Toniolo and Linder, Equation (13)

Average Error: 34.3 → 29.2

Time: 39.4s

Precision: binary64

Cost: 33476

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ t_2 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+182}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* n 2.0)
           (*
            U
            (+
             t
             (+
              (* U* (* l (/ (/ l (/ Om n)) Om)))
              (* (/ l Om) (* l -2.0))))))))
        (t_2 (* U (* n 2.0))))
   (if (<= l -5e+182)
     (*
      (pow
       (pow (* (fma (* n (pow Om -2.0)) (- U* U) (/ -2.0 Om)) (* n U)) 0.25)
       2.0)
      (* (sqrt 2.0) (- l)))
     (if (<= l -4e+53)
       t_1
       (if (<= l -3.7e-60)
         (sqrt
          (*
           (* (* n U) 2.0)
           (+
            t
            (+ (* (/ (* n (* l l)) Om) (/ U* Om)) (* l (* -2.0 (/ l Om)))))))
         (if (<= l -1.2e-102)
           t_1
           (if (<= l 3.2e-305)
             (sqrt
              (*
               t_2
               (+
                (+ t (* -2.0 (/ (* l l) Om)))
                (* (* n (pow (/ l Om) 2.0)) (- U* U)))))
             (if (<= l 5.9e-117)
               (sqrt (* 2.0 (* U (* n t))))
               (if (<= l 3.8e+220)
                 (sqrt
                  (*
                   t_2
                   (+
                    (+ t (* -2.0 (* l (* l (/ 1.0 Om)))))
                    (* (* (* l (/ l Om)) (/ n Om)) (- U* U)))))
                 (*
                  (* l (sqrt 2.0))
                  (sqrt
                   (*
                    (* n U)
                    (+ (/ -2.0 Om) (* (/ n (* Om Om)) (- U* U)))))))))))))))

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 = sqrt(((n * 2.0) * (U * (t + ((U_42_ * (l * ((l / (Om / n)) / Om))) + ((l / Om) * (l * -2.0)))))));
	double t_2 = U * (n * 2.0);
	double tmp;
	if (l <= -5e+182) {
		tmp = pow(pow((fma((n * pow(Om, -2.0)), (U_42_ - U), (-2.0 / Om)) * (n * U)), 0.25), 2.0) * (sqrt(2.0) * -l);
	} else if (l <= -4e+53) {
		tmp = t_1;
	} else if (l <= -3.7e-60) {
		tmp = sqrt((((n * U) * 2.0) * (t + ((((n * (l * l)) / Om) * (U_42_ / Om)) + (l * (-2.0 * (l / Om)))))));
	} else if (l <= -1.2e-102) {
		tmp = t_1;
	} else if (l <= 3.2e-305) {
		tmp = sqrt((t_2 * ((t + (-2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	} else if (l <= 5.9e-117) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else if (l <= 3.8e+220) {
		tmp = sqrt((t_2 * ((t + (-2.0 * (l * (l * (1.0 / Om))))) + (((l * (l / Om)) * (n / Om)) * (U_42_ - U)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * U) * ((-2.0 / Om) + ((n / (Om * Om)) * (U_42_ - U)))));
	}
	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 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(U_42_ * Float64(l * Float64(Float64(l / Float64(Om / n)) / Om))) + Float64(Float64(l / Om) * Float64(l * -2.0)))))))
	t_2 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (l <= -5e+182)
		tmp = Float64(((Float64(fma(Float64(n * (Om ^ -2.0)), Float64(U_42_ - U), Float64(-2.0 / Om)) * Float64(n * U)) ^ 0.25) ^ 2.0) * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -4e+53)
		tmp = t_1;
	elseif (l <= -3.7e-60)
		tmp = sqrt(Float64(Float64(Float64(n * U) * 2.0) * Float64(t + Float64(Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om)) + Float64(l * Float64(-2.0 * Float64(l / Om)))))));
	elseif (l <= -1.2e-102)
		tmp = t_1;
	elseif (l <= 3.2e-305)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))));
	elseif (l <= 5.9e-117)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	elseif (l <= 3.8e+220)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t + Float64(-2.0 * Float64(l * Float64(l * Float64(1.0 / Om))))) + Float64(Float64(Float64(l * Float64(l / Om)) * Float64(n / Om)) * Float64(U_42_ - U)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U))))));
	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[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ * N[(l * N[(N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+182], N[(N[Power[N[Power[N[(N[(N[(n * N[Power[Om, -2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e+53], t$95$1, If[LessEqual[l, -3.7e-60], N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t + N[(N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(l * N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -1.2e-102], t$95$1, If[LessEqual[l, 3.2e-305], N[Sqrt[N[(t$95$2 * 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.9e-117], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e+220], N[Sqrt[N[(t$95$2 * N[(N[(t + N[(-2.0 * N[(l * N[(l * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -4.99999999999999973e182

