Toniolo and Linder, Equation (13)

Average Error: 34.5 → 29.9

Time: 51.8s

Precision: binary64

Cost: 21132

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{1}{\frac{1}{-1 \cdot t_2}}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{\frac{\frac{-0.5}{n}}{U}}{t_1 \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\ell \cdot \frac{2}{\frac{Om}{\ell}} - t\right)}}}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot t_1\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_2}}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2
         (*
          (* (sqrt 2.0) l)
          (sqrt (* n (* U (- (* (- U* U) (/ n (pow Om 2.0))) (/ 2.0 Om))))))))
   (if (<= l -3.3e+172)
     (/ 1.0 (/ 1.0 (* -1.0 t_2)))
     (if (<= l -4.4e-101)
       (sqrt
        (/
         1.0
         (/
          (/ (/ -0.5 n) U)
          (+ (* t_1 (* n (- U U*))) (- (* l (/ 2.0 (/ Om l))) t)))))
       (if (<= l 7.2e+110)
         (sqrt
          (*
           (* 2.0 n)
           (* (- (- t (* 2.0 (* l (/ l Om)))) (* (- U U*) (* n t_1))) U)))
         (/ 1.0 (/ 1.0 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 = pow((l / Om), 2.0);
	double t_2 = (sqrt(2.0) * l) * sqrt((n * (U * (((U_42_ - U) * (n / pow(Om, 2.0))) - (2.0 / Om)))));
	double tmp;
	if (l <= -3.3e+172) {
		tmp = 1.0 / (1.0 / (-1.0 * t_2));
	} else if (l <= -4.4e-101) {
		tmp = sqrt((1.0 / (((-0.5 / n) / U) / ((t_1 * (n * (U - U_42_))) + ((l * (2.0 / (Om / l))) - t)))));
	} else if (l <= 7.2e+110) {
		tmp = sqrt(((2.0 * n) * (((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * (n * t_1))) * U)));
	} else {
		tmp = 1.0 / (1.0 / 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 / om) ** 2.0d0
    t_2 = (sqrt(2.0d0) * l) * sqrt((n * (u * (((u_42 - u) * (n / (om ** 2.0d0))) - (2.0d0 / om)))))
    if (l <= (-3.3d+172)) then
        tmp = 1.0d0 / (1.0d0 / ((-1.0d0) * t_2))
    else if (l <= (-4.4d-101)) then
        tmp = sqrt((1.0d0 / ((((-0.5d0) / n) / u) / ((t_1 * (n * (u - u_42))) + ((l * (2.0d0 / (om / l))) - t)))))
    else if (l <= 7.2d+110) then
        tmp = sqrt(((2.0d0 * n) * (((t - (2.0d0 * (l * (l / om)))) - ((u - u_42) * (n * t_1))) * u)))
    else
        tmp = 1.0d0 / (1.0d0 / 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 = Math.pow((l / Om), 2.0);
	double t_2 = (Math.sqrt(2.0) * l) * Math.sqrt((n * (U * (((U_42_ - U) * (n / Math.pow(Om, 2.0))) - (2.0 / Om)))));
	double tmp;
	if (l <= -3.3e+172) {
		tmp = 1.0 / (1.0 / (-1.0 * t_2));
	} else if (l <= -4.4e-101) {
		tmp = Math.sqrt((1.0 / (((-0.5 / n) / U) / ((t_1 * (n * (U - U_42_))) + ((l * (2.0 / (Om / l))) - t)))));
	} else if (l <= 7.2e+110) {
		tmp = Math.sqrt(((2.0 * n) * (((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * (n * t_1))) * U)));
	} else {
		tmp = 1.0 / (1.0 / 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 = math.pow((l / Om), 2.0)
	t_2 = (math.sqrt(2.0) * l) * math.sqrt((n * (U * (((U_42_ - U) * (n / math.pow(Om, 2.0))) - (2.0 / Om)))))
	tmp = 0
	if l <= -3.3e+172:
		tmp = 1.0 / (1.0 / (-1.0 * t_2))
	elif l <= -4.4e-101:
		tmp = math.sqrt((1.0 / (((-0.5 / n) / U) / ((t_1 * (n * (U - U_42_))) + ((l * (2.0 / (Om / l))) - t)))))
	elif l <= 7.2e+110:
		tmp = math.sqrt(((2.0 * n) * (((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * (n * t_1))) * U)))
	else:
		tmp = 1.0 / (1.0 / 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 / Om) ^ 2.0
	t_2 = Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(U_42_ - U) * Float64(n / (Om ^ 2.0))) - Float64(2.0 / Om))))))
	tmp = 0.0
	if (l <= -3.3e+172)
		tmp = Float64(1.0 / Float64(1.0 / Float64(-1.0 * t_2)));
	elseif (l <= -4.4e-101)
		tmp = sqrt(Float64(1.0 / Float64(Float64(Float64(-0.5 / n) / U) / Float64(Float64(t_1 * Float64(n * Float64(U - U_42_))) + Float64(Float64(l * Float64(2.0 / Float64(Om / l))) - t)))));
	elseif (l <= 7.2e+110)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(Float64(U - U_42_) * Float64(n * t_1))) * U)));
	else
		tmp = Float64(1.0 / Float64(1.0 / 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 / Om) ^ 2.0;
	t_2 = (sqrt(2.0) * l) * sqrt((n * (U * (((U_42_ - U) * (n / (Om ^ 2.0))) - (2.0 / Om)))));
	tmp = 0.0;
	if (l <= -3.3e+172)
		tmp = 1.0 / (1.0 / (-1.0 * t_2));
	elseif (l <= -4.4e-101)
		tmp = sqrt((1.0 / (((-0.5 / n) / U) / ((t_1 * (n * (U - U_42_))) + ((l * (2.0 / (Om / l))) - t)))));
	elseif (l <= 7.2e+110)
		tmp = sqrt(((2.0 * n) * (((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * (n * t_1))) * U)));
	else
		tmp = 1.0 / (1.0 / 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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.3e+172], N[(1.0 / N[(1.0 / N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.4e-101], N[Sqrt[N[(1.0 / N[(N[(N[(-0.5 / n), $MachinePrecision] / U), $MachinePrecision] / N[(N[(t$95$1 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.2e+110], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.29999999999999983e172

