Toniolo and Linder, Equation (13)

Average Error: 34.0 → 30.7

Time: 1.1min

Precision: binary64

Cost: 34068

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{Om}{\ell}}\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := n \cdot \left(U \cdot t\right)\\ t_4 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_5 := n \cdot \left(\left(U - U*\right) \cdot t_4\right)\\ t_6 := \frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{t_6 \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(2, t_2, t_5\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot t_2\right) + \left(U* - U\right) \cdot \left(n \cdot t_4\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, t_5\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, \frac{\frac{\left(\ell \cdot n\right) \cdot \left(U \cdot -4\right)}{{t_1}^{2}}}{t_1}\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot t_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(2, t_3, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)\right)}^{0.25}\right)}^{2}\\ \end{array} \]

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 (cbrt (/ Om l)))
        (t_2 (/ (* l l) Om))
        (t_3 (* n (* U t)))
        (t_4 (pow (/ l Om) 2.0))
        (t_5 (* n (* (- U U*) t_4)))
        (t_6 (+ (* (/ n (* Om Om)) (- U* U)) (/ -2.0 Om))))
   (if (<= l -1.75e+83)
     (* (sqrt (* t_6 (* n U))) (* l (- (sqrt 2.0))))
     (if (<= l -7.4e-146)
       (sqrt (* -2.0 (* U (* n (- (fma 2.0 t_2 t_5) t)))))
       (if (<= l 6.4e-105)
         (sqrt
          (* (* U (* n 2.0)) (+ (- t (* 2.0 t_2)) (* (- U* U) (* n t_4)))))
         (if (<= l 5.9e-55)
           (* (sqrt (* n 2.0)) (sqrt (* U (- t (fma 2.0 (/ l (/ Om l)) t_5)))))
           (if (<= l 7.5e+217)
             (sqrt
              (fma 2.0 t_3 (/ (/ (* (* l n) (* U -4.0)) (pow t_1 2.0)) t_1)))
             (if (<= l 4.1e+260)
               (* (sqrt 2.0) (* l (sqrt (* U (* n t_6)))))
               (pow
                (pow (fma 2.0 t_3 (* (* U -4.0) (/ (* l n) (/ Om l)))) 0.25)
                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 = cbrt((Om / l));
	double t_2 = (l * l) / Om;
	double t_3 = n * (U * t);
	double t_4 = pow((l / Om), 2.0);
	double t_5 = n * ((U - U_42_) * t_4);
	double t_6 = ((n / (Om * Om)) * (U_42_ - U)) + (-2.0 / Om);
	double tmp;
	if (l <= -1.75e+83) {
		tmp = sqrt((t_6 * (n * U))) * (l * -sqrt(2.0));
	} else if (l <= -7.4e-146) {
		tmp = sqrt((-2.0 * (U * (n * (fma(2.0, t_2, t_5) - t)))));
	} else if (l <= 6.4e-105) {
		tmp = sqrt(((U * (n * 2.0)) * ((t - (2.0 * t_2)) + ((U_42_ - U) * (n * t_4)))));
	} else if (l <= 5.9e-55) {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, (l / (Om / l)), t_5))));
	} else if (l <= 7.5e+217) {
		tmp = sqrt(fma(2.0, t_3, ((((l * n) * (U * -4.0)) / pow(t_1, 2.0)) / t_1)));
	} else if (l <= 4.1e+260) {
		tmp = sqrt(2.0) * (l * sqrt((U * (n * t_6))));
	} else {
		tmp = pow(pow(fma(2.0, t_3, ((U * -4.0) * ((l * n) / (Om / l)))), 0.25), 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 = cbrt(Float64(Om / l))
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(n * Float64(U * t))
	t_4 = Float64(l / Om) ^ 2.0
	t_5 = Float64(n * Float64(Float64(U - U_42_) * t_4))
	t_6 = Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) + Float64(-2.0 / Om))
	tmp = 0.0
	if (l <= -1.75e+83)
		tmp = Float64(sqrt(Float64(t_6 * Float64(n * U))) * Float64(l * Float64(-sqrt(2.0))));
	elseif (l <= -7.4e-146)
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(2.0, t_2, t_5) - t)))));
	elseif (l <= 6.4e-105)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * t_2)) + Float64(Float64(U_42_ - U) * Float64(n * t_4)))));
	elseif (l <= 5.9e-55)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l / Float64(Om / l)), t_5)))));
	elseif (l <= 7.5e+217)
		tmp = sqrt(fma(2.0, t_3, Float64(Float64(Float64(Float64(l * n) * Float64(U * -4.0)) / (t_1 ^ 2.0)) / t_1)));
	elseif (l <= 4.1e+260)
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U * Float64(n * t_6)))));
	else
		tmp = (fma(2.0, t_3, Float64(Float64(U * -4.0) * Float64(Float64(l * n) / Float64(Om / l)))) ^ 0.25) ^ 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[Power[N[(Om / l), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(n * N[(N[(U - U$42$), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.75e+83], N[(N[Sqrt[N[(t$95$6 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -7.4e-146], N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(2.0 * t$95$2 + t$95$5), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.4e-105], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.9e-55], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+217], N[Sqrt[N[(2.0 * t$95$3 + N[(N[(N[(N[(l * n), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.1e+260], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U * N[(n * t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * t$95$3 + N[(N[(U * -4.0), $MachinePrecision] * N[(N[(l * n), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -1.74999999999999989e83

