Toniolo and Linder, Equation (13)

Average Error: 34.7 → 29.0

Time: 46.8s

Precision: binary64

Cost: 14412

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_2 \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{U* \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)}{Om} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_1 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (sqrt (* U (* n (+ (* (/ n Om) (/ U* Om)) (/ -2.0 Om)))))))
   (if (<= l -4e+126)
     (* (sqrt 2.0) (* t_2 (- l)))
     (if (<= l 2.6e-72)
       (sqrt
        (*
         2.0
         (*
          n
          (* U (+ t (+ (/ (* U* (* (/ l Om) (* l n))) Om) (* -2.0 t_1)))))))
       (if (<= l 6.3e+209)
         (sqrt
          (* 2.0 (* (* U n) (+ t (* t_1 (- -2.0 (* (/ n Om) (- U U*))))))))
         (* (sqrt 2.0) (* l t_2)))))))

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 = l * (l / Om);
	double t_2 = sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))));
	double tmp;
	if (l <= -4e+126) {
		tmp = sqrt(2.0) * (t_2 * -l);
	} else if (l <= 2.6e-72) {
		tmp = sqrt((2.0 * (n * (U * (t + (((U_42_ * ((l / Om) * (l * n))) / Om) + (-2.0 * t_1)))))));
	} else if (l <= 6.3e+209) {
		tmp = sqrt((2.0 * ((U * n) * (t + (t_1 * (-2.0 - ((n / Om) * (U - U_42_))))))));
	} else {
		tmp = sqrt(2.0) * (l * t_2);
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = l * (l / om)
    t_2 = sqrt((u * (n * (((n / om) * (u_42 / om)) + ((-2.0d0) / om)))))
    if (l <= (-4d+126)) then
        tmp = sqrt(2.0d0) * (t_2 * -l)
    else if (l <= 2.6d-72) then
        tmp = sqrt((2.0d0 * (n * (u * (t + (((u_42 * ((l / om) * (l * n))) / om) + ((-2.0d0) * t_1)))))))
    else if (l <= 6.3d+209) then
        tmp = sqrt((2.0d0 * ((u * n) * (t + (t_1 * ((-2.0d0) - ((n / om) * (u - u_42))))))))
    else
        tmp = sqrt(2.0d0) * (l * t_2)
    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 = l * (l / Om);
	double t_2 = Math.sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))));
	double tmp;
	if (l <= -4e+126) {
		tmp = Math.sqrt(2.0) * (t_2 * -l);
	} else if (l <= 2.6e-72) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (((U_42_ * ((l / Om) * (l * n))) / Om) + (-2.0 * t_1)))))));
	} else if (l <= 6.3e+209) {
		tmp = Math.sqrt((2.0 * ((U * n) * (t + (t_1 * (-2.0 - ((n / Om) * (U - U_42_))))))));
	} else {
		tmp = Math.sqrt(2.0) * (l * t_2);
	}
	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 = l * (l / Om)
	t_2 = math.sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))))
	tmp = 0
	if l <= -4e+126:
		tmp = math.sqrt(2.0) * (t_2 * -l)
	elif l <= 2.6e-72:
		tmp = math.sqrt((2.0 * (n * (U * (t + (((U_42_ * ((l / Om) * (l * n))) / Om) + (-2.0 * t_1)))))))
	elif l <= 6.3e+209:
		tmp = math.sqrt((2.0 * ((U * n) * (t + (t_1 * (-2.0 - ((n / Om) * (U - U_42_))))))))
	else:
		tmp = math.sqrt(2.0) * (l * t_2)
	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(l * Float64(l / Om))
	t_2 = sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / Om)))))
	tmp = 0.0
	if (l <= -4e+126)
		tmp = Float64(sqrt(2.0) * Float64(t_2 * Float64(-l)));
	elseif (l <= 2.6e-72)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(Float64(U_42_ * Float64(Float64(l / Om) * Float64(l * n))) / Om) + Float64(-2.0 * t_1)))))));
	elseif (l <= 6.3e+209)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * n) * Float64(t + Float64(t_1 * Float64(-2.0 - Float64(Float64(n / Om) * Float64(U - U_42_))))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * t_2));
	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 = l * (l / Om);
	t_2 = sqrt((U * (n * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om)))));
	tmp = 0.0;
	if (l <= -4e+126)
		tmp = sqrt(2.0) * (t_2 * -l);
	elseif (l <= 2.6e-72)
		tmp = sqrt((2.0 * (n * (U * (t + (((U_42_ * ((l / Om) * (l * n))) / Om) + (-2.0 * t_1)))))));
	elseif (l <= 6.3e+209)
		tmp = sqrt((2.0 * ((U * n) * (t + (t_1 * (-2.0 - ((n / Om) * (U - U_42_))))))));
	else
		tmp = sqrt(2.0) * (l * t_2);
	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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e+126], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e-72], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[(U$42$ * N[(N[(l / Om), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.3e+209], N[Sqrt[N[(2.0 * N[(N[(U * n), $MachinePrecision] * N[(t + N[(t$95$1 * N[(-2.0 - N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.9999999999999997e126

