

    5. Applied associate-*r*38.3

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

    6. Taylor expanded around inf 39.4

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(t \cdot \left(U \cdot n\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

    7. Simplified39.6

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t \cdot 2\right) - \frac{4 \cdot U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}}}\]

    if 7.786798218553426e-276 < n

    1. Initial program 34.3

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

    2. Simplified31.4

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

    4. Applied sqr-pow31.4

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

    5. Applied associate-*r*30.6

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

    7. Applied associate-*l*30.5

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

    9. Applied sqrt-prod23.7

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

  4. Final simplification28.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.776184263616495583834663238899089510483 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \le 7.786798218553425520162161300387875418626 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(2 \cdot t\right) \cdot \left(U \cdot n\right) - \frac{U \cdot 4}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(U* - U, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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*))))))