Toniolo and Linder, Equation (13)

Average Error: 34.1 → 30.1

Time: 1.0min

Precision: binary64

Cost: 15652

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ t_3 := \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot t_1\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.06 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5.6 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

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 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (+ (* (/ l (/ Om l)) -2.0) (* n (* t_1 (- U* U)))))))))
        (t_3
         (sqrt
          (*
           (* U (* n -2.0))
           (+ (* (* n t_1) (- U U*)) (- (* 2.0 (/ (* l l) Om)) t))))))
   (if (<= l -1.02e+247)
     (*
      (sqrt 2.0)
      (*
       (sqrt (* (* n U) (+ (/ -2.0 Om) (* (/ n Om) (- (/ U* Om) (/ U Om))))))
       (- l)))
     (if (<= l -1.06e+18)
       (sqrt (fma 2.0 (* n (* U t)) (* (/ (* n l) (/ Om l)) (* U -4.0))))
       (if (<= l -2.05e-104)
         t_3
         (if (<= l -1.2e-175)
           t_2
           (if (<= l -5.6e-278)
             (* (sqrt 2.0) (sqrt (* U (* n t))))
             (if (<= l 1.2e-296)
               t_2
               (if (<= l 1.05e-235)
                 (* (sqrt (* 2.0 n)) (sqrt (* U t)))
                 (if (<= l 6.4e-226)
                   (* (sqrt (* 2.0 (* n U))) (sqrt t))
                   (if (<= l 3.6e-162)
                     t_3
                     (if (<= l 1.15e+63)
                       (sqrt
                        (*
                         (* 2.0 n)
                         (*
                          U
                          (+
                           t
                           (+
                            (* (/ (* l (* n l)) Om) (/ U* Om))
                            (/ (* l -2.0) (/ Om l)))))))
                       (*
                        (sqrt 2.0)
                        (*
                         l
                         (sqrt
                          (*
                           (* n U)
                           (+
                            (/ -2.0 Om)
                            (* (/ U* Om) (/ n 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 t_1 = pow((l / Om), 2.0);
	double t_2 = sqrt(((2.0 * n) * (U * (t + (((l / (Om / l)) * -2.0) + (n * (t_1 * (U_42_ - U))))))));
	double t_3 = sqrt(((U * (n * -2.0)) * (((n * t_1) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t))));
	double tmp;
	if (l <= -1.02e+247) {
		tmp = sqrt(2.0) * (sqrt(((n * U) * ((-2.0 / Om) + ((n / Om) * ((U_42_ / Om) - (U / Om)))))) * -l);
	} else if (l <= -1.06e+18) {
		tmp = sqrt(fma(2.0, (n * (U * t)), (((n * l) / (Om / l)) * (U * -4.0))));
	} else if (l <= -2.05e-104) {
		tmp = t_3;
	} else if (l <= -1.2e-175) {
		tmp = t_2;
	} else if (l <= -5.6e-278) {
		tmp = sqrt(2.0) * sqrt((U * (n * t)));
	} else if (l <= 1.2e-296) {
		tmp = t_2;
	} else if (l <= 1.05e-235) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (l <= 6.4e-226) {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	} else if (l <= 3.6e-162) {
		tmp = t_3;
	} else if (l <= 1.15e+63) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((((l * (n * l)) / Om) * (U_42_ / Om)) + ((l * -2.0) / (Om / l)))))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((n * U) * ((-2.0 / Om) + ((U_42_ / Om) * (n / 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_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l / Float64(Om / l)) * -2.0) + Float64(n * Float64(t_1 * Float64(U_42_ - U))))))))
	t_3 = sqrt(Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * t_1) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t))))
	tmp = 0.0
	if (l <= -1.02e+247)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(Float64(U_42_ / Om) - Float64(U / Om)))))) * Float64(-l)));
	elseif (l <= -1.06e+18)
		tmp = sqrt(fma(2.0, Float64(n * Float64(U * t)), Float64(Float64(Float64(n * l) / Float64(Om / l)) * Float64(U * -4.0))));
	elseif (l <= -2.05e-104)
		tmp = t_3;
	elseif (l <= -1.2e-175)
		tmp = t_2;
	elseif (l <= -5.6e-278)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * t))));
	elseif (l <= 1.2e-296)
		tmp = t_2;
	elseif (l <= 1.05e-235)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (l <= 6.4e-226)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	elseif (l <= 3.6e-162)
		tmp = t_3;
	elseif (l <= 1.15e+63)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(Float64(l * Float64(n * l)) / Om) * Float64(U_42_ / Om)) + Float64(Float64(l * -2.0) / Float64(Om / l)))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n * U) * Float64(Float64(-2.0 / Om) + Float64(Float64(U_42_ / Om) * Float64(n / 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$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.02e+247], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ / Om), $MachinePrecision] - N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.06e+18], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n * l), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -2.05e-104], t$95$3, If[LessEqual[l, -1.2e-175], t$95$2, If[LessEqual[l, -5.6e-278], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e-296], t$95$2, If[LessEqual[l, 1.05e-235], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.4e-226], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e-162], t$95$3, If[LessEqual[l, 1.15e+63], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(U$42$ / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if l < -1.02e247

