Toniolo and Linder, Equation (13)

Average Error: 34.5 → 27.5

Time: 38.3s

Precision: binary64

Cost: 20944

Math

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

↓

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t_2 \cdot \sqrt{2}\right)\\ \end{array} \]

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 (* -2.0 (/ l (/ Om l))))
        (t_2 (sqrt (* (* n U) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))))
   (if (<= l -1.095e+120)
     (* t_2 (* (sqrt 2.0) (- l)))
     (if (<= l -9.5e-242)
       (sqrt
        (* (* n 2.0) (* U (+ t (+ (* (/ n Om) (/ (* l (* l U*)) Om)) t_1)))))
       (if (<= l 4.5e-73)
         (sqrt
          (*
           (* U (* n -2.0))
           (+
            (* (* n (pow (/ l Om) 2.0)) (- U U*))
            (- (* 2.0 (/ (* l l) Om)) t))))
         (if (<= l 5.4e+145)
           (sqrt
            (*
             (* n 2.0)
             (* U (+ t (+ (/ n (* (/ Om (* l l)) (/ Om U*))) t_1)))))
           (* l (* t_2 (sqrt 2.0)))))))))

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 = -2.0 * (l / (Om / l));
	double t_2 = sqrt(((n * U) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))));
	double tmp;
	if (l <= -1.095e+120) {
		tmp = t_2 * (sqrt(2.0) * -l);
	} else if (l <= -9.5e-242) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((n / Om) * ((l * (l * U_42_)) / Om)) + t_1)))));
	} else if (l <= 4.5e-73) {
		tmp = sqrt(((U * (n * -2.0)) * (((n * pow((l / Om), 2.0)) * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t))));
	} else if (l <= 5.4e+145) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / ((Om / (l * l)) * (Om / U_42_))) + t_1)))));
	} else {
		tmp = l * (t_2 * sqrt(2.0));
	}
	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(-2.0 * Float64(l / Float64(Om / l)))
	t_2 = sqrt(Float64(Float64(n * U) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))
	tmp = 0.0
	if (l <= -1.095e+120)
		tmp = Float64(t_2 * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -9.5e-242)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(n / Om) * Float64(Float64(l * Float64(l * U_42_)) / Om)) + t_1)))));
	elseif (l <= 4.5e-73)
		tmp = sqrt(Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t))));
	elseif (l <= 5.4e+145)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n / Float64(Float64(Om / Float64(l * l)) * Float64(Om / U_42_))) + t_1)))));
	else
		tmp = Float64(l * Float64(t_2 * sqrt(2.0)));
	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[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.095e+120], N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -9.5e-242], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(l * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.5e-73], N[Sqrt[N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $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, 5.4e+145], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.09500000000000003e120

   1. Initial program 57.0

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

   2. Simplified 45.4

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

      [Start] 57.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 57.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 57.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 57.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around inf 58.9

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

   4. Simplified 58.7

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

      [Start] 58.9	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      *-commutative [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      cancel-sign-sub-inv [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/l* [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/r/ [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)\right)} \]
      unpow2 [=>] 58.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right)} \]

   5. Taylor expanded in l around -inf 34.0

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

   6. Simplified 29.4

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

      [Start] 34.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      associate-*r* [=>] 34.0	\[ \color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [=>] 34.0	\[ \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right)} \]

   if -1.09500000000000003e120 < l < -9.4999999999999997e-242

   1. Initial program 28.5

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

   2. Simplified 29.1

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

      [Start] 28.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      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* [=>] 29.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 33.3

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

   4. Simplified 28.4

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

      [Start] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      *-commutative [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 33.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left({\ell}^{2} \cdot U*\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 29.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{{\ell}^{2} \cdot U*}{Om} \cdot \frac{n}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 29.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 28.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\ell \cdot \left(\ell \cdot U*\right)}}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]

   if -9.4999999999999997e-242 < l < 4.5e-73

   1. Initial program 24.9

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

   if 4.5e-73 < l < 5.40000000000000044e145

   1. Initial program 30.8

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

   2. Simplified 31.1

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

      [Start] 30.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 30.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 31.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 30.3

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

   4. Simplified 30.3

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

      [Start] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(-U*\right)\right)\right)\right)\right)} \]
      distribute-lft-out [=>] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U + \left(-U*\right)\right)}\right)\right)\right)} \]
      sub-neg [<=] 30.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 31.9

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

   6. Simplified 27.9

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

      [Start] 31.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 31.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 31.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      times-frac [=>] 27.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 27.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 27.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 27.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. Applied egg-rr 28.0

