Toniolo and Linder, Equation (13)

Average Error: 34.5 → 29.9

Time: 29.3s

Precision: binary64

Cost: 14860

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \frac{1}{Om}\\ t_2 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+172}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(t_1 + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - t_2\right) - t_3 \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \left(U - U*\right) \cdot \left(t_3 \cdot n\right)\right) - t_2\right)\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{-{Om}^{2}} + t_1\right)\right)\right) \cdot -2}\\ \end{array} \]

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 (* 2.0 (/ 1.0 Om)))
        (t_2 (* 2.0 (* l (/ l Om))))
        (t_3 (pow (/ l Om) 2.0)))
   (if (<= l -3.8e+172)
     (*
      l
      (- (sqrt (* (* n (* U (+ t_1 (* n (/ (- U U*) (pow Om 2.0)))))) -2.0))))
     (if (<= l -5e-96)
       (sqrt (* (* 2.0 (* n U)) (- (- t t_2) (* t_3 (* n (- U U*))))))
       (if (<= l 4.9e+110)
         (sqrt (* (* U (- (- t (* (- U U*) (* t_3 n))) t_2)) (+ n n)))
         (*
          l
          (sqrt
           (* (* n (* U (+ (* U* (/ n (- (pow Om 2.0)))) t_1))) -2.0))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (1.0 / Om);
	double t_2 = 2.0 * (l * (l / Om));
	double t_3 = pow((l / Om), 2.0);
	double tmp;
	if (l <= -3.8e+172) {
		tmp = l * -sqrt(((n * (U * (t_1 + (n * ((U - U_42_) / pow(Om, 2.0)))))) * -2.0));
	} else if (l <= -5e-96) {
		tmp = sqrt(((2.0 * (n * U)) * ((t - t_2) - (t_3 * (n * (U - U_42_))))));
	} else if (l <= 4.9e+110) {
		tmp = sqrt(((U * ((t - ((U - U_42_) * (t_3 * n))) - t_2)) * (n + n)));
	} else {
		tmp = l * sqrt(((n * (U * ((U_42_ * (n / -pow(Om, 2.0))) + t_1))) * -2.0));
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * (1.0d0 / om)
    t_2 = 2.0d0 * (l * (l / om))
    t_3 = (l / om) ** 2.0d0
    if (l <= (-3.8d+172)) then
        tmp = l * -sqrt(((n * (u * (t_1 + (n * ((u - u_42) / (om ** 2.0d0)))))) * (-2.0d0)))
    else if (l <= (-5d-96)) then
        tmp = sqrt(((2.0d0 * (n * u)) * ((t - t_2) - (t_3 * (n * (u - u_42))))))
    else if (l <= 4.9d+110) then
        tmp = sqrt(((u * ((t - ((u - u_42) * (t_3 * n))) - t_2)) * (n + n)))
    else
        tmp = l * sqrt(((n * (u * ((u_42 * (n / -(om ** 2.0d0))) + t_1))) * (-2.0d0)))
    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 = 2.0 * (1.0 / Om);
	double t_2 = 2.0 * (l * (l / Om));
	double t_3 = Math.pow((l / Om), 2.0);
	double tmp;
	if (l <= -3.8e+172) {
		tmp = l * -Math.sqrt(((n * (U * (t_1 + (n * ((U - U_42_) / Math.pow(Om, 2.0)))))) * -2.0));
	} else if (l <= -5e-96) {
		tmp = Math.sqrt(((2.0 * (n * U)) * ((t - t_2) - (t_3 * (n * (U - U_42_))))));
	} else if (l <= 4.9e+110) {
		tmp = Math.sqrt(((U * ((t - ((U - U_42_) * (t_3 * n))) - t_2)) * (n + n)));
	} else {
		tmp = l * Math.sqrt(((n * (U * ((U_42_ * (n / -Math.pow(Om, 2.0))) + t_1))) * -2.0));
	}
	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 = 2.0 * (1.0 / Om)
	t_2 = 2.0 * (l * (l / Om))
	t_3 = math.pow((l / Om), 2.0)
	tmp = 0
	if l <= -3.8e+172:
		tmp = l * -math.sqrt(((n * (U * (t_1 + (n * ((U - U_42_) / math.pow(Om, 2.0)))))) * -2.0))
	elif l <= -5e-96:
		tmp = math.sqrt(((2.0 * (n * U)) * ((t - t_2) - (t_3 * (n * (U - U_42_))))))
	elif l <= 4.9e+110:
		tmp = math.sqrt(((U * ((t - ((U - U_42_) * (t_3 * n))) - t_2)) * (n + n)))
	else:
		tmp = l * math.sqrt(((n * (U * ((U_42_ * (n / -math.pow(Om, 2.0))) + t_1))) * -2.0))
	return tmp

