Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 64.9%

Time: 29.7s

Precision: binary64

Cost: 21068

Math

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

↓

\[\begin{array}{l} t_1 := t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t_1}\\ \end{array} \]

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
 (let* ((t_1 (+ t (* (/ l Om) (fma l -2.0 (* (* n (/ l Om)) (- U* U)))))))
   (if (<= n -1.45e-117)
     (sqrt (* 2.0 (* (* n U) t_1)))
     (if (<= n -1.5e-230)
       (sqrt
        (*
         (* n 2.0)
         (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
       (if (<= n -1e-309)
         (sqrt
          (*
           2.0
           (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))
         (* (sqrt (* n 2.0)) (sqrt (* U t_1))))))))

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 t_1 = t + ((l / Om) * fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U))));
	double tmp;
	if (n <= -1.45e-117) {
		tmp = sqrt((2.0 * ((n * U) * t_1)));
	} else if (n <= -1.5e-230) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))))))));
	} else if (n <= -1e-309) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * t_1));
	}
	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_)
	t_1 = Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (n <= -1.45e-117)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * t_1)));
	elseif (n <= -1.5e-230)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (n <= -1e-309)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t_1)));
	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$42$_] := Block[{t$95$1 = N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.45e-117], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1.5e-230], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1e-309], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.45e-117

   1. Initial program 55.2%

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

   2. Simplified 51.5%

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

      [Start] 55.2%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 53.6%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 53.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 53.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 53.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 53.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 53.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 53.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 53.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 53.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 47.6%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 47.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 51.5%

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

      [Start] 51.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      *-un-lft-identity [=>] 51.5%	\[ \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
      associate-*l* [=>] 51.5%	\[ 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

   4. Simplified 59.8%

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

      [Start] 51.5%	\[ 1 \cdot \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)} \]
      *-lft-identity [=>] 51.5%	\[ \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]
      associate-*r* [=>] 55.9%	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
      *-commutative [=>] 55.9%	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)\right)} \]
      associate-*r* [=>] 59.8%	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)\right)} \]

   if -1.45e-117 < n < -1.5e-230

   1. Initial program 39.0%

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

   2. Simplified 69.7%

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

      [Start] 39.0%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 60.9%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 60.9%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 69.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 69.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 69.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 69.8%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 69.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 69.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 69.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   if -1.5e-230 < n < -1.000000000000002e-309

   1. Initial program 27.4%

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

   2. Simplified 40.2%

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

      [Start] 27.4%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 21.7%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 21.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 21.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 21.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 21.7%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 40.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 40.2%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U* around 0 21.2%

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

   4. Simplified 64.4%

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

      [Start] 21.2%	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 27.3%	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right)\right) \cdot U\right)}} \]
      +-commutative [<=] 27.3%	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right)}\right) \cdot U\right)} \]

   if -1.000000000000002e-309 < n

   1. Initial program 55.4%

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

   2. Simplified 58.5%

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

      [Start] 55.4%	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 53.0%	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      sub-neg [=>] 53.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]
      associate--l+ [=>] 53.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      *-commutative [=>] 53.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 53.0%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l/ [<=] 56.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 56.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [<=] 56.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 56.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 52.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]
      unpow2 [=>] 52.1%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]
      associate-*l* [=>] 54.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 66.9%

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

      [Start] 58.5%	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)} \]
      sqrt-prod [=>] 66.9%	\[ \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}} \]
      *-commutative [=>] 66.9%	\[ \sqrt{\color{blue}{n \cdot 2}} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)} \]

   4. Simplified 72.3%

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

      [Start] 66.9%	\[ \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)} \]
      *-commutative [=>] 66.9%	\[ \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)} \]
      *-commutative [=>] 66.9%	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)} \]
      associate-*r* [=>] 72.3%	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	64.9%
Cost	21068

\[\begin{array}{l} t_1 := t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t_1}\\ \end{array} \]

Alternative 2
Accuracy	65.2%
Cost	43336

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

Alternative 3
Accuracy	61.1%
Cost	14537

\[\begin{array}{l} \mathbf{if}\;U \leq -3.3 \cdot 10^{-141} \lor \neg \left(U \leq 1.05 \cdot 10^{-150}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 4
Accuracy	61.1%
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 - \left(U - U*\right) \cdot \frac{n}{Om}\right)}}}\\ \end{array} \]

Alternative 5
Accuracy	60.7%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 6
Accuracy	46.3%
Cost	8536

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	56.6%
Cost	8393

\[\begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-204} \lor \neg \left(n \leq 5.8 \cdot 10^{-195}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	56.3%
Cost	8388

\[\begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	46.7%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\frac{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	49.5%
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{\frac{Om}{U* - U}} - 2\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-136}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	48.2%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-37} \lor \neg \left(n \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	49.7%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;n \leq -1.78 \cdot 10^{-37} \lor \neg \left(n \leq 2.85 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	56.1%
Cost	8137

\[\begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-230} \lor \neg \left(n \leq 2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	43.2%
Cost	8077

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+161}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-12} \lor \neg \left(t \leq -1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot \left(-\ell\right)}{\frac{Om}{U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 15
Accuracy	43.0%
Cost	7885

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-12} \lor \neg \left(t \leq -5.2 \cdot 10^{-47}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{n}{\frac{Om}{U} \cdot \frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \]

Alternative 16
Accuracy	44.2%
Cost	7561

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 2.3 \cdot 10^{+45}\right):\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17
Accuracy	39.2%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5800 \lor \neg \left(\ell \leq 7.2 \cdot 10^{+50}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18
Accuracy	41.5%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 1.75 \cdot 10^{+49}\right):\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 19
Accuracy	44.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	37.2%
Cost	6912

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

Alternative 21
Accuracy	35.7%
Cost	6848

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

Reproduce

herbie shell --seed 2023178 
(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*))))))