Toniolo and Linder, Equation (13)

Average Error: 34.2 → 28.7

Time: 44.9s

Precision: binary64

Cost: 21268

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+141}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.06 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;{\left(\sqrt[3]{n} \cdot \sqrt[3]{U \cdot \left(2 \cdot t\right)}\right)}^{1.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

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
           (* U (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om))))))))
   (if (<= l -2.1e+141)
     (* t_1 (* (sqrt 2.0) (- l)))
     (if (<= l -1.06e-20)
       (sqrt (* 2.0 (* U (* n (- t (* (/ l Om) (* l 2.0)))))))
       (if (<= l 1.05e-62)
         (pow (* (cbrt n) (cbrt (* U (* 2.0 t)))) 1.5)
         (if (<= l 2.6e+66)
           (sqrt
            (*
             (* 2.0 n)
             (*
              U
              (+
               t
               (+
                (* (/ (* l (* l U*)) Om) (/ n Om))
                (/ -2.0 (/ (/ Om l) l)))))))
           (if (<= l 1.06e+183)
             (sqrt
              (*
               (* 2.0 n)
               (* (* l (* l U)) (+ (/ -2.0 Om) (* (/ n Om) (/ (- U* U) Om))))))
             (* (* l (sqrt 2.0)) t_1))))))))

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 * (U * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	double tmp;
	if (l <= -2.1e+141) {
		tmp = t_1 * (sqrt(2.0) * -l);
	} else if (l <= -1.06e-20) {
		tmp = sqrt((2.0 * (U * (n * (t - ((l / Om) * (l * 2.0)))))));
	} else if (l <= 1.05e-62) {
		tmp = pow((cbrt(n) * cbrt((U * (2.0 * t)))), 1.5);
	} else if (l <= 2.6e+66) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((((l * (l * U_42_)) / Om) * (n / Om)) + (-2.0 / ((Om / l) / l)))))));
	} else if (l <= 1.06e+183) {
		tmp = sqrt(((2.0 * n) * ((l * (l * U)) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))));
	} else {
		tmp = (l * sqrt(2.0)) * t_1;
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((n * (U * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	double tmp;
	if (l <= -2.1e+141) {
		tmp = t_1 * (Math.sqrt(2.0) * -l);
	} else if (l <= -1.06e-20) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - ((l / Om) * (l * 2.0)))))));
	} else if (l <= 1.05e-62) {
		tmp = Math.pow((Math.cbrt(n) * Math.cbrt((U * (2.0 * t)))), 1.5);
	} else if (l <= 2.6e+66) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((((l * (l * U_42_)) / Om) * (n / Om)) + (-2.0 / ((Om / l) / l)))))));
	} else if (l <= 1.06e+183) {
		tmp = Math.sqrt(((2.0 * n) * ((l * (l * U)) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * t_1;
	}
	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(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om))))))
	tmp = 0.0
	if (l <= -2.1e+141)
		tmp = Float64(t_1 * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -1.06e-20)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64(l / Om) * Float64(l * 2.0)))))));
	elseif (l <= 1.05e-62)
		tmp = Float64(cbrt(n) * cbrt(Float64(U * Float64(2.0 * t)))) ^ 1.5;
	elseif (l <= 2.6e+66)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(Float64(l * Float64(l * U_42_)) / Om) * Float64(n / Om)) + Float64(-2.0 / Float64(Float64(Om / l) / l)))))));
	elseif (l <= 1.06e+183)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(l * Float64(l * U)) * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * t_1);
	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[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.1e+141], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.06e-20], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.05e-62], N[Power[N[(N[Power[n, 1/3], $MachinePrecision] * N[Power[N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], If[LessEqual[l, 2.6e+66], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(N[(l * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.06e+183], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -2.0999999999999998e141

   1. Initial program 61.7

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

   2. Simplified 46.5

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

      [Start] 61.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.7	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 61.7	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 61.7	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around -inf 34.4

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

   if -2.0999999999999998e141 < l < -1.06e-20

   1. Initial program 32.4

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

   2. Taylor expanded in n around 0 34.8

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

   3. Simplified 33.7

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

      [Start] 34.8	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 33.7	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 33.7	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      unpow2 [=>] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]
      cancel-sign-sub-inv [=>] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)\right)} \]
      metadata-eval [=>] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)} \]
      *-commutative [<=] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2}\right)\right)\right)} \]
      *-commutative [=>] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right)\right)\right)} \]
      associate-*l* [=>] 33.7	\[ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)}\right)\right)\right)} \]

   if -1.06e-20 < l < 1.05e-62

   1. Initial program 26.0

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

   2. Simplified 28.4

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

      [Start] 26.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 26.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 26.0	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 26.0	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 30.6

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

   4. Applied egg-rr 31.0

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot \left(2 \cdot t\right)}\right)}^{1.5}} \]

   5. Applied egg-rr 24.0

      \[\leadsto {\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{U \cdot \left(2 \cdot t\right)}\right)}}^{1.5} \]

   if 1.05e-62 < l < 2.60000000000000012e66

   1. Initial program 29.0

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

   2. Simplified 30.6

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

      [Start] 29.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 29.5	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 30.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 31.8

