Toniolo and Linder, Equation (13)

Average Error: 35.1 → 28.1

Time: 34.7s

Precision: binary64

Cost: 14412

Math

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

↓

\[\begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U*}{Om}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(t_1 + \frac{-2}{Om}\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ n Om) (/ U* Om))))
   (if (<= l -4.8e+162)
     (* (sqrt 2.0) (* l (- (sqrt (* (* n U) (+ t_1 (/ -2.0 Om)))))))
     (if (<= l 1.66e-46)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (+ (* (/ n (/ Om l)) (* U* (/ l Om))) (* -2.0 (/ l (/ Om l)))))))
       (if (<= l 3.1e+199)
         (sqrt (* 2.0 (* n (- (* U t) (* (* l (* l U)) (- (/ 2.0 Om) t_1))))))
         (*
          (* l (sqrt 2.0))
          (sqrt (* n (* U (+ (/ n (/ (* Om Om) U*)) (/ -2.0 Om)))))))))))

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 = (n / Om) * (U_42_ / Om);
	double tmp;
	if (l <= -4.8e+162) {
		tmp = sqrt(2.0) * (l * -sqrt(((n * U) * (t_1 + (-2.0 / Om)))));
	} else if (l <= 1.66e-46) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (((n / (Om / l)) * (U_42_ * (l / Om))) + (-2.0 * (l / (Om / l)))))));
	} else if (l <= 3.1e+199) {
		tmp = sqrt((2.0 * (n * ((U * t) - ((l * (l * U)) * ((2.0 / Om) - t_1))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * ((n / ((Om * Om) / U_42_)) + (-2.0 / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

↓

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (n / om) * (u_42 / om)
    if (l <= (-4.8d+162)) then
        tmp = sqrt(2.0d0) * (l * -sqrt(((n * u) * (t_1 + ((-2.0d0) / om)))))
    else if (l <= 1.66d-46) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (((n / (om / l)) * (u_42 * (l / om))) + ((-2.0d0) * (l / (om / l)))))))
    else if (l <= 3.1d+199) then
        tmp = sqrt((2.0d0 * (n * ((u * t) - ((l * (l * u)) * ((2.0d0 / om) - t_1))))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n * (u * ((n / ((om * om) / u_42)) + ((-2.0d0) / om)))))
    end if
    code = tmp
end function

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 = (n / Om) * (U_42_ / Om);
	double tmp;
	if (l <= -4.8e+162) {
		tmp = Math.sqrt(2.0) * (l * -Math.sqrt(((n * U) * (t_1 + (-2.0 / Om)))));
	} else if (l <= 1.66e-46) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (((n / (Om / l)) * (U_42_ * (l / Om))) + (-2.0 * (l / (Om / l)))))));
	} else if (l <= 3.1e+199) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) - ((l * (l * U)) * ((2.0 / Om) - t_1))))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n * (U * ((n / ((Om * Om) / U_42_)) + (-2.0 / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, 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_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = (n / Om) * (U_42_ / Om)
	tmp = 0
	if l <= -4.8e+162:
		tmp = math.sqrt(2.0) * (l * -math.sqrt(((n * U) * (t_1 + (-2.0 / Om)))))
	elif l <= 1.66e-46:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (((n / (Om / l)) * (U_42_ * (l / Om))) + (-2.0 * (l / (Om / l)))))))
	elif l <= 3.1e+199:
		tmp = math.sqrt((2.0 * (n * ((U * t) - ((l * (l * U)) * ((2.0 / Om) - t_1))))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n * (U * ((n / ((Om * Om) / U_42_)) + (-2.0 / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n / Om) * Float64(U_42_ / Om))
	tmp = 0.0
	if (l <= -4.8e+162)
		tmp = Float64(sqrt(2.0) * Float64(l * Float64(-sqrt(Float64(Float64(n * U) * Float64(t_1 + Float64(-2.0 / Om)))))));
	elseif (l <= 1.66e-46)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ * Float64(l / Om))) + Float64(-2.0 * Float64(l / Float64(Om / l)))))));
	elseif (l <= 3.1e+199)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) - Float64(Float64(l * Float64(l * U)) * Float64(Float64(2.0 / Om) - t_1))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(U * Float64(Float64(n / Float64(Float64(Om * Om) / U_42_)) + Float64(-2.0 / Om))))));
	end
	return tmp
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n / Om) * (U_42_ / Om);
	tmp = 0.0;
	if (l <= -4.8e+162)
		tmp = sqrt(2.0) * (l * -sqrt(((n * U) * (t_1 + (-2.0 / Om)))));
	elseif (l <= 1.66e-46)
		tmp = sqrt(((2.0 * (n * U)) * (t + (((n / (Om / l)) * (U_42_ * (l / Om))) + (-2.0 * (l / (Om / l)))))));
	elseif (l <= 3.1e+199)
		tmp = sqrt((2.0 * (n * ((U * t) - ((l * (l * U)) * ((2.0 / Om) - t_1))))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * ((n / ((Om * Om) / U_42_)) + (-2.0 / Om)))));
	end
	tmp_2 = 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[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * (-N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t$95$1 + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.66e-46], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.1e+199], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] - N[(N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.80000000000000018e162