   1. Initial program 59.0

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

   2. Simplified 45.7

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

      [Start] 59.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 58.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 58.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 58.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 58.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 58.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 58.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 45.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 45.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 45.7

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

   4. Simplified 45.7

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

      [Start] 45.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(-U*\right)\right)\right)\right)\right)} \]
      distribute-lft-out [=>] 45.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U + \left(-U*\right)\right)}\right)\right)\right)} \]
      sub-neg [<=] 45.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 60.1

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

   6. Simplified 45.6

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

      [Start] 60.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 60.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 60.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      times-frac [=>] 58.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 58.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 45.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 45.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. Taylor expanded in U around 0 60.7

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

   8. Simplified 45.6

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

      [Start] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/r/ [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      *-commutative [=>] 60.7	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 60.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \color{blue}{\frac{{\ell}^{2} \cdot U*}{\frac{{Om}^{2}}{n}}}\right)\right)\right)\right)} \]
      unpow2 [=>] 60.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{{\ell}^{2} \cdot U*}{\frac{\color{blue}{Om \cdot Om}}{n}}\right)\right)\right)\right)} \]
      associate-*r/ [<=] 60.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{{\ell}^{2} \cdot U*}{\color{blue}{Om \cdot \frac{Om}{n}}}\right)\right)\right)\right)} \]
      *-commutative [=>] 60.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{\color{blue}{U* \cdot {\ell}^{2}}}{Om \cdot \frac{Om}{n}}\right)\right)\right)\right)} \]

   9. Taylor expanded in l around -inf 35.9

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

   10. Simplified 33.1

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

       [Start] 35.9	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
       mul-1-neg [=>] 35.9	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
       associate-*l* [=>] 35.9	\[ -\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
       distribute-rgt-neg-in [=>] 35.9	\[ \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
       associate-*r* [=>] 38.2	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}}\right) \]
       *-commutative [=>] 38.2	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
       cancel-sign-sub-inv [=>] 38.2	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)}\right) \]
       unpow2 [=>] 38.2	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
       times-frac [=>] 33.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
       metadata-eval [=>] 33.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)}\right) \]
       associate-*r/ [=>] 33.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)}\right) \]
       metadata-eval [=>] 33.1	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)}\right) \]

   if -3.9999999999999997e126 < l < 2.59999999999999996e-72

   1. Initial program 27.5

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

   2. Simplified 28.6

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

      [Start] 27.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 28.2	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 28.6	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 28.2

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

   4. Simplified 28.2

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

      [Start] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(-U*\right)\right)\right)\right)\right)} \]
      distribute-lft-out [=>] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U + \left(-U*\right)\right)}\right)\right)\right)} \]
      sub-neg [<=] 28.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 33.2

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

   6. Simplified 28.7

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

      [Start] 33.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 33.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 33.2	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      times-frac [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 28.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. Taylor expanded in U around 0 33.6

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

   8. Simplified 28.8

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

      [Start] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/r/ [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      *-commutative [=>] 33.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 34.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \color{blue}{\frac{{\ell}^{2} \cdot U*}{\frac{{Om}^{2}}{n}}}\right)\right)\right)\right)} \]
      unpow2 [=>] 34.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{{\ell}^{2} \cdot U*}{\frac{\color{blue}{Om \cdot Om}}{n}}\right)\right)\right)\right)} \]
      associate-*r/ [<=] 32.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{{\ell}^{2} \cdot U*}{\color{blue}{Om \cdot \frac{Om}{n}}}\right)\right)\right)\right)} \]
      *-commutative [=>] 32.0	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - \frac{\color{blue}{U* \cdot {\ell}^{2}}}{Om \cdot \frac{Om}{n}}\right)\right)\right)\right)} \]

   9. Applied egg-rr 27.7

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

   if 2.59999999999999996e-72 < l < 6.30000000000000002e209

   1. Initial program 36.2

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

   2. Simplified 34.0

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

      [Start] 36.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 37.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 37.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 33.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 34.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 33.0

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

   4. Simplified 33.0

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

      [Start] 33.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(-U*\right)\right)\right)\right)\right)} \]
      distribute-lft-out [=>] 33.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U + \left(-U*\right)\right)}\right)\right)\right)} \]
      sub-neg [<=] 33.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 38.9

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

   6. Simplified 31.5

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

      [Start] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 38.9	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      times-frac [=>] 35.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 35.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 31.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. Applied egg-rr 47.3

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

   8. Simplified 30.8

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

      [Start] 47.3	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{\ell \cdot \ell}{Om} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\right)} - 1 \]
      expm1-def [=>] 35.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{\ell \cdot \ell}{Om} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 34.8	\[ \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{\ell \cdot \ell}{Om} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      associate-/l* [=>] 30.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 30.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 30.8	\[ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(2 + \color{blue}{\left(U - U*\right) \cdot \frac{n}{Om}}\right)\right)\right)} \]

   if 6.30000000000000002e209 < l

   1. Initial program 64.0

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

   2. Simplified 55.7

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

      [Start] 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	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 55.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Applied egg-rr 55.4