   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 50.1

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

      [Start] 64.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 50.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 50.0

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

   4. Taylor expanded in l around -inf 33.3

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

   5. Simplified 33.5

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

      [Start] 33.3	\[ \frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)}} \]
      rational.json-simplify-50 [=>] 33.3	\[ \frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right)}} \]
      rational.json-simplify-43 [=>] 33.5	\[ \frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)}} \]
      rational.json-simplify-41 [=>] 33.5	\[ \frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \color{blue}{\frac{2}{\frac{Om}{1}}}\right)\right)}\right)}} \]
      rational.json-simplify-26 [=>] 33.5	\[ \frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{\color{blue}{Om}}\right)\right)}\right)}} \]

   if -3.29999999999999983e172 < l < -4.3999999999999998e-101

   1. Initial program 32.5

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

   2. Simplified 32.6

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

      [Start] 32.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-47 [=>] 32.5	\[ \sqrt{\color{blue}{\left(-\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]
      rational.json-simplify-17 [=>] 32.5	\[ \sqrt{\color{blue}{\frac{\left(2 \cdot n\right) \cdot U}{-1}} \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-43 [=>] 32.5	\[ \sqrt{\color{blue}{\left(U \cdot \frac{2 \cdot n}{-1}\right)} \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-43 [=>] 32.5	\[ \sqrt{\left(U \cdot \color{blue}{\left(n \cdot \frac{2}{-1}\right)}\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      metadata-eval [=>] 32.5	\[ \sqrt{\left(U \cdot \left(n \cdot \color{blue}{-2}\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-50 [=>] 32.5	\[ \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-24 [=>] 33.9	\[ \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\color{blue}{n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-50 [=>] 33.9	\[ \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} - \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]
      rational.json-simplify-43 [=>] 32.6	\[ \sqrt{\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) - \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   3. Applied egg-rr 33.8

