Average Error: 33.6 → 28.8
Time: 3.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.718468025429329 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* \cdot n - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot 2\right) \cdot U\right)}^{\frac{1}{2}}\\ \end{array}\]
double f(double n, double U, double t, double l, double Om, double U_) {
        double r18232585 = 2.0;
        double r18232586 = n;
        double r18232587 = r18232585 * r18232586;
        double r18232588 = U;
        double r18232589 = r18232587 * r18232588;
        double r18232590 = t;
        double r18232591 = l;
        double r18232592 = r18232591 * r18232591;
        double r18232593 = Om;
        double r18232594 = r18232592 / r18232593;
        double r18232595 = r18232585 * r18232594;
        double r18232596 = r18232590 - r18232595;
        double r18232597 = r18232591 / r18232593;
        double r18232598 = pow(r18232597, r18232585);
        double r18232599 = r18232586 * r18232598;
        double r18232600 = U_;
        double r18232601 = r18232588 - r18232600;
        double r18232602 = r18232599 * r18232601;
        double r18232603 = r18232596 - r18232602;
        double r18232604 = r18232589 * r18232603;
        double r18232605 = sqrt(r18232604);
        return r18232605;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r18232606 = n;
        double r18232607 = -3.718468025429329e+78;
        bool r18232608 = r18232606 <= r18232607;
        double r18232609 = 2.0;
        double r18232610 = U;
        double r18232611 = r18232610 * r18232606;
        double r18232612 = l;
        double r18232613 = Om;
        double r18232614 = r18232612 / r18232613;
        double r18232615 = r18232612 * r18232614;
        double r18232616 = -2.0;
        double r18232617 = t;
        double r18232618 = fma(r18232615, r18232616, r18232617);
        double r18232619 = r18232614 * r18232614;
        double r18232620 = U_;
        double r18232621 = r18232610 - r18232620;
        double r18232622 = r18232619 * r18232621;
        double r18232623 = r18232606 * r18232622;
        double r18232624 = r18232618 - r18232623;
        double r18232625 = r18232611 * r18232624;
        double r18232626 = r18232609 * r18232625;
        double r18232627 = sqrt(r18232626);
        double r18232628 = r18232618 * r18232606;
        double r18232629 = r18232606 * r18232614;
        double r18232630 = r18232629 * r18232629;
        double r18232631 = r18232630 * r18232621;
        double r18232632 = r18232628 - r18232631;
        double r18232633 = r18232632 * r18232609;
        double r18232634 = r18232633 * r18232610;
        double r18232635 = 0.5;
        double r18232636 = pow(r18232634, r18232635);
        double r18232637 = r18232608 ? r18232627 : r18232636;
        return r18232637;
}


\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 -3.718468025429329 \cdot 10^{+78}:\\
\;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;{\left(\left(\left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* \cdot n - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot 2\right) \cdot U\right)}^{\frac{1}{2}}\\

\end{array}

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 2 regimes

  2. if n < -3.718468025429329e+78

    1. Initial program 34.5

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

    2. Simplified39.6

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

    4. Applied *-un-lft-identity39.6

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

    5. Applied times-frac39.2

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

    6. Simplified39.2

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

    8. Applied associate-*r*31.7

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

    if -3.718468025429329e+78 < n

    1. Initial program 33.5

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

    2. Simplified33.9

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

    4. Applied *-un-lft-identity33.9

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

    5. Applied times-frac31.0

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

    6. Simplified31.0

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

    8. Applied associate-*r*29.6

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

    10. Applied sub-neg29.6

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

    11. Applied distribute-rgt-in29.6

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

    12. Applied distribute-rgt-in29.6

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

    13. Simplified28.3

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

    15. Applied pow128.3

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

    16. Applied sqrt-pow128.3

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

    17. Simplified28.3

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

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.718468025429329 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_* \cdot n - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot 2\right) \cdot U\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(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*))))))