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

   8. Taylor expanded in U around 0 34.7

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

   9. Simplified 29.3

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

      [Start] 34.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + -1 \cdot \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)\right)} \]
      associate-*r/ [=>] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\frac{-1 \cdot n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)\right)} \]
      neg-mul-1 [<=] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{-n}}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}\right)\right)\right)} \]
      unpow2 [=>] 34.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{-n}{\frac{\color{blue}{Om \cdot Om}}{{\ell}^{2} \cdot U*}}\right)\right)\right)} \]
      times-frac [=>] 29.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{-n}{\color{blue}{\frac{Om}{{\ell}^{2}} \cdot \frac{Om}{U*}}}\right)\right)\right)} \]
      unpow2 [=>] 29.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{-n}{\frac{Om}{\color{blue}{\ell \cdot \ell}} \cdot \frac{Om}{U*}}\right)\right)\right)} \]

   if 5.40000000000000044e145 < l

   1. Initial program 62.0

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

   2. Simplified 49.1

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

      [Start] 62.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 62.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 62.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around inf 62.9

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

   4. Simplified 62.8

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

      [Start] 62.9	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      *-commutative [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}\right)} \]
      cancel-sign-sub-inv [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/l* [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-/r/ [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      *-commutative [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)\right)} \]
      unpow2 [=>] 62.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right)} \]

   5. Taylor expanded in l around 0 33.6

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

   6. Simplified 27.3

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

      [Start] 33.6	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      *-commutative [=>] 33.6	\[ \color{blue}{\left(\ell \cdot \sqrt{2}\right)} \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      *-commutative [=>] 33.6	\[ \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      associate-*l* [=>] 33.6	\[ \color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [<=] 33.6	\[ \ell \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      associate-*r* [=>] 31.8	\[ \ell \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      sub-neg [=>] 31.8	\[ \ell \cdot \left(\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      distribute-rgt-in [=>] 31.8	\[ \ell \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} \cdot \left(n \cdot U\right) + \left(-2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \sqrt{2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.0
Cost	20944

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2
Error	30.1
Cost	14992

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;n \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left(t_3 \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\left(n \cdot t_3\right) \cdot \left(U* - U\right) + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \end{array} \]

Alternative 3
Error	28.5
Cost	14860

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)}\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot t_2\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_2\\ \end{array} \]

Alternative 4
Error	31.4
Cost	14676

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right) + \frac{-2}{Om} \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 10^{-183}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+190}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{-2}{Om} - \frac{n}{\frac{Om \cdot Om}{U}}\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	30.2
Cost	14668

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;n \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_2 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(t_2 + \left(\frac{n}{Om} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n + n}\\ \end{array} \]

Alternative 6
Error	30.9
Cost	14540

\[\begin{array}{l} t_1 := \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\\ t_2 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_3 := \left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 \cdot t_1 + \frac{-2}{Om} \cdot t_3\right)}\\ \mathbf{elif}\;\ell \leq 10^{-184}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + t_1\right)\right)}\right)\\ \end{array} \]

Alternative 7
Error	30.6
Cost	14540

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right) + \frac{-2}{Om} \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 10^{-184}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)}\\ \end{array} \]

Alternative 8
Error	28.8
Cost	14408

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U* - U}}\right)}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\ell \cdot \frac{\ell}{\frac{Om}{n}}}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

Alternative 9
Error	31.9
Cost	8780

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \left(\ell \cdot U\right) \cdot \left(\ell \cdot n\right)\\ t_3 := \sqrt{2 \cdot \left(t_2 \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right) + \frac{-2}{Om} \cdot t_2\right)}\\ \mathbf{if}\;\ell \leq -1.095 \cdot 10^{+120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 10^{-183}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	33.1
Cost	8656

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot t_1 - \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 11
Error	33.1
Cost	8656

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 12
Error	32.8
Cost	8524

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := -2 \cdot t_1\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot t_1 - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-184}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + t_2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 13
Error	32.2
Cost	8521

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;Om \leq 1.9 \cdot 10^{-103} \lor \neg \left(Om \leq 6.5 \cdot 10^{+22}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{n}{Om} \cdot t_1\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(2 \cdot t_1 - \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right) - t\right)\right)}\\ \end{array} \]

Alternative 14
Error	32.5
Cost	8400

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right) - t\right)\right)}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 15
Error	34.6
Cost	8008

\[\begin{array}{l} \mathbf{if}\;U \leq -7.1 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.1 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	34.4
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-220} \lor \neg \left(\ell \leq 1.9 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \end{array} \]

Alternative 17
Error	34.5
Cost	7625

\[\begin{array}{l} \mathbf{if}\;Om \leq -7.2 \cdot 10^{+70} \lor \neg \left(Om \leq 3.2 \cdot 10^{+47}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	34.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot 2}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	34.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 20
Error	41.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 21
Error	40.7
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 22
Error	39.2
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-20} \lor \neg \left(n \leq 3 \cdot 10^{-152}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Error	40.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.55 \cdot 10^{-23} \lor \neg \left(Om \leq 1.4 \cdot 10^{+51}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 24
Error	40.4
Cost	6848

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

Reproduce

herbie shell --seed 2023056 
(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*))))))