Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 61.1%

Time: 36.4s

Precision: binary64

Cost: 14796

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{\frac{Om}{n \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -2.3e+95)
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (* Om (/ 1.0 (* U (* n l)))))))
    0.5)
   (if (<= l -4.5e-187)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (if (<= l 8.2e+111)
       (sqrt
        (*
         2.0
         (*
          U
          (+
           (* n t)
           (/ (fma (* l (/ n Om)) (- U* U) (* l -2.0)) (/ Om (* n l)))))))
       (*
        (* l (sqrt 2.0))
        (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -2.3e+95) {
		tmp = pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om * (1.0 / (U * (n * l))))))), 0.5);
	} else if (l <= -4.5e-187) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 8.2e+111) {
		tmp = sqrt((2.0 * (U * ((n * t) + (fma((l * (n / Om)), (U_42_ - U), (l * -2.0)) / (Om / (n * l)))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -2.3e+95)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om * Float64(1.0 / Float64(U * Float64(n * l))))))) ^ 0.5;
	elseif (l <= -4.5e-187)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 8.2e+111)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(fma(Float64(l * Float64(n / Om)), Float64(U_42_ - U), Float64(l * -2.0)) / Float64(Om / Float64(n * l)))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -2.3e+95], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * N[(1.0 / N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, -4.5e-187], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.2e+111], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.29999999999999997e95

   1. Initial program 20.2%

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

   2. Simplified 47.0%

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

      [Start] 20.2%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 23.1%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 23.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 23.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 23.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 23.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 37.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 57.8%

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

   4. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 57.8%	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      pow1/2 [=>] 58.2%	\[ \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}^{0.5}} \]
      distribute-lft-out [=>] 58.2%	\[ {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)\right)}}^{0.5} \]
      associate-/l* [=>] 61.0%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)\right)}^{0.5} \]
      associate-/l* [=>] 61.0%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5} \]
      *-commutative [=>] 61.0%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5} \]
      *-commutative [=>] 61.0%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \color{blue}{\left(U \cdot \ell\right)}}}\right)\right)}^{0.5} \]

   5. Taylor expanded in U* around inf 61.3%

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\color{blue}{\frac{Om}{\ell \cdot U*}}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]

   6. Simplified 61.4%

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\color{blue}{\frac{\frac{Om}{\ell}}{U*}}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]
      Step-by-step derivation

      [Start] 61.3%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]
      associate-/r* [=>] 61.4%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\color{blue}{\frac{\frac{Om}{\ell}}{U*}}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]

   7. Applied egg-rr 69.9%

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{\frac{Om}{\ell}}{U*}} + \ell \cdot -2}{\color{blue}{Om \cdot \frac{1}{\left(n \cdot \ell\right) \cdot U}}}\right)\right)}^{0.5} \]
      Step-by-step derivation

      [Start] 61.4%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{\frac{Om}{\ell}}{U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]
      div-inv [=>] 61.4%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{\frac{Om}{\ell}}{U*}} + \ell \cdot -2}{\color{blue}{Om \cdot \frac{1}{n \cdot \left(U \cdot \ell\right)}}}\right)\right)}^{0.5} \]
      *-commutative [<=] 61.4%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{\frac{Om}{\ell}}{U*}} + \ell \cdot -2}{Om \cdot \frac{1}{n \cdot \color{blue}{\left(\ell \cdot U\right)}}}\right)\right)}^{0.5} \]
      associate-*r* [=>] 69.9%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{\frac{Om}{\ell}}{U*}} + \ell \cdot -2}{Om \cdot \frac{1}{\color{blue}{\left(n \cdot \ell\right) \cdot U}}}\right)\right)}^{0.5} \]

   if -2.29999999999999997e95 < l < -4.4999999999999998e-187

   1. Initial program 54.9%

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

   2. Simplified 61.1%

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

      [Start] 54.9%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.3%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 61.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 61.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 61.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 61.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 61.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 61.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 61.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 61.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 56.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 56.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 58.3%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 65.1%