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\frac{\frac{-0.5}{n}}{U}}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\ell \cdot \frac{2}{\frac{Om}{\ell}} - t\right)}}}} \]

   if -4.3999999999999998e-101 < l < 7.1999999999999994e110

   1. Initial program 26.5

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

   2. Simplified 27.1

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

      [Start] 26.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-50 [=>] 26.5	\[ \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]
      rational.json-simplify-24 [=>] 26.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]
      rational.json-simplify-43 [=>] 26.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)} \]
      rational.json-simplify-50 [=>] 26.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right) \cdot U\right)} \]
      rational.json-simplify-24 [=>] 27.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right) \cdot U\right)} \]

   3. Applied egg-rr 29.5

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + 0\right)}\right) \cdot U\right)} \]

   4. Simplified 26.8

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

      [Start] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + 0\right)\right) \cdot U\right)} \]
      rational.json-simplify-8 [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right) \cdot U\right)} \]
      rational.json-simplify-50 [=>] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot U\right)} \]
      rational.json-simplify-26 [<=] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \color{blue}{\frac{{\left(\frac{\ell}{Om}\right)}^{2}}{1}}\right) \cdot U\right)} \]
      rational.json-simplify-27 [<=] 29.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\frac{n \cdot \left(U - U*\right)}{\frac{1}{{\left(\frac{\ell}{Om}\right)}^{2}}}}\right) \cdot U\right)} \]
      rational.json-simplify-43 [=>] 26.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \frac{n}{\frac{1}{{\left(\frac{\ell}{Om}\right)}^{2}}}}\right) \cdot U\right)} \]
      rational.json-simplify-27 [=>] 26.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left(n \cdot \frac{{\left(\frac{\ell}{Om}\right)}^{2}}{1}\right)}\right) \cdot U\right)} \]
      rational.json-simplify-26 [=>] 26.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right) \cdot U\right)} \]

   if 7.1999999999999994e110 < l

   1. Initial program 57.9

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

   2. Simplified 47.0

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

      [Start] 57.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 47.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 47.0

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

   4. Taylor expanded in l around inf 35.8

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

   5. Simplified 35.6

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

      [Start] 35.8	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}} \]
      rational.json-simplify-50 [<=] 35.8	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}}}} \]
      rational.json-simplify-50 [=>] 35.8	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}}} \]
      rational.json-simplify-43 [=>] 35.6	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\color{blue}{\left(U* - U\right) \cdot \frac{n}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)\right)}}} \]
      rational.json-simplify-41 [=>] 35.6	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \color{blue}{\frac{2}{\frac{Om}{1}}}\right)\right)}}} \]
      rational.json-simplify-26 [=>] 35.6	\[ \frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{\color{blue}{Om}}\right)\right)}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 29.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{1}{\frac{1}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\right)}}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{\frac{\frac{-0.5}{n}}{U}}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\ell \cdot \frac{2}{\frac{Om}{\ell}} - t\right)}}}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	28.6
Cost	39500

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t_2\right) \cdot \left(U - U*\right)\\ t_4 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_3\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(U - \left(U - t \cdot \left(U + U\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_3\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right) + n \cdot \left(t_2 \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(U* - U\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{n \cdot \left(\ell \cdot \sqrt{2}\right)}{\frac{Om}{\sqrt{U \cdot \left(U* - U\right)}}}}}\\ \end{array} \]

Alternative 2
Error	28.9
Cost	30728

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(U - \left(U - t \cdot \left(U + U\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{n \cdot \left(\ell \cdot \sqrt{2}\right)}{\frac{Om}{\sqrt{U \cdot \left(U* - U\right)}}}}}\\ \end{array} \]

Alternative 3
Error	30.6
Cost	21132

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+210}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot \sqrt{\frac{-n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -5.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right) + n \cdot \left(t_1 \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(U* - U\right)\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot t_1\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2} \cdot \left(\sqrt{U \cdot \left(n \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \ell\right)}}\\ \end{array} \]