   1. Initial program 64.0

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

   2. Simplified 57.2

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

      [Start] 64.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 64.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around inf 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \sqrt{2 \cdot \left(\left({\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{2 \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]
      associate-*r/ [=>] 64.0	\[ \sqrt{2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right) \cdot \left(n \cdot U\right)\right)} \]
      metadata-eval [=>] 64.0	\[ \sqrt{2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]

   5. Taylor expanded in l around -inf 32.1

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

   6. Simplified 33.2

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}\right)} \]
      Proof

      [Start] 32.1	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right) \]
      mul-1-neg [=>] 32.1	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-*l* [=>] 31.9	\[ -\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      distribute-rgt-neg-in [=>] 31.9	\[ \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      associate-*r* [=>] 30.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]

   7. Taylor expanded in U* around 0 31.9

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

   8. Simplified 24.7

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

      [Start] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right) - 2 \cdot \frac{1}{Om}\right)\right)}\right) \]
      cancel-sign-sub-inv [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \color{blue}{\left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right) + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)}\right) \]
      +-commutative [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \color{blue}{\left(\left(-2\right) \cdot \frac{1}{Om} + \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right)\right)}\right)}\right) \]
      metadata-eval [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\color{blue}{-2} \cdot \frac{1}{Om} + \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\right) \]
      associate-*r/ [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\color{blue}{\frac{-2 \cdot 1}{Om}} + \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\right) \]
      metadata-eval [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{\color{blue}{-2}}{Om} + \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} + \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\right) \]
      +-commutative [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} + \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + -1 \cdot \frac{n \cdot U}{{Om}^{2}}\right)}\right)\right)}\right) \]
      mul-1-neg [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} + \left(\frac{n \cdot U*}{{Om}^{2}} + \color{blue}{\left(-\frac{n \cdot U}{{Om}^{2}}\right)}\right)\right)\right)}\right) \]
      unsub-neg [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} + \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} - \frac{n \cdot U}{{Om}^{2}}\right)}\right)\right)}\right) \]
      associate-+r- [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \color{blue}{\left(\left(\frac{-2}{Om} + \frac{n \cdot U*}{{Om}^{2}}\right) - \frac{n \cdot U}{{Om}^{2}}\right)}\right)}\right) \]
      unpow2 [=>] 31.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\left(\frac{-2}{Om} + \frac{n \cdot U*}{\color{blue}{Om \cdot Om}}\right) - \frac{n \cdot U}{{Om}^{2}}\right)\right)}\right) \]
      times-frac [=>] 29.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\left(\frac{-2}{Om} + \color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}}\right) - \frac{n \cdot U}{{Om}^{2}}\right)\right)}\right) \]
      unpow2 [=>] 29.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right) - \frac{n \cdot U}{\color{blue}{Om \cdot Om}}\right)\right)}\right) \]
      times-frac [=>] 24.7	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right) - \color{blue}{\frac{n}{Om} \cdot \frac{U}{Om}}\right)\right)}\right) \]