   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 U around 0 64.0

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

   3. Simplified 50.2

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

      [Start] 64.0	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 64.0	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   4. Taylor expanded in l around -inf 32.9

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

   5. Simplified 28.6

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

      [Start] 32.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      mul-1-neg [=>] 32.9	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      associate-*l* [=>] 32.9	\[ -\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      distribute-rgt-neg-in [=>] 32.9	\[ \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [=>] 32.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      associate-*r* [=>] 32.8	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      cancel-sign-sub-inv [=>] 32.8	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}}\right) \]
      unpow2 [=>] 32.8	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      times-frac [=>] 28.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      metadata-eval [=>] 28.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)}\right) \]
      associate-*r/ [=>] 28.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)}\right) \]
      metadata-eval [=>] 28.6	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)}\right) \]

   if -4.80000000000000018e162 < l < 1.6599999999999999e-46

   1. Initial program 27.7

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

   2. Taylor expanded in U around 0 33.4

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

   3. Simplified 28.0

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

      [Start] 33.4	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 33.4	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 33.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 33.2	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 33.2	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 33.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 33.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 33.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 33.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 33.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 33.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 32.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   4. Applied egg-rr 26.6

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

   if 1.6599999999999999e-46 < l < 3.09999999999999986e199

   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. Taylor expanded in U around 0 41.7

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

   3. Simplified 34.7

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

      [Start] 41.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 41.7	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 41.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 42.1	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 42.1	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 42.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 42.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 42.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 42.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 42.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 41.6	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 39.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   4. Taylor expanded in l around 0 38.6

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

   5. Simplified 31.4

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

      [Start] 38.6	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \left(n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      distribute-lft-out [=>] 38.6	\[ \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right) + n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)}} \]
      distribute-lft-out [=>] 38.6	\[ \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U + \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)}} \]
      *-commutative [=>] 38.6	\[ \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{U \cdot t} + \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      cancel-sign-sub-inv [=>] 38.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)} \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 38.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      times-frac [=>] 35.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 35.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 35.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 35.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 35.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right)\right)\right)} \]
      associate-*l* [=>] 31.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot U\right)\right)}\right)\right)} \]

   if 3.09999999999999986e199 < 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 U around 0 64.0

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

   3. Simplified 55.6

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

      [Start] 64.0	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 64.0	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 64.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   4. Taylor expanded in l around inf 32.7

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

   5. Simplified 33.9

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

      [Start] 32.7	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      *-commutative [=>] 32.7	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]
      associate-/l* [=>] 33.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U*}}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      unpow2 [=>] 33.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U*}} - 2 \cdot \frac{1}{Om}\right)\right)} \]
      associate-*r/ [=>] 33.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]
      metadata-eval [=>] 33.9	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{\color{blue}{2}}{Om}\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	28.8
Cost	14412

\[\begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U*}{Om}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{-2 \cdot \frac{n \cdot U}{Om}} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t_1 + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 2
Error	29.2
Cost	14412

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{-2 \cdot \frac{n \cdot U}{Om}} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 3
Error	30.7
Cost	13908

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;U \leq 1.3 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 4.9 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t \cdot \left(2 \cdot n\right)}\\ \end{array} \]

Alternative 4
Error	30.7
Cost	13908

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;U \leq 8.2 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.15 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t + \ell \cdot \left(-2 \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \leq 1.38 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 5
Error	29.5
Cost	13900

\[\begin{array}{l} t_1 := \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+242}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_1 \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot t_1\\ \end{array} \]

Alternative 6
Error	31.4
Cost	8656

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ t_3 := \sqrt{t_2 \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(U* \cdot \frac{\ell}{Om}\right) + t_1\right)\right)}\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-160}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1.1 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + t_1\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-232}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	33.1
Cost	8528

\[\begin{array}{l} t_1 := n \cdot \left(2 \cdot U\right)\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)\right)\right) \cdot t_1}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{n}{\frac{Om}{U*} \cdot \frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t - \left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	35.4
Cost	8272

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{n}{\frac{Om}{U*} \cdot \frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	34.3
Cost	8272

\[\begin{array}{l} t_1 := \sqrt{\left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	34.2
Cost	8272

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot \left(-2 + n \cdot \frac{U*}{Om}\right)\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.7 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(U \cdot -2\right)\right) \cdot \left(\ell \cdot \frac{\ell \cdot \left(2 - \frac{n}{Om} \cdot U*\right)}{Om} - t\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	35.1
Cost	8144

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 45000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 - \frac{n}{Om} \cdot U*\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.1
Cost	8144

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{n}{\frac{Om}{U*} \cdot \frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 - \frac{n}{Om} \cdot U*\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	33.7
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+157} \lor \neg \left(\ell \leq 5.3 \cdot 10^{+153}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	34.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	36.7
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+48} \lor \neg \left(\ell \leq 3.3 \cdot 10^{+86}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\left(n \cdot U\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Error	40.0
Cost	6848

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

Reproduce

herbie shell --seed 2023053 
(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*))))))