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

   if -4.4999999999999998e-187 < l < 8.19999999999999973e111

   1. Initial program 58.1%

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

   2. Simplified 54.2%

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

      [Start] 58.1%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 63.4%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 63.4%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 51.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 51.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 57.1%

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

   4. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 57.1%	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}} \]
      pow1/2 [=>] 57.1%	\[ \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}^{0.5}} \]
      distribute-lft-out [=>] 57.1%	\[ {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\left(\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}\right)\right)}}^{0.5} \]
      associate-/l* [=>] 57.1%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \color{blue}{\frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}}\right)\right)}^{0.5} \]
      associate-/l* [=>] 57.9%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\color{blue}{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}}} + -2 \cdot \ell}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5} \]
      *-commutative [=>] 57.9%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \color{blue}{\ell \cdot -2}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5} \]
      *-commutative [=>] 57.9%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \color{blue}{\left(U \cdot \ell\right)}}}\right)\right)}^{0.5} \]

   5. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\left(n \cdot \ell\right) \cdot U}}\right)}} \]
      Step-by-step derivation

      [Start] 57.9%	\[ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]
      *-un-lft-identity [=>] 57.9%	\[ \color{blue}{1 \cdot {\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}} \]
      unpow1/2 [=>] 57.9%	\[ 1 \cdot \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}} \]
      fma-def [=>] 58.8%	\[ 1 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{\frac{Om}{\ell \cdot \left(U* - U\right)}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}} \]
      associate-/r/ [=>] 60.4%	\[ 1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)} \]
      *-commutative [<=] 60.4%	\[ 1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{n \cdot \color{blue}{\left(\ell \cdot U\right)}}}\right)} \]
      associate-*r* [=>] 60.4%	\[ 1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\color{blue}{\left(n \cdot \ell\right) \cdot U}}}\right)} \]

   6. Simplified 67.2%

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

      [Start] 60.4%	\[ 1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\left(n \cdot \ell\right) \cdot U}}\right)} \]
      *-lft-identity [=>] 60.4%	\[ \color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(n, t \cdot U, \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\left(n \cdot \ell\right) \cdot U}}\right)}} \]
      fma-udef [=>] 59.6%	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\left(n \cdot \ell\right) \cdot U}}\right)}} \]
      associate-*r* [=>] 63.8%	\[ \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot t\right) \cdot U} + \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{\frac{Om}{\left(n \cdot \ell\right) \cdot U}}\right)} \]
      associate-/r/ [=>] 62.1%	\[ \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot U\right)}\right)} \]
      associate-*r* [=>] 66.2%	\[ \sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U + \color{blue}{\left(\frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot U}\right)} \]
      distribute-rgt-out [=>] 67.2%	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + \frac{\frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right) + \ell \cdot -2}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}} \]

   7. Applied egg-rr 65.5%

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

      [Start] 67.2%	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\frac{n}{Om} \cdot \ell, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)} \]
      associate-*l/ [=>] 65.5%	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \color{blue}{\frac{\mathsf{fma}\left(\frac{n}{Om} \cdot \ell, U* - U, \ell \cdot -2\right) \cdot \left(n \cdot \ell\right)}{Om}}\right)\right)} \]

   8. Simplified 69.0%

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

      [Start] 65.5%	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\frac{n}{Om} \cdot \ell, U* - U, \ell \cdot -2\right) \cdot \left(n \cdot \ell\right)}{Om}\right)\right)} \]
      associate-/l* [=>] 69.0%	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \color{blue}{\frac{\mathsf{fma}\left(\frac{n}{Om} \cdot \ell, U* - U, \ell \cdot -2\right)}{\frac{Om}{n \cdot \ell}}}\right)\right)} \]
      *-commutative [=>] 69.0%	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\color{blue}{\ell \cdot \frac{n}{Om}}, U* - U, \ell \cdot -2\right)}{\frac{Om}{n \cdot \ell}}\right)\right)} \]

   if 8.19999999999999973e111 < l

   1. Initial program 16.7%

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

   2. Simplified 44.7%

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

      [Start] 16.7%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 16.8%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 16.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 16.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 16.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 16.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 25.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 25.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 25.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 25.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 25.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 25.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 25.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 50.0%