   9. Applied egg-rr 57.5

      \[\leadsto \sqrt{2} \cdot \left(-\ell \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(n \cdot U\right)}\right)} - 1\right)}\right) \]

   10. Simplified 24.6

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

       [Start] 57.5	\[ \sqrt{2} \cdot \left(-\ell \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(n \cdot U\right)}\right)} - 1\right)\right) \]
       expm1-def [=>] 24.7	\[ \sqrt{2} \cdot \left(-\ell \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(n \cdot U\right)}\right)\right)}\right) \]
       expm1-log1p [=>] 24.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \color{blue}{\sqrt{\left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(n \cdot U\right)}}\right) \]
       *-commutative [=>] 24.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)}}\right) \]

   if -1.02e247 < l < -1.06e18

   1. Initial program 45.5

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

   2. Taylor expanded in Om around inf 46.7

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

   3. Simplified 45.6

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot \ell}}{U}}\right)}} \]
      Proof

      [Start] 46.7	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 46.7	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 46.7	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 46.7	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r* [=>] 45.6	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{{\ell}^{2}}}{U}}}\right)} \]
      unpow2 [=>] 45.6	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot \ell}}}{U}}\right)} \]

   4. Applied egg-rr 50.2

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)} - 1} \]

   5. Simplified 33.4

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}} \]
      Proof

      [Start] 50.2	\[ e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)} - 1 \]
      expm1-def [=>] 38.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 37.9	\[ \color{blue}{\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}} \]
      associate-*r/ [=>] 33.4	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \color{blue}{\frac{n \cdot \ell}{\frac{Om}{\ell}}} \cdot \left(U \cdot -4\right)\right)} \]

   if -1.06e18 < l < -2.04999999999999992e-104 or 6.39999999999999965e-226 < l < 3.5999999999999998e-162

   1. Initial program 27.7

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

   if -2.04999999999999992e-104 < l < -1.2e-175 or -5.60000000000000015e-278 < l < 1.19999999999999998e-296

   1. Initial program 24.3

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

   2. Simplified 24.4

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

      [Start] 24.3	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 23.7	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 23.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 24.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   if -1.2e-175 < l < -5.60000000000000015e-278

   1. Initial program 23.6

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

   2. Simplified 23.4

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

      [Start] 23.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 23.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 23.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 23.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 42.3

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

   4. Simplified 42.3

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

      [Start] 42.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 42.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      *-commutative [=>] 42.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)\right)} \]

   5. Taylor expanded in n around 0 41.3

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

   6. Simplified 41.3

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

      [Start] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]
      *-commutative [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
      unpow2 [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]
      associate-/l* [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right)} \]
      *-commutative [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\frac{\ell}{\frac{Om}{\ell}} \cdot 2}\right)} \]
      associate-/r/ [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot 2\right)} \]
      associate-*l* [=>] 41.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)}\right)} \]

   7. Taylor expanded in l around 0 26.5

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

   8. Simplified 28.2

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(t \cdot n\right)}} \]
      Proof

      [Start] 26.5	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)} \]
      associate-*r* [=>] 28.2	\[ \sqrt{2} \cdot \sqrt{\color{blue}{\left(n \cdot t\right) \cdot U}} \]
      *-commutative [=>] 28.2	\[ \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]
      *-commutative [=>] 28.2	\[ \sqrt{2} \cdot \sqrt{U \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 1.19999999999999998e-296 < l < 1.05e-235

   1. Initial program 23.2

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

   2. Simplified 24.1

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

      [Start] 23.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 24.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 24.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 41.8

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

   4. Simplified 41.8

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

      [Start] 41.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      associate-/l* [=>] 41.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)} \]
      *-commutative [=>] 41.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)\right)} \]