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(1.0 / Om))
	t_2 = Float64(2.0 * Float64(l * Float64(l / Om)))
	t_3 = Float64(l / Om) ^ 2.0
	tmp = 0.0
	if (l <= -3.8e+172)
		tmp = Float64(l * Float64(-sqrt(Float64(Float64(n * Float64(U * Float64(t_1 + Float64(n * Float64(Float64(U - U_42_) / (Om ^ 2.0)))))) * -2.0))));
	elseif (l <= -5e-96)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - t_2) - Float64(t_3 * Float64(n * Float64(U - U_42_))))));
	elseif (l <= 4.9e+110)
		tmp = sqrt(Float64(Float64(U * Float64(Float64(t - Float64(Float64(U - U_42_) * Float64(t_3 * n))) - t_2)) * Float64(n + n)));
	else
		tmp = Float64(l * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(U_42_ * Float64(n / Float64(-(Om ^ 2.0)))) + t_1))) * -2.0)));
	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 = 2.0 * (1.0 / Om);
	t_2 = 2.0 * (l * (l / Om));
	t_3 = (l / Om) ^ 2.0;
	tmp = 0.0;
	if (l <= -3.8e+172)
		tmp = l * -sqrt(((n * (U * (t_1 + (n * ((U - U_42_) / (Om ^ 2.0)))))) * -2.0));
	elseif (l <= -5e-96)
		tmp = sqrt(((2.0 * (n * U)) * ((t - t_2) - (t_3 * (n * (U - U_42_))))));
	elseif (l <= 4.9e+110)
		tmp = sqrt(((U * ((t - ((U - U_42_) * (t_3 * n))) - t_2)) * (n + n)));
	else
		tmp = l * sqrt(((n * (U * ((U_42_ * (n / -(Om ^ 2.0))) + t_1))) * -2.0));
	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[(2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -3.8e+172], N[(l * (-N[Sqrt[N[(N[(n * N[(U * N[(t$95$1 + N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-96], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - t$95$2), $MachinePrecision] - N[(t$95$3 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.9e+110], N[Sqrt[N[(N[(U * N[(N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$3 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[Sqrt[N[(N[(n * N[(U * N[(N[(U$42$ * N[(n / (-N[Power[Om, 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.7999999999999997e172

   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-49 [=>] 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. Taylor expanded in l around inf 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \sqrt{\color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \sqrt{\left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      rational.json-simplify-49 [=>] 64.0	\[ \sqrt{\left(\color{blue}{n \cdot \frac{U - U*}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ \sqrt{\left(n \cdot \frac{U - U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \color{blue}{\left(-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \sqrt{\left(n \cdot \frac{U - U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)\right)} \]

   5. Taylor expanded in l around -inf 64.0

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

   6. Simplified 35.2

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

      [Start] 64.0	\[ -1 \cdot \left(\sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{-2}\right)\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ -1 \cdot \color{blue}{\left(\ell \cdot \left(\sqrt{-2} \cdot \sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{\ell \cdot \left(\left(\sqrt{-2} \cdot \sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right) \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 64.0	\[ \ell \cdot \color{blue}{\left(-\sqrt{-2} \cdot \sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right)} \]
      exponential.json-simplify-20 [=>] 32.3	\[ \ell \cdot \left(-\color{blue}{\sqrt{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right) \cdot -2}}\right) \]

   if -3.7999999999999997e172 < l < -4.99999999999999995e-96

   1. Initial program 32.6

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

   2. Simplified 33.3

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

      [Start] 32.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 32.6	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 32.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 31.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 31.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 31.2	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 33.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   if -4.99999999999999995e-96 < l < 4.90000000000000002e110

   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 28.7

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}} \]
      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-2 [=>] 26.5	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 26.5	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 26.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 26.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 26.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 28.7	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Applied egg-rr 29.6

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

   4. Simplified 26.8

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

      [Start] 29.6	\[ \sqrt{n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) + n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]
      rational.json-simplify-2 [=>] 29.6	\[ \sqrt{\color{blue}{\left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot n} + n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]
      rational.json-simplify-51 [=>] 29.5	\[ \sqrt{\color{blue}{\left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)}} \]
      rational.json-simplify-43 [=>] 27.1	\[ \sqrt{\left(U \cdot \left(\left(t - \color{blue}{n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)} \]
      rational.json-simplify-43 [=>] 26.8	\[ \sqrt{\left(U \cdot \left(\left(t - \color{blue}{\left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)} \]

   if 4.90000000000000002e110 < l

   1. Initial program 57.8

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

   2. Simplified 46.9

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

      [Start] 57.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 46.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Taylor expanded in l around inf 59.3