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

   4. Simplified 55.4

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

      [Start] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(-U*\right)\right)\right)\right)\right)} \]
      distribute-lft-out [=>] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U + \left(-U*\right)\right)}\right)\right)\right)} \]
      sub-neg [<=] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 64.0

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

   6. Simplified 55.4

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

      [Start] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      *-commutative [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)\right)} \]
      times-frac [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]
      unpow2 [=>] 64.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/l* [=>] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\frac{\ell}{\frac{Om}{\ell}}} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-/r/ [=>] 55.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. Taylor expanded in U around 0 64.0

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

   8. Simplified 55.4

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

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

   9. Taylor expanded in l around inf 33.7

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

   10. Simplified 28.4

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

       [Start] 33.7	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
       associate-*l* [=>] 33.7	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
       associate-*r* [=>] 35.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}}\right) \]
       *-commutative [=>] 35.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
       cancel-sign-sub-inv [=>] 35.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right)}\right) \]
       unpow2 [=>] 35.6	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
       times-frac [=>] 28.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)\right)}\right) \]
       metadata-eval [=>] 28.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)\right)}\right) \]
       associate-*r/ [=>] 28.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)\right)}\right) \]
       metadata-eval [=>] 28.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{U* \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)}{Om} + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	30.2
Cost	14412

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_1 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{U* \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)}{Om} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+209}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	29.4
Cost	14412

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} + \frac{-2}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{U* \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)}{Om} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_1 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	32.6
Cost	14284

\[\begin{array}{l} t_1 := \sqrt{\left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{\frac{\frac{Om}{n}}{U*}} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U* \leq -2.25 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_2 - U* \cdot \left(\frac{n}{Om} \cdot t_2\right)\right) - t\right)\right)\right)}\\ \mathbf{elif}\;U* \leq -5.7 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq -7.5 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, n \cdot \left(U \cdot t\right), \frac{\left(n \cdot -4\right) \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)}{Om}\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq 4.2 \cdot 10^{+224}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{n} \cdot \frac{Om}{\ell}} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_2 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	32.5
Cost	8917

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ t_3 := \frac{n}{Om} \cdot t_2\\ \mathbf{if}\;U* \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_2 - U* \cdot t_3\right) - t\right)\right)\right)}\\ \mathbf{elif}\;U* \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{\frac{\frac{Om}{n}}{U*}} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;U* \leq 7 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1\right)\right)}\\ \mathbf{elif}\;U* \leq 5 \cdot 10^{-64} \lor \neg \left(U* \leq 6.2 \cdot 10^{+225}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_2 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 - \left(U - U*\right) \cdot t_3\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	32.4
Cost	8917

\[\begin{array}{l} t_1 := -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U* \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_2 - U* \cdot \left(\frac{n}{Om} \cdot t_2\right)\right) - t\right)\right)\right)}\\ \mathbf{elif}\;U* \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{\frac{\frac{Om}{n}}{U*}} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;U* \leq 5 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1\right)\right)}\\ \mathbf{elif}\;U* \leq 6 \cdot 10^{-59} \lor \neg \left(U* \leq 7.5 \cdot 10^{+225}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_2 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t_1 + \frac{\ell}{\frac{Om}{n} \cdot \frac{Om}{\ell}} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	32.5
Cost	8789

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - U* \cdot \left(\frac{n}{Om} \cdot t_1\right)\right) - t\right)\right)\right)}\\ \mathbf{if}\;U* \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U* \leq -4.2 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{\frac{\frac{Om}{n}}{U*}} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;U* \leq 1.26 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 5.4 \cdot 10^{-64} \lor \neg \left(U* \leq 3.4 \cdot 10^{+224}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + t_1 \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	32.4
Cost	8137

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-95} \lor \neg \left(n \leq 1.36 \cdot 10^{-280}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	32.3
Cost	8136

\[\begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell}{Om} \cdot \left(\frac{\ell}{\frac{\frac{Om}{n}}{U*}} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]

Alternative 9
Error	37.4
Cost	7892

\[\begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;U* \leq -3 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 - \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 3.2 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{\ell \cdot 2}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;U* \leq 9.8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U* \leq 1.3 \cdot 10^{+259}:\\ \;\;\;\;n \cdot \frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left(\ell \cdot \ell\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 10
Error	34.6
Cost	7625

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-173} \lor \neg \left(\ell \leq -1.15 \cdot 10^{-292}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 11
Error	33.9
Cost	7492

\[\begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 12
Error	35.0
Cost	7492

\[\begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(U \cdot n\right)\right) \cdot \left(\frac{\ell \cdot 2}{\frac{Om}{\ell}} - t\right)}\\ \end{array} \]

Alternative 13
Error	40.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]

Alternative 14
Error	40.6
Cost	6848

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

Reproduce

herbie shell --seed 2023038 
(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*))))))