   5. Taylor expanded in l around 0 39.2

      \[\leadsto \sqrt{2 \cdot n} \cdot \color{blue}{\sqrt{t \cdot U}} \]

   if 1.05e-235 < l < 6.39999999999999965e-226

   1. Initial program 21.4

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

   2. Simplified 23.1

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

      [Start] 21.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 21.4	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 21.6	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 21.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 28.2

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

   4. Applied egg-rr 45.1

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

   if 3.5999999999999998e-162 < l < 1.14999999999999997e63

   1. Initial program 29.5

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

   2. Simplified 29.9

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

      [Start] 29.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 28.7	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 29.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 31.5

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

   4. Simplified 28.7

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

      [Start] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot \ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 31.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 31.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 28.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 28.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]

   if 1.14999999999999997e63 < l

   1. Initial program 50.8

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

   2. Simplified 42.0

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

      [Start] 50.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 50.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 50.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 41.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 42.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 53.9

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

   4. Simplified 47.9

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

      [Start] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [=>] 53.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot \ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 53.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 53.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 51.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]
      associate-*r* [=>] 47.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2 \cdot \ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)} \]

   5. Taylor expanded in t around 0 57.3

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

   6. Simplified 54.0

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

      [Start] 57.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]
      *-commutative [=>] 57.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      cancel-sign-sub-inv [=>] 57.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]
      unpow2 [=>] 57.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      unpow2 [=>] 57.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      times-frac [=>] 56.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}} + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      associate-*l* [=>] 56.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot U*\right)}}{Om} + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      metadata-eval [=>] 56.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      unpow2 [=>] 56.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]
      associate-/l* [=>] 54.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right)\right)} \]

   7. Taylor expanded in l around 0 37.6

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

   8. Simplified 33.3

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

      [Start] 37.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      associate-*l* [=>] 37.5	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right) \cdot n}}\right) \]
      associate-*l* [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left(U \cdot n\right)}}\right) \]
      cancel-sign-sub-inv [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(U \cdot n\right)}\right) \]
      unpow2 [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(U \cdot n\right)}\right) \]
      times-frac [=>] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(U \cdot n\right)}\right) \]
      metadata-eval [=>] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left(U \cdot n\right)}\right) \]
      associate-*r/ [=>] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left(U \cdot n\right)}\right) \]
      metadata-eval [=>] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right) \cdot \left(U \cdot n\right)}\right) \]
      *-commutative [<=] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \color{blue}{\left(n \cdot U\right)}}\right) \]

3. Recombined 9 regimes into one program.

4. Final simplification 30.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.06 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.6 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	25.8
Cost	51532

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot t_1\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+303}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 2
Error	29.4
Cost	27668

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t_1, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -6000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 3
Error	29.2
Cost	21196

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -140:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot \frac{Om}{\ell}}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + n \cdot \left(t_1 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot t_1\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 4
Error	30.6
Cost	14728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 5
Error	30.1
Cost	14676

\[\begin{array}{l} t_1 := \sqrt{2 \cdot n}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;n \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-299}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, t_2, -4 \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 6.1 \cdot 10^{-220}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\ell \cdot \left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \frac{n}{\frac{\frac{-Om}{\ell}}{U \cdot \left(-\ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \end{array} \]

Alternative 6
Error	32.8
Cost	14676

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 7
Error	32.8
Cost	14676

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 8
Error	30.4
Cost	14544

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\left(t + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{Om} \cdot \frac{n}{Om}\right)}\right)\\ \end{array} \]

Alternative 9
Error	33.7
Cost	13644

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 10
Error	32.7
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot -2 + t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 11
Error	32.8
Cost	8524

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 12
Error	33.7
Cost	7756

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\left(t + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 13
Error	36.0
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+177} \lor \neg \left(\ell \leq 1.8 \cdot 10^{+102}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 14
Error	33.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.65 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 16
Error	38.6
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	38.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -5000000:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 18
Error	39.5
Cost	6848

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

Reproduce

herbie shell --seed 2023060 
(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*))))))