Alternative 4
Error	30.5
Cost	21132

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+209}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot \sqrt{\frac{-n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right) + n \cdot \left(t_1 \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(U* - U\right)\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot t_1\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\left(U* - U\right) \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 5
Error	31.6
Cost	15124

\[\begin{array}{l} t_1 := n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := n \cdot t_2\\ t_4 := t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;U \leq -5.8 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{-U \cdot \left(\left(n + n\right) \cdot \left(\left(U - U*\right) \cdot t_3 + \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 3 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t_4 - n \cdot \left(\left(U - U*\right) \cdot t_2\right)\right) \cdot U\right)}\\ \mathbf{elif}\;U \leq 10^{-198}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right) \cdot \frac{n}{0.5} - \left(U - U*\right) \cdot \left(\left(n + n\right) \cdot t_3\right)\right)}\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{t_1 + -4 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \frac{n}{Om}\right)}\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t_4 - t_3 \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1}\\ \end{array} \]

Alternative 6
Error	32.5
Cost	14992

\[\begin{array}{l} t_1 := t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t_1 - \left(U - U*\right) \cdot \left(n \cdot t_2\right)\right) \cdot U\right)}\\ \mathbf{if}\;Om \leq -1.05 \cdot 10^{+84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(t - \frac{\ell \cdot \left(2 \cdot \ell\right)}{Om}\right) - t_2 \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;Om \leq -2.7 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t_1 - n \cdot \left(\left(U - U*\right) \cdot t_2\right)\right) \cdot U\right)}\\ \mathbf{elif}\;Om \leq 2.9 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\frac{\sqrt{2}}{Om}}{\frac{1}{n \cdot \ell} \cdot \frac{1}{\sqrt{U \cdot \left(U* - U\right)}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	31.7
Cost	14860

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left(\left(U - U*\right) \cdot t_1\right)\right) \cdot U\right)}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+209}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot \sqrt{\frac{-n \cdot U}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.12 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(t - \frac{\ell \cdot \left(2 \cdot \ell\right)}{Om}\right) - t_1 \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	32.8
Cost	14728

\[\begin{array}{l} t_1 := \sqrt{\frac{-n \cdot U}{Om}}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(t - \frac{\ell \cdot \left(2 \cdot \ell\right)}{Om}\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \ell\right)\\ \end{array} \]

Alternative 9
Error	32.9
Cost	13960

\[\begin{array}{l} t_1 := \sqrt{\frac{-n \cdot U}{Om}}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \ell\right)\\ \end{array} \]

Alternative 10
Error	36.3
Cost	7304

\[\begin{array}{l} t_1 := \sqrt{\frac{-n \cdot U}{Om}}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+180}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \ell\right)\\ \end{array} \]

Alternative 11
Error	39.7
Cost	7240

\[\begin{array}{l} \mathbf{if}\;n \leq 9 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U}{\frac{1}{t \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 12
Error	39.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;n \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{0.5}{\frac{\frac{0.25}{t \cdot U}}{n}}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U}{\frac{1}{t \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 13
Error	39.7
Cost	7240

\[\begin{array}{l} \mathbf{if}\;n \leq 10^{-265}:\\ \;\;\;\;\sqrt{\frac{\frac{U}{\frac{0.5}{t}}}{\frac{1}{n}}}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U}{\frac{1}{t \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 14
Error	38.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+180}:\\ \;\;\;\;-2 \cdot \left(\ell \cdot \sqrt{\frac{-n \cdot U}{Om}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\frac{\frac{0.25}{t \cdot U}}{n}}}\\ \end{array} \]

Alternative 15
Error	39.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{U \cdot \frac{t}{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 16
Error	39.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq 3.2 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{U}{\frac{0.5}{t \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 17
Error	40.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{U \cdot \frac{t}{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Error	40.1
Cost	6848

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

Reproduce

herbie shell --seed 2023073 
(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*))))))