Toniolo and Linder, Equation (13)

Average Error: 34.5 → 29.3

Time: 41.5s

Precision: binary64

Cost: 21132

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+145}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

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 (* (* n U) (+ (* (/ n Om) (/ U* Om)) (/ -2.0 Om))))))
   (if (<= l -1.8e+145)
     (* t_2 (* l (- (sqrt 2.0))))
     (if (<= l -1.02e-249)
       (sqrt
        (*
         -2.0
         (*
          n
          (*
           U
           (-
            (- (* 2.0 (/ l (/ Om l))) (* (/ U* Om) (/ (* l (* l n)) Om)))
            t)))))
       (if (<= l 5.6e-236)
         (sqrt
          (*
           U
           (*
            (- t (fma 2.0 (* l (/ l Om)) (* n (* t_1 (- U U*)))))
            (* n 2.0))))
         (if (<= l 9.5e+147)
           (sqrt
            (*
             (* U (* n 2.0))
             (+ (+ t (* -2.0 (/ (* l l) Om))) (* (* n t_1) (- U* U)))))
           (* (sqrt 2.0) (* l t_2))))))))

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(((n * U) * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om))));
	double tmp;
	if (l <= -1.8e+145) {
		tmp = t_2 * (l * -sqrt(2.0));
	} else if (l <= -1.02e-249) {
		tmp = sqrt((-2.0 * (n * (U * (((2.0 * (l / (Om / l))) - ((U_42_ / Om) * ((l * (l * n)) / Om))) - t)))));
	} else if (l <= 5.6e-236) {
		tmp = sqrt((U * ((t - fma(2.0, (l * (l / Om)), (n * (t_1 * (U - U_42_))))) * (n * 2.0))));
	} else if (l <= 9.5e+147) {
		tmp = sqrt(((U * (n * 2.0)) * ((t + (-2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	} else {
		tmp = sqrt(2.0) * (l * t_2);
	}
	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(n * U) * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / Om))))
	tmp = 0.0
	if (l <= -1.8e+145)
		tmp = Float64(t_2 * Float64(l * Float64(-sqrt(2.0))));
	elseif (l <= -1.02e-249)
		tmp = sqrt(Float64(-2.0 * Float64(n * Float64(U * Float64(Float64(Float64(2.0 * Float64(l / Float64(Om / l))) - Float64(Float64(U_42_ / Om) * Float64(Float64(l * Float64(l * n)) / Om))) - t)))));
	elseif (l <= 5.6e-236)
		tmp = sqrt(Float64(U * Float64(Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_))))) * Float64(n * 2.0))));
	elseif (l <= 9.5e+147)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * t_2));
	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[(n * U), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.8e+145], N[(t$95$2 * N[(l * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.02e-249], N[Sqrt[N[(-2.0 * N[(n * N[(U * N[(N[(N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.6e-236], N[Sqrt[N[(U * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.5e+147], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.79999999999999987e145

   1. Initial program 62.6

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

   2. Simplified 50.0

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

      [Start] 62.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.9	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 62.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 62.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 62.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 62.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 62.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 49.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 50.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 63.5

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

   4. Simplified 56.6

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

      [Start] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 63.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-*r* [=>] 63.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 63.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 63.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 63.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 56.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Taylor expanded in l around -inf 35.1

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

   6. Simplified 29.4

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

      [Start] 35.1	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      associate-*r* [=>] 35.1	\[ \color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [=>] 35.1	\[ \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right)} \]
      *-commutative [=>] 35.1	\[ \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      associate-*r* [=>] 34.5	\[ \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      cancel-sign-sub-inv [=>] 34.5	\[ \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      associate-/l* [=>] 35.7	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U*}}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      unpow2 [=>] 35.7	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U*}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      associate-/l* [=>] 32.5	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\color{blue}{\frac{Om}{\frac{U*}{Om}}}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      associate-/r/ [=>] 29.4	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      metadata-eval [=>] 29.4	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      associate-*r/ [=>] 29.4	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      metadata-eval [=>] 29.4	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \]
      *-commutative [=>] 29.4	\[ \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)} \cdot \left(-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{2}\right)}\right) \]

   if -1.79999999999999987e145 < l < -1.02e-249

   1. Initial program 29.1

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

   2. Simplified 30.1

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

      [Start] 29.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.2	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 29.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 30.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 34.4

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

   4. Simplified 30.7

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

      [Start] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 34.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-*r* [=>] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 33.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 31.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 31.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 30.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   if -1.02e-249 < l < 5.59999999999999973e-236