   1. Initial program 52.8

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

   2. Simplified 43.4

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

      [Start] 52.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 52.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 52.8	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 52.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around -inf 36.0

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

   4. Simplified 35.2

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

      [Start] 36.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      associate-*r* [=>] 36.0	\[ \color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      *-commutative [=>] 36.0	\[ \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \ell\right)\right)} \]

   if -1.74999999999999989e83 < l < -7.39999999999999973e-146

   1. Initial program 28.7

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

   2. Simplified 29.8

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

      [Start] 28.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 29.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 29.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 45.4

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

   4. Simplified 30.4

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

      [Start] 45.4	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)}\right)} - 1 \]
      expm1-def [=>] 31.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 30.3	\[ \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)}} \]
      *-commutative [=>] 30.3	\[ \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)} \]
      associate-*l* [=>] 30.4	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)\right)\right)}} \]
      associate-*r* [=>] 30.4	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n}\right)\right)\right)\right)} \]
      *-commutative [=>] 30.4	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)\right)} \]
      *-commutative [=>] 30.4	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)\right)\right)} \]

   if -7.39999999999999973e-146 < l < 6.39999999999999962e-105

   1. Initial program 24.5

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

   if 6.39999999999999962e-105 < l < 5.8999999999999998e-55

   1. Initial program 27.2

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

   2. Applied egg-rr 40.9

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

   3. Simplified 40.8

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

      [Start] 40.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r/ [=>] 40.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell \cdot \ell}{Om}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 40.9	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*r* [=>] 39.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)} \]
      *-commutative [<=] 39.7	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)} \]
      associate-*l* [=>] 40.8	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   if 5.8999999999999998e-55 < l < 7.5000000000000001e217

   1. Initial program 38.5

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

   2. Taylor expanded in Om around inf 40.3

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

   3. Simplified 39.8

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

      [Start] 40.3	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 40.3	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 40.3	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 39.9	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r* [=>] 39.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{{\ell}^{2}}}{U}}}\right)} \]
      unpow2 [=>] 39.8	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot \ell}}}{U}}\right)} \]

   4. Applied egg-rr 50.6

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

   5. Simplified 33.4

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

      [Start] 50.6	\[ e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)} - 1 \]
      expm1-def [=>] 36.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 35.2	\[ \color{blue}{\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}} \]
      associate-*r/ [=>] 33.4	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \color{blue}{\frac{n \cdot \ell}{\frac{Om}{\ell}}} \cdot \left(U \cdot -4\right)\right)} \]