   1. Initial program 64.0

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

   2. Simplified 53.1

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

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

   3. Taylor expanded in l around -inf 33.4

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

   4. Simplified 33.3

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

      [Start] 33.4	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      associate-*r* [=>] 33.4	\[ \color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [=>] 33.4	\[ \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right)} \]

   5. Applied egg-rr 33.7

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)\right)}^{0.25}\right)}^{2}} \cdot \left(-\sqrt{2} \cdot \ell\right) \]

   if -4.99999999999999973e182 < l < -4e53 or -3.70000000000000025e-60 < l < -1.2e-102

   1. Initial program 34.1

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

   2. Applied egg-rr 34.1

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

   3. Simplified 30.9

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

      [Start] 34.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 34.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\color{blue}{\left(\left(-\ell\right) \cdot \ell\right)} \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{-Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      neg-mul-1 [=>] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{\color{blue}{-1 \cdot Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-/r* [=>] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{\frac{1}{-1}}{Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      metadata-eval [=>] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{\color{blue}{-1}}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 37.4

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

   5. Simplified 30.1

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

      [Start] 37.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 37.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 37.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)} \]
      times-frac [=>] 33.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 33.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*r/ [<=] 30.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in U around 0 39.6

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

   7. Simplified 29.7

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

      [Start] 39.6	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 39.6	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unpow2 [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r/ [<=] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-*r* [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      *-commutative [=>] 39.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 39.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \color{blue}{\frac{{\ell}^{2} \cdot U*}{\frac{{Om}^{2}}{n}}}\right)\right)\right)} \]
      *-commutative [=>] 39.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \frac{\color{blue}{U* \cdot {\ell}^{2}}}{\frac{{Om}^{2}}{n}}\right)\right)\right)} \]
      unpow2 [=>] 39.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \frac{U* \cdot {\ell}^{2}}{\frac{\color{blue}{Om \cdot Om}}{n}}\right)\right)\right)} \]
      associate-*r/ [<=] 37.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \frac{U* \cdot {\ell}^{2}}{\color{blue}{Om \cdot \frac{Om}{n}}}\right)\right)\right)} \]
      associate-*r/ [<=] 33.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - \color{blue}{U* \cdot \frac{{\ell}^{2}}{Om \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   if -4e53 < l < -3.70000000000000025e-60

   1. Initial program 29.9

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

   2. Applied egg-rr 29.9

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

   3. Simplified 29.9

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

      [Start] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\color{blue}{\left(\left(-\ell\right) \cdot \ell\right)} \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{-Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      neg-mul-1 [=>] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{\color{blue}{-1 \cdot Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-/r* [=>] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{\frac{1}{-1}}{Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      metadata-eval [=>] 29.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{\color{blue}{-1}}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in U around 0 33.1

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

   5. Simplified 28.9

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

      [Start] 33.1	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 33.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 33.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 32.9	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 32.9	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      *-commutative [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      unpow2 [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot 2 - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-*r/ [<=] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot 2 - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-*l* [=>] 32.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot 2\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-*r* [=>] 32.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \left(\frac{\ell}{Om} \cdot 2\right) - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)} \]
      unpow2 [=>] 32.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \left(\frac{\ell}{Om} \cdot 2\right) - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)} \]

   if -1.2e-102 < l < 3.20000000000000009e-305

   1. Initial program 25.4

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

   if 3.20000000000000009e-305 < l < 5.9000000000000003e-117

   1. Initial program 25.4

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

   2. Simplified 28.1

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

      [Start] 25.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 25.4	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 25.4	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 25.4	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 29.4

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

   4. Applied egg-rr 46.6

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

   5. Simplified 29.3

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

      [Start] 46.6	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + -1\right)} \]
      metadata-eval [<=] 46.6	\[ \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 46.6	\[ \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)} - 1\right)}} \]
      expm1-def [=>] 30.4	\[ \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left(t \cdot U\right)\right)\right)}} \]
      expm1-log1p [=>] 29.4	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U\right)\right)}} \]
      associate-*r* [=>] 29.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 29.3	\[ \sqrt{2 \cdot \left(\color{blue}{\left(t \cdot n\right)} \cdot U\right)} \]

   if 5.9000000000000003e-117 < l < 3.79999999999999984e220

   1. Initial program 35.1

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

   2. Applied egg-rr 35.2

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

   3. Simplified 31.2

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

      [Start] 35.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 35.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\color{blue}{\left(\left(-\ell\right) \cdot \ell\right)} \cdot \frac{1}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 31.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{-Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      neg-mul-1 [=>] 31.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{1}{\color{blue}{-1 \cdot Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-/r* [=>] 31.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{\frac{1}{-1}}{Om}}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      metadata-eval [=>] 31.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{\color{blue}{-1}}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 36.9

