Average Error: 34.9 → 30.7
Time: 41.2s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 5.3087767675532778 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 3.5853438188507749 \cdot 10^{-68} \lor \neg \left(t \le 1.1743060886816904 \cdot 10^{137}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]
\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}\;t \le 5.3087767675532778 \cdot 10^{-246}:\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\

\mathbf{elif}\;t \le 3.5853438188507749 \cdot 10^{-68} \lor \neg \left(t \le 1.1743060886816904 \cdot 10^{137}\right):\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r170926 = 2.0;
        double r170927 = n;
        double r170928 = r170926 * r170927;
        double r170929 = U;
        double r170930 = r170928 * r170929;
        double r170931 = t;
        double r170932 = l;
        double r170933 = r170932 * r170932;
        double r170934 = Om;
        double r170935 = r170933 / r170934;
        double r170936 = r170926 * r170935;
        double r170937 = r170931 - r170936;
        double r170938 = r170932 / r170934;
        double r170939 = pow(r170938, r170926);
        double r170940 = r170927 * r170939;
        double r170941 = U_;
        double r170942 = r170929 - r170941;
        double r170943 = r170940 * r170942;
        double r170944 = r170937 - r170943;
        double r170945 = r170930 * r170944;
        double r170946 = sqrt(r170945);
        return r170946;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r170947 = t;
        double r170948 = 5.308776767553278e-246;
        bool r170949 = r170947 <= r170948;
        double r170950 = 2.0;
        double r170951 = n;
        double r170952 = r170950 * r170951;
        double r170953 = U;
        double r170954 = r170952 * r170953;
        double r170955 = l;
        double r170956 = Om;
        double r170957 = r170956 / r170955;
        double r170958 = r170955 / r170957;
        double r170959 = r170950 * r170958;
        double r170960 = r170947 - r170959;
        double r170961 = r170951 * r170955;
        double r170962 = 1.0;
        double r170963 = 1.0;
        double r170964 = pow(r170956, r170963);
        double r170965 = r170962 / r170964;
        double r170966 = pow(r170965, r170963);
        double r170967 = r170961 * r170966;
        double r170968 = r170955 / r170956;
        double r170969 = 2.0;
        double r170970 = r170950 / r170969;
        double r170971 = pow(r170968, r170970);
        double r170972 = r170967 * r170971;
        double r170973 = U_;
        double r170974 = r170953 - r170973;
        double r170975 = r170972 * r170974;
        double r170976 = r170960 - r170975;
        double r170977 = r170954 * r170976;
        double r170978 = sqrt(r170977);
        double r170979 = 3.585343818850775e-68;
        bool r170980 = r170947 <= r170979;
        double r170981 = 1.1743060886816904e+137;
        bool r170982 = r170947 <= r170981;
        double r170983 = !r170982;
        bool r170984 = r170980 || r170983;
        double r170985 = sqrt(r170954);
        double r170986 = pow(r170968, r170950);
        double r170987 = r170951 * r170986;
        double r170988 = r170987 * r170974;
        double r170989 = r170960 - r170988;
        double r170990 = sqrt(r170989);
        double r170991 = r170985 * r170990;
        double r170992 = r170953 * r170989;
        double r170993 = r170952 * r170992;
        double r170994 = sqrt(r170993);
        double r170995 = r170984 ? r170991 : r170994;
        double r170996 = r170949 ? r170978 : r170995;
        return r170996;
}

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < 5.308776767553278e-246

    1. Initial program 35.2

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

    3. Applied associate-/l*32.2

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

    5. Applied sqr-pow32.2

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

    6. Applied associate-*r*31.2

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

    7. Taylor expanded around 0 32.4

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

    if 5.308776767553278e-246 < t < 3.585343818850775e-68 or 1.1743060886816904e+137 < t

    1. Initial program 37.0

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

    3. Applied associate-/l*34.3

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

    5. Applied sqrt-prod29.6

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

    if 3.585343818850775e-68 < t < 1.1743060886816904e+137

    1. Initial program 30.8

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

    3. Applied associate-/l*28.1

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

    5. Applied associate-*l*27.3

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

  4. Final simplification30.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 5.3087767675532778 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 3.5853438188507749 \cdot 10^{-68} \lor \neg \left(t \le 1.1743060886816904 \cdot 10^{137}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))