   6. Applied egg-rr 34.0

      \[\leadsto \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \color{blue}{\frac{\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -4\right)}{{\left(\sqrt[3]{\frac{Om}{\ell}}\right)}^{2}}}{\sqrt[3]{\frac{Om}{\ell}}}}\right)} \]

   if 7.5000000000000001e217 < l < 4.10000000000000025e260

   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 51.3

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
      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	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      cancel-sign-sub-inv [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]
      +-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]
      distribute-lft-neg-in [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(-\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{n \cdot \left(-{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} \]
      fma-def [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n, -{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\right)} \]

   3. Taylor expanded in l around inf 33.4

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

   4. Simplified 37.5

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

      [Start] 33.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*l* [=>] 33.3	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      *-commutative [=>] 33.3	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}}\right) \]
      associate-*r* [=>] 37.0	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}}\right) \]
      *-commutative [=>] 37.0	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      cancel-sign-sub-inv [=>] 37.0	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)}\right) \]
      associate-/l* [=>] 39.0	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-/r/ [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{{Om}^{2}} \cdot \left(U* - U\right)} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      unpow2 [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\color{blue}{Om \cdot Om}} \cdot \left(U* - U\right) + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
      metadata-eval [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-*r/ [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)}\right) \]
      metadata-eval [=>] 37.5	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{\color{blue}{-2}}{Om}\right)\right)}\right) \]

   if 4.10000000000000025e260 < 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. Taylor expanded in Om around inf 64.0

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

   3. Simplified 64.0

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

      [Start] 64.0	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      fma-def [=>] 64.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(2, n \cdot \left(t \cdot U\right), -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
      *-commutative [=>] 64.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \color{blue}{\left(U \cdot t\right)}, -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]
      associate-/r* [=>] 64.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{{\ell}^{2}}}{U}}}\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{\frac{Om}{\color{blue}{\ell \cdot \ell}}}{U}}\right)} \]

   4. Applied egg-rr 62.4

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

   5. Simplified 50.1

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

      [Start] 62.4	\[ e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)} - 1 \]
      expm1-def [=>] 60.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 60.4	\[ \color{blue}{\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(n \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot -4\right)\right)}} \]
      associate-*r/ [=>] 50.1	\[ \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \color{blue}{\frac{n \cdot \ell}{\frac{Om}{\ell}}} \cdot \left(U \cdot -4\right)\right)} \]

   6. Applied egg-rr 46.0

      \[\leadsto \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \color{blue}{\frac{\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -4\right)}{{\left(\sqrt[3]{\frac{Om}{\ell}}\right)}^{2}}}{\sqrt[3]{\frac{Om}{\ell}}}}\right)} \]

   7. Applied egg-rr 50.2

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{\ell \cdot n}{\frac{Om}{\ell}} \cdot \left(U \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 30.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{\frac{\left(\ell \cdot n\right) \cdot \left(U \cdot -4\right)}{{\left(\sqrt[3]{\frac{Om}{\ell}}\right)}^{2}}}{\sqrt[3]{\frac{Om}{\ell}}}\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)\right)}^{0.25}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.4
Cost	57612

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(U* - U\right) \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{-\frac{Om}{\ell}}{-\ell \cdot U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(\ell \cdot \left(n \cdot -2\right)\right) \cdot \left(\ell \cdot U\right)\right)}\\ \end{array} \]

Alternative 2
Error	26.1
Cost	57612

\[\begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \sqrt{U \cdot t}} \cdot {\left(U \cdot t\right)}^{0.25}\\ \mathbf{elif}\;t_1 \leq 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \frac{n}{\frac{-\frac{Om}{\ell}}{-\ell \cdot U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(\ell \cdot \left(n \cdot -2\right)\right) \cdot \left(\ell \cdot U\right)\right)}\\ \end{array} \]