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

   4. Simplified 28.5

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

      [Start] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \]
      *-commutative [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      +-commutative [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      mul-1-neg [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unsub-neg [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      associate-*r/ [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2 \cdot {\ell}^{2}}{Om}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/l* [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\color{blue}{\frac{2}{\frac{Om}{{\ell}^{2}}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{Om}{\color{blue}{\ell \cdot \ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      associate-/r* [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\color{blue}{\frac{\frac{Om}{\ell}}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)} \]
      *-commutative [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)} \]
      unpow2 [=>] 31.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - \frac{\left({\ell}^{2} \cdot U*\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]
      times-frac [=>] 28.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - \color{blue}{\frac{{\ell}^{2} \cdot U*}{Om} \cdot \frac{n}{Om}}\right)\right)\right)} \]
      unpow2 [=>] 28.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]
      associate-*l* [=>] 28.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\frac{2}{\frac{\frac{Om}{\ell}}{\ell}} - \frac{\color{blue}{\ell \cdot \left(\ell \cdot U*\right)}}{Om} \cdot \frac{n}{Om}\right)\right)\right)} \]

   if 2.60000000000000012e66 < l < 1.06e183

   1. Initial program 38.6

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

   2. Simplified 35.1

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

      [Start] 38.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 39.3	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 39.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 39.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 39.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 39.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 39.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 34.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 35.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in l around inf 48.3

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

   4. Simplified 40.6

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

      [Start] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      mul-1-neg [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}} \]
      distribute-rgt-neg-in [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)}} \]
      *-commutative [<=] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)} \]
      unpow2 [=>] 48.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)} \]
      times-frac [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)} \]
      associate-*r/ [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)} \]
      metadata-eval [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{\color{blue}{2}}{Om}\right) \cdot \left(-{\ell}^{2} \cdot U\right)\right)} \]
      unpow2 [=>] 45.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(-\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)} \]
      associate-*l* [=>] 40.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\frac{U - U*}{Om} \cdot \frac{n}{Om} + \frac{2}{Om}\right) \cdot \left(-\color{blue}{\ell \cdot \left(\ell \cdot U\right)}\right)\right)} \]

   if 1.06e183 < 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 51.4

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

      [Start] 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 34.4

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.06 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;{\left(\sqrt[3]{n} \cdot \sqrt[3]{U \cdot \left(2 \cdot t\right)}\right)}^{1.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.1
Cost	30728

\[\begin{array}{l} t_1 := \left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot U*}} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	28.8
Cost	21268

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{-62}:\\ \;\;\;\;{\left(\sqrt[3]{n} \cdot \sqrt[3]{U \cdot \left(2 \cdot t\right)}\right)}^{1.5}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+178}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]

Alternative 3
Error	28.8
Cost	20108

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+140}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;{\left(\sqrt[3]{n} \cdot \sqrt[3]{U \cdot \left(2 \cdot t\right)}\right)}^{1.5}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

Alternative 4
Error	33.2
Cost	15520

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ t_2 := \sqrt{n + n}\\ \mathbf{if}\;n \leq -4.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(n \cdot -4\right)}{Om}}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-99}:\\ \;\;\;\;t_2 \cdot \sqrt{U \cdot t - U \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot \ell\right)\right)}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(-2 \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{U \cdot t}\\ \end{array} \]

Alternative 5
Error	33.3
Cost	14813

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq -8.4 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{-275}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(n \cdot -4\right)}{Om}}\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-99} \lor \neg \left(n \leq 3.5 \cdot 10^{+38}\right):\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t - U \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right) - t\right)\right)}\\ \end{array} \]

Alternative 6
Error	32.8
Cost	14417

\[\begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-99} \lor \neg \left(n \leq 3.5 \cdot 10^{+38}\right):\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t - U \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right) - t\right)\right)}\\ \end{array} \]

Alternative 7
Error	30.6
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot U*}} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 8
Error	31.5
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 10^{+41}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot U*}} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)}\right)\\ \end{array} \]

Alternative 9
Error	30.5
Cost	14280

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+140}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot U*}} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

Alternative 10
Error	32.2
Cost	8524

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} \cdot \frac{n}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	31.9
Cost	8392

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell \cdot U*}} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	34.4
Cost	8268

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	34.4
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	34.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+140} \lor \neg \left(\ell \leq 1.05 \cdot 10^{+32}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	36.7
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+52} \lor \neg \left(\ell \leq 1.85 \cdot 10^{+33}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{-2}} \cdot \left(\ell \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 16
Error	40.6
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+55} \lor \neg \left(\ell \leq 2.4 \cdot 10^{+35}\right):\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 17
Error	38.8
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+56} \lor \neg \left(\ell \leq 2.4 \cdot 10^{+35}\right):\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(n \cdot -4\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 18
Error	39.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+120} \lor \neg \left(n \leq 9.2 \cdot 10^{-153}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 19
Error	39.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+138} \lor \neg \left(n \leq 9.2 \cdot 10^{-153}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]

Alternative 20
Error	40.5
Cost	6848

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

Reproduce

herbie shell --seed 2023032 
(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*))))))