Average Error: 34.5 → 27.6
Time: 1.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 6.135153741442139648885952268970329687829 \cdot 10^{146}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \end{array}\]
\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}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\
\;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\

\mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 6.135153741442139648885952268970329687829 \cdot 10^{146}:\\
\;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2774005 = 2.0;
        double r2774006 = n;
        double r2774007 = r2774005 * r2774006;
        double r2774008 = U;
        double r2774009 = r2774007 * r2774008;
        double r2774010 = t;
        double r2774011 = l;
        double r2774012 = r2774011 * r2774011;
        double r2774013 = Om;
        double r2774014 = r2774012 / r2774013;
        double r2774015 = r2774005 * r2774014;
        double r2774016 = r2774010 - r2774015;
        double r2774017 = r2774011 / r2774013;
        double r2774018 = pow(r2774017, r2774005);
        double r2774019 = r2774006 * r2774018;
        double r2774020 = U_;
        double r2774021 = r2774008 - r2774020;
        double r2774022 = r2774019 * r2774021;
        double r2774023 = r2774016 - r2774022;
        double r2774024 = r2774009 * r2774023;
        double r2774025 = sqrt(r2774024);
        return r2774025;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2774026 = 2.0;
        double r2774027 = n;
        double r2774028 = r2774026 * r2774027;
        double r2774029 = U;
        double r2774030 = r2774028 * r2774029;
        double r2774031 = t;
        double r2774032 = l;
        double r2774033 = r2774032 * r2774032;
        double r2774034 = Om;
        double r2774035 = r2774033 / r2774034;
        double r2774036 = r2774035 * r2774026;
        double r2774037 = r2774031 - r2774036;
        double r2774038 = r2774032 / r2774034;
        double r2774039 = pow(r2774038, r2774026);
        double r2774040 = r2774027 * r2774039;
        double r2774041 = U_;
        double r2774042 = r2774029 - r2774041;
        double r2774043 = r2774040 * r2774042;
        double r2774044 = r2774037 - r2774043;
        double r2774045 = r2774030 * r2774044;
        double r2774046 = sqrt(r2774045);
        double r2774047 = 0.0;
        bool r2774048 = r2774046 <= r2774047;
        double r2774049 = r2774029 * r2774044;
        double r2774050 = sqrt(r2774049);
        double r2774051 = sqrt(r2774028);
        double r2774052 = r2774050 * r2774051;
        double r2774053 = 6.13515374144214e+146;
        bool r2774054 = r2774046 <= r2774053;
        double r2774055 = 2.0;
        double r2774056 = r2774026 / r2774055;
        double r2774057 = pow(r2774038, r2774056);
        double r2774058 = r2774057 * r2774027;
        double r2774059 = r2774057 * r2774058;
        double r2774060 = r2774042 * r2774059;
        double r2774061 = r2774037 - r2774060;
        double r2774062 = r2774061 * r2774030;
        double r2774063 = sqrt(r2774062);
        double r2774064 = r2774034 / r2774032;
        double r2774065 = r2774032 / r2774064;
        double r2774066 = r2774026 * r2774065;
        double r2774067 = r2774031 - r2774066;
        double r2774068 = r2774067 * r2774029;
        double r2774069 = r2774028 * r2774068;
        double r2774070 = sqrt(r2774069);
        double r2774071 = r2774054 ? r2774063 : r2774070;
        double r2774072 = r2774048 ? r2774052 : r2774071;
        return r2774072;
}

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 (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))) < 0.0

    1. Initial program 56.9

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

    3. Applied associate-*l*38.3

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

    5. Applied sqrt-prod37.6

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

    if 0.0 < (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))) < 6.13515374144214e+146

    1. Initial program 1.6

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

    3. Applied sqr-pow1.6

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

    4. Applied associate-*r*1.1

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

    if 6.13515374144214e+146 < (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))

    1. Initial program 62.9

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

    3. Applied associate-*l*61.7

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

    5. Applied associate-/l*54.9

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

    6. Taylor expanded around 0 53.3

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{0}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 6.135153741442139648885952268970329687829 \cdot 10^{146}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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*))))))