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

   4. Simplified 59.5

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

      [Start] 59.3	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      rational.json-simplify-43 [=>] 59.3	\[ \sqrt{\color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)}} \]
      rational.json-simplify-2 [=>] 59.3	\[ \sqrt{\left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      rational.json-simplify-49 [=>] 59.5	\[ \sqrt{\left(\color{blue}{n \cdot \frac{U - U*}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(\left(n \cdot \left({\ell}^{2} \cdot U\right)\right) \cdot -2\right)} \]
      rational.json-simplify-2 [=>] 59.5	\[ \sqrt{\left(n \cdot \frac{U - U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \color{blue}{\left(-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)}} \]
      rational.json-simplify-2 [=>] 59.5	\[ \sqrt{\left(n \cdot \frac{U - U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-2 \cdot \left(n \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)\right)} \]

   5. Taylor expanded in l around 0 64.0

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

   6. Simplified 36.8

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

      [Start] 64.0	\[ \sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{-2}\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{\ell \cdot \left(\sqrt{-2} \cdot \sqrt{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)}\right)} \]
      exponential.json-simplify-20 [=>] 35.1	\[ \ell \cdot \color{blue}{\sqrt{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right) \cdot -2}} \]
      rational.json-simplify-43 [=>] 35.7	\[ \ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \cdot -2} \]
      rational.json-simplify-1 [=>] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \color{blue}{\left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)}\right)\right) \cdot -2} \]
      rational.json-simplify-2 [=>] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(2 \cdot \frac{1}{Om} + \frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-49 [=>] 36.8	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(2 \cdot \frac{1}{Om} + \color{blue}{n \cdot \frac{U - U*}{{Om}^{2}}}\right)\right)\right) \cdot -2} \]

   7. Taylor expanded in U around 0 36.1

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

   8. Simplified 35.7

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

      [Start] 36.1	\[ \ell \cdot \sqrt{\left(n \cdot \left(\left(-1 \cdot \frac{n \cdot U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right) \cdot -2} \]
      rational.json-simplify-2 [=>] 36.1	\[ \ell \cdot \sqrt{\left(n \cdot \color{blue}{\left(U \cdot \left(-1 \cdot \frac{n \cdot U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\right) \cdot -2} \]
      rational.json-simplify-49 [=>] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(-1 \cdot \color{blue}{\left(U* \cdot \frac{n}{{Om}^{2}}\right)} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-43 [=>] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\color{blue}{U* \cdot \left(\frac{n}{{Om}^{2}} \cdot -1\right)} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-8 [<=] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \color{blue}{\left(-\frac{n}{{Om}^{2}}\right)} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-10 [=>] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \color{blue}{\frac{\frac{n}{{Om}^{2}}}{-1}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-46 [<=] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \color{blue}{\frac{n}{{Om}^{2} \cdot -1}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]
      rational.json-simplify-8 [<=] 35.7	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{\color{blue}{-{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]

3. Recombined 4 regimes into one program.

4. Final simplification 29.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+172}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(2 \cdot \frac{1}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{-{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.0
Cost	14728

\[\begin{array}{l} t_1 := 2 \cdot \frac{1}{Om}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+174}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(t_1 + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{-{Om}^{2}} + t_1\right)\right)\right) \cdot -2}\\ \end{array} \]

Alternative 2
Error	32.7
Cost	14672

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;-\ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{-{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\\ \end{array} \]

Alternative 3
Error	32.3
Cost	14672

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ t_2 := 2 \cdot \frac{1}{Om}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(t_2 + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(U* \cdot \frac{n}{-{Om}^{2}} + t_2\right)\right)\right) \cdot -2}\\ \end{array} \]

Alternative 4
Error	32.8
Cost	14224

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)\right)}\\ t_2 := \ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	33.5
Cost	14224

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-\ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 6
Error	36.8
Cost	13960

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ t_2 := \sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{-4 \cdot \left(n \cdot \frac{\frac{1}{Om}}{\frac{\frac{1}{U}}{{\ell}^{2}}}\right)}\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \sqrt{U* \cdot U}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(U \cdot U*\right) \cdot 2}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	36.8
Cost	13900

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ t_2 := \sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\ell \leq -5.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -0.000105:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \sqrt{U* \cdot U}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(U \cdot U*\right) \cdot 2}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	36.6
Cost	7504

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ t_2 := \ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+180}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(U \cdot U*\right) \cdot 2}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	36.4
Cost	7240

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+180}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	39.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq 10^{-266}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 10000000:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}\\ \end{array} \]

Alternative 11
Error	38.3
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq 5.9 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 12
Error	38.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.85 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \end{array} \]

Alternative 13
Error	39.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(U \cdot t\right) \cdot 2\right)}\\ \end{array} \]

Alternative 14
Error	40.3
Cost	6848

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

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*))))))