Average Error: 33.6 → 28.8
Time: 2.0m
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 r16726675 = 2.0;
        double r16726676 = n;
        double r16726677 = r16726675 * r16726676;
        double r16726678 = U;
        double r16726679 = r16726677 * r16726678;
        double r16726680 = t;
        double r16726681 = l;
        double r16726682 = r16726681 * r16726681;
        double r16726683 = Om;
        double r16726684 = r16726682 / r16726683;
        double r16726685 = r16726675 * r16726684;
        double r16726686 = r16726680 - r16726685;
        double r16726687 = r16726681 / r16726683;
        double r16726688 = pow(r16726687, r16726675);
        double r16726689 = r16726676 * r16726688;
        double r16726690 = U_;
        double r16726691 = r16726678 - r16726690;
        double r16726692 = r16726689 * r16726691;
        double r16726693 = r16726686 - r16726692;
        double r16726694 = r16726679 * r16726693;
        double r16726695 = sqrt(r16726694);
        return r16726695;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r16726696 = n;
        double r16726697 = -3.718468025429329e+78;
        bool r16726698 = r16726696 <= r16726697;
        double r16726699 = 2.0;
        double r16726700 = U;
        double r16726701 = r16726700 * r16726696;
        double r16726702 = l;
        double r16726703 = Om;
        double r16726704 = r16726702 / r16726703;
        double r16726705 = r16726702 * r16726704;
        double r16726706 = -2.0;
        double r16726707 = t;
        double r16726708 = fma(r16726705, r16726706, r16726707);
        double r16726709 = r16726704 * r16726704;
        double r16726710 = U_;
        double r16726711 = r16726700 - r16726710;
        double r16726712 = r16726709 * r16726711;
        double r16726713 = r16726696 * r16726712;
        double r16726714 = r16726708 - r16726713;
        double r16726715 = r16726701 * r16726714;
        double r16726716 = r16726699 * r16726715;
        double r16726717 = sqrt(r16726716);
        double r16726718 = r16726708 * r16726696;
        double r16726719 = r16726696 * r16726704;
        double r16726720 = r16726719 * r16726719;
        double r16726721 = r16726720 * r16726711;
        double r16726722 = r16726718 - r16726721;
        double r16726723 = r16726722 * r16726699;
        double r16726724 = r16726723 * r16726700;
        double r16726725 = 0.5;
        double r16726726 = pow(r16726724, r16726725);
        double r16726727 = r16726698 ? r16726717 : r16726726;
        return r16726727;
}


\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*))))))