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

   5. Simplified 30.0

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

      [Start] 36.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 36.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 36.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)} \]
      times-frac [=>] 34.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 34.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*r/ [<=] 30.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\left(-\ell\right) \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right) - \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]

   if 3.79999999999999984e220 < l

   1. Initial program 64.0

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

   2. Simplified 55.9

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

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

   3. Taylor expanded in l around inf 32.4

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

   4. Simplified 32.2

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

      [Start] 32.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*r* [=>] 31.8	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]
      *-commutative [=>] 31.8	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}} \]
      cancel-sign-sub-inv [=>] 31.8	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left(n \cdot U\right)} \]
      associate-/l* [=>] 32.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \]
      associate-/r/ [=>] 32.2	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \]
      unpow2 [=>] 32.2	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \]
      metadata-eval [=>] 32.2	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \]
      associate-*r/ [=>] 32.2	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left(n \cdot U\right)} \]
      metadata-eval [=>] 32.2	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right) \cdot \left(n \cdot U\right)} \]

3. Recombined 7 regimes into one program.

4. Final simplification 29.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+182}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	29.2
Cost	15124

\[\begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+176}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U*}}\right)}\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 2
Error	29.7
Cost	15068

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+182}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U*}}\right)}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om \cdot Om} \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 3
Error	30.9
Cost	14212

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(-\ell\right)\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om \cdot Om}{U*}}\right)}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-308} \lor \neg \left(\ell \leq 3.9 \cdot 10^{-113}\right):\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 4
Error	31.2
Cost	13700

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.32 \cdot 10^{+172}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{-2 \cdot \frac{n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + \left(\frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-282} \lor \neg \left(\ell \leq 1.85 \cdot 10^{-119}\right):\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 5
Error	31.2
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq 1.9 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{n \cdot U}\\ \end{array} \]

Alternative 6
Error	31.2
Cost	13512

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;t \leq 3 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left(t_1 \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]

Alternative 7
Error	33.9
Cost	9052

\[\begin{array}{l} t_1 := \sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -1.55 \cdot 10^{+219}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -1.42 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -6 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 9 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\ell \cdot n}{\frac{\frac{Om}{U}}{U* - U} \cdot \frac{Om}{\ell \cdot n}}}\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 5.3 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right) - t\right)\right)}\\ \mathbf{elif}\;Om \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	33.9
Cost	9052

\[\begin{array}{l} t_1 := \left(n \cdot U\right) \cdot 2\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ t_3 := n \cdot \left(\ell \cdot \ell\right)\\ \mathbf{if}\;Om \leq -6.1 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\frac{t_3}{Om} \cdot \frac{U*}{Om} + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\ell \cdot n}{\frac{\frac{Om}{U}}{U* - U} \cdot \frac{Om}{\ell \cdot n}}}\\ \mathbf{elif}\;Om \leq 2.55 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 8.6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - \frac{U* \cdot t_3}{Om \cdot Om}\right) - t\right)\right)}\\ \mathbf{elif}\;Om \leq 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 9
Error	29.5
Cost	8649

\[\begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{+88} \lor \neg \left(U \leq 3.9 \cdot 10^{-103}\right):\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(U* \cdot \left(\ell \cdot \frac{\frac{\ell}{\frac{Om}{n}}}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	36.3
Cost	8400

\[\begin{array}{l} t_1 := \left(n \cdot U\right) \cdot 2\\ t_2 := \sqrt{t_1 \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\ell \cdot n}{\frac{\frac{Om}{U}}{U* - U} \cdot \frac{Om}{\ell \cdot n}}}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-304}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 11
Error	35.3
Cost	8012

\[\begin{array}{l} t_1 := \sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;Om \leq -1.55 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -3.1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 6.2 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\ell \cdot n}{\frac{\frac{Om}{U}}{U* - U} \cdot \frac{Om}{\ell \cdot n}}}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	34.8
Cost	7757

\[\begin{array}{l} \mathbf{if}\;Om \leq -9.8 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -8 \cdot 10^{-56} \lor \neg \left(Om \leq 5 \cdot 10^{+20}\right):\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	35.0
Cost	7625

\[\begin{array}{l} \mathbf{if}\;U \leq -1.6 \cdot 10^{-98} \lor \neg \left(U \leq -2.7 \cdot 10^{-280}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 14
Error	34.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -1.95 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;U \leq -2.8 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \ell \cdot \frac{\ell \cdot 2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	39.8
Cost	7245

\[\begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-70} \lor \neg \left(n \leq 8.6 \cdot 10^{-134}\right) \land n \leq 6.5 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	39.8
Cost	7245

\[\begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-133} \lor \neg \left(n \leq 2.25 \cdot 10^{+244}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	39.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \end{array} \]

Alternative 18
Error	40.6
Cost	6848

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

Reproduce

herbie shell --seed 2023059 
(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*))))))