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

   4. Taylor expanded in l around inf 70.6%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{\frac{Om}{n \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	61.1%
Cost	14796

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{\frac{Om}{n \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 2
Accuracy	63.8%
Cost	45512

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ t_2 := {\left(2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{\ell \cdot -2 + \frac{n}{Om} \cdot \left(\ell \cdot \left(U* - U\right)\right)}{\frac{Om}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.25}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]

Alternative 3
Accuracy	65.3%
Cost	45256

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ t_2 := {\left(2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{\ell \cdot -2 + U* \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.25}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]

Alternative 4
Accuracy	63.7%
Cost	30728

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 5
Accuracy	60.4%
Cost	14796

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 6
Accuracy	61.6%
Cost	14412

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.05 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 7
Accuracy	61.4%
Cost	14284

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+101}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 8
Accuracy	59.0%
Cost	8652

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+98}:\\ \;\;\;\;{\left(2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot U*}}}{\frac{Om}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n}{\frac{-Om}{U \cdot \ell}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 9
Accuracy	59.6%
Cost	8652

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+99}:\\ \;\;\;\;{\left(2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{Om \cdot \frac{1}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t_1 + \frac{\ell \cdot -2 + \frac{n}{\frac{-Om}{U \cdot \ell}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 10
Accuracy	58.4%
Cost	8324

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot U*}}}{\frac{Om}{U \cdot \left(n \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 10^{-20}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 11
Accuracy	57.2%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 12
Accuracy	58.7%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.4 \cdot 10^{+74} \lor \neg \left(Om \leq 7.2 \cdot 10^{+71}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	54.1%
Cost	7944

\[\begin{array}{l} \mathbf{if}\;n \leq -7.7 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \frac{n}{\frac{Om \cdot Om}{\ell \cdot U*}}\right)\right)}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{n \cdot -2}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14
Accuracy	52.0%
Cost	7881

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.5 \cdot 10^{-189} \lor \neg \left(Om \leq 9.6 \cdot 10^{-83}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 15
Accuracy	51.0%
Cost	7817

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.55 \cdot 10^{-188} \lor \neg \left(Om \leq 1.28 \cdot 10^{-82}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-n}{\frac{Om}{U \cdot U*}}\right)}{Om}}\\ \end{array} \]

Alternative 16
Accuracy	51.1%
Cost	7817

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.8 \cdot 10^{-190} \lor \neg \left(Om \leq 6.8 \cdot 10^{-83}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\left(U \cdot U*\right) \cdot \left(-n\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 17
Accuracy	52.2%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;n \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{n \cdot -2}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18
Accuracy	50.0%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{-190} \lor \neg \left(Om \leq 1.06 \cdot 10^{-82}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)}{Om \cdot Om}}\\ \end{array} \]

Alternative 19
Accuracy	49.9%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;Om \leq -3.1 \cdot 10^{-189} \lor \neg \left(Om \leq 8.5 \cdot 10^{-83}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}}\\ \end{array} \]

Alternative 20
Accuracy	39.6%
Cost	7628

\[\begin{array}{l} t_1 := \sqrt{\frac{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 \cdot U\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -470000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{-214}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Accuracy	43.6%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;t \leq -0.00105:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	50.3%
Cost	7488

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

Alternative 23
Accuracy	36.4%
Cost	6912

\[{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \]

Alternative 24
Accuracy	36.0%
Cost	6848

\[\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023271 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))