Alternative 3
Error	30.2
Cost	27664

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := n \cdot \left(\left(U - U*\right) \cdot t_2\right)\\ \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(2, t_1, t_3\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot t_1\right) + \left(U* - U\right) \cdot \left(n \cdot t_2\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, t_3\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 4
Error	29.6
Cost	21000

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(2, t_1, n \cdot \left(\left(U - U*\right) \cdot t_2\right)\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 15.2:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot t_1\right) + \left(U* - U\right) \cdot \left(n \cdot t_2\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 5
Error	32.3
Cost	15332

\[\begin{array}{l} t_1 := n \cdot \frac{U*}{Om}\\ t_2 := \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ t_3 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.32 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.4 \cdot 10^{-99}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 17:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + t_1\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\frac{2 - t_1}{Om} \cdot \left(\left(\ell \cdot \left(n \cdot -2\right)\right) \cdot \left(\ell \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	31.7
Cost	15200

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -3.55 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 15.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 7
Error	31.7
Cost	15200

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 14:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 8
Error	30.8
Cost	15200

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 0.104:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+120}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_2, -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 9
Error	30.7
Cost	15200

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ t_2 := \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq -9.6 \cdot 10^{+84}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 15.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_3, -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 10
Error	30.0
Cost	14936

\[\begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := \sqrt{\left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+84}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 16:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+120}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(2 - n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, t_1, -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 11
Error	32.4
Cost	14812

\[\begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \left(U \cdot -4\right) \cdot \frac{\ell \cdot n}{\frac{Om}{\ell}}\right)}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 16.5:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{\left(U* - U\right) \cdot \left(U \cdot 2\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	32.5
Cost	14548

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 15.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, \frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right), n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{2}{Om} - \frac{U*}{\frac{Om \cdot Om}{n}}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	32.5
Cost	14548

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 17:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), -4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.7 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{2}{Om} - \frac{U*}{\frac{Om \cdot Om}{n}}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	32.5
Cost	13644

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2 \cdot \left|\left(n \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 10:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \frac{n \cdot \left(2 - n \cdot \frac{U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(\frac{2}{Om} - \frac{U*}{\frac{Om \cdot Om}{n}}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	32.4
Cost	8784

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;\left(\ell \cdot \frac{n}{Om}\right) \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	33.4
Cost	8144

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot t_1\right)\right)\right)}\\ \mathbf{if}\;U \leq -5.8 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;U \leq -7.8 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{\ell \cdot U*} \cdot \frac{Om}{U}}{n}}}\\ \mathbf{elif}\;U \leq 3.7 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t - \left(n \cdot U\right) \cdot \left(2 \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 17
Error	34.5
Cost	8012

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 - n \cdot \frac{U*}{Om}}{Om} \cdot \left(\left(\ell \cdot \left(n \cdot -2\right)\right) \cdot \left(\ell \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Error	34.2
Cost	7888

\[\begin{array}{l} t_1 := \left(n \cdot U\right) \cdot 2\\ t_2 := \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{if}\;U \leq -2.35 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot t_1}\\ \mathbf{elif}\;U \leq -7.2 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 1.95 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{\ell \cdot U*} \cdot \frac{Om}{U}}{n}}}\\ \mathbf{elif}\;U \leq 1.7 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \end{array} \]

Alternative 19
Error	35.1
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.86 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-227}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	35.1
Cost	7756

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.82 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	33.8
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 22
Error	36.0
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+77} \lor \neg \left(\ell \leq 2.2 \cdot 10^{+16}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot \frac{U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \end{array} \]

Alternative 23
Error	39.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-167} \lor \neg \left(t \leq 2.9 \cdot 10^{+161}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 24
Error	39.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \end{array} \]

Alternative 25
Error	39.8
Cost	6848

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

Reproduce

herbie shell --seed 2023041 
(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*))))))