   1. Initial program 23.9

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

   2. Simplified 24.8

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

      [Start] 23.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 24.7	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 24.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      fma-def [=>] 24.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-*l* [=>] 24.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 46.7

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

   4. Simplified 26.8

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

      [Start] 46.7	\[ e^{\mathsf{log1p}\left(\sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}\right)} - 1 \]
      expm1-def [=>] 30.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 29.8	\[ \color{blue}{\sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}} \]
      associate-*r* [=>] 26.8	\[ \sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right) \cdot \left(2 \cdot n\right)\right)} \]

   if 5.59999999999999973e-236 < l < 9.4999999999999996e147

   1. Initial program 29.1

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

   if 9.4999999999999996e147 < l

   1. Initial program 62.7

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

   2. Simplified 50.3

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

      [Start] 62.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.6	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 62.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 62.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 62.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 62.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 62.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 49.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 50.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 63.0

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

   4. Simplified 57.8

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

      [Start] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-*r* [=>] 63.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 63.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 63.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 57.8	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Taylor expanded in l around inf 34.0

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

   6. Simplified 29.3

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

      [Start] 34.0	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      associate-*l* [=>] 33.9	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [=>] 33.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      associate-*r* [=>] 33.8	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      cancel-sign-sub-inv [=>] 33.8	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}}\right) \]
      associate-/l* [=>] 34.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U*}}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      unpow2 [=>] 34.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U*}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      associate-/l* [=>] 31.1	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\color{blue}{\frac{Om}{\frac{U*}{Om}}}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      associate-/r/ [=>] 29.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      metadata-eval [=>] 29.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)}\right) \]
      associate-*r/ [=>] 29.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)}\right) \]
      metadata-eval [=>] 29.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 29.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	30.2
Cost	14992

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+149}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

Alternative 2
Error	28.4
Cost	14728

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+153}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_1\right)\\ \end{array} \]

Alternative 3
Error	30.5
Cost	14676

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om \cdot \frac{Om}{U*}} + \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -3.4 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell \cdot U}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 4
Error	31.4
Cost	14544

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell \cdot U}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om \cdot \frac{Om}{U*}} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	31.2
Cost	14544

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell \cdot U}}\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 6
Error	30.3
Cost	14544

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+144}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_1\right)\\ \end{array} \]

Alternative 7
Error	34.0
Cost	14416

\[\begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\ell \cdot \left(n \cdot \frac{\ell}{\frac{Om}{U}}\right)\right)\right)}\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell \cdot U}{Om} \cdot \left(\ell \cdot -4\right) + 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right) - U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \end{array} \]

Alternative 8
Error	34.0
Cost	14416

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;n \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell \cdot U}{Om} \cdot \left(\ell \cdot -4\right) + 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\ell \cdot \left(n \cdot \frac{\ell}{\frac{Om}{U}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right) - U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \end{array} \]

Alternative 9
Error	34.0
Cost	14416

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;n \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell \cdot U}{Om} \cdot \left(\ell \cdot -4\right) + 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{\ell \cdot U}}\right)\right)}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right) - U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \end{array} \]

Alternative 10
Error	32.7
Cost	8656

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(\frac{\ell \cdot U}{Om} \cdot \left(\ell \cdot -4\right) + 2 \cdot \left(U \cdot t\right)\right)}\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U*}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}\right) - t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	35.3
Cost	7824

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{if}\;n \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq -1.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+215}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{+302}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.3
Cost	7824

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(-2 \cdot \left(U \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{+299}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 13
Error	31.3
Cost	7753

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-18} \lor \neg \left(U \leq 5 \cdot 10^{-100}\right):\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\frac{\ell \cdot U}{Om} \cdot \left(\ell \cdot -4\right) + 2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 14
Error	33.0
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-18} \lor \neg \left(U \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	35.6
Cost	7624

\[\begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{\frac{Om}{\ell \cdot n}}\\ \end{array} \]

Alternative 16
Error	39.4
Cost	7560

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+254}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \left(-\sqrt{\left(U - U*\right) \cdot \left(U \cdot -2\right)}\right)\right)\\ \end{array} \]

Alternative 17
Error	40.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U* \leq -4.5 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \end{array} \]

Alternative 18
Error	40.8
Cost	6848

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

Alternative 19
Error	40.8
Cost	6848

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

Reproduce

herbie shell --seed 2023018 
(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*))))))