Toniolo and Linder, Equation (13)

Average Error: 35.2 → 31.9

Time: 30.1s

Precision: binary64

Cost: 15124

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ t_2 := \sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+171}:\\ \;\;\;\;\ell \cdot \left(-t_2\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - t_1\right) - t_3 \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \left(U \cdot {\ell}^{2}\right) \cdot \left(\frac{n}{Om} \cdot -4\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;\ell \cdot t_2\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+263}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \left(U - U*\right) \cdot \left(n \cdot t_3\right)\right) - t_1\right)\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

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 (* l (/ l Om))))
        (t_2
         (sqrt
          (*
           (* n (* U (+ (* (- U U*) (/ n (pow Om 2.0))) (* 2.0 (/ 1.0 Om)))))
           -2.0)))
        (t_3 (pow (/ l Om) 2.0)))
   (if (<= l -4.1e+171)
     (* l (- t_2))
     (if (<= l 2.1e-83)
       (sqrt (* (* 2.0 (* n U)) (- (- t t_1) (* t_3 (* n (- U U*))))))
       (if (<= l 7.6e+80)
         (sqrt
          (+ (* n (* 2.0 (* U t))) (* (* U (pow l 2.0)) (* (/ n Om) -4.0))))
         (if (<= l 2.3e+247)
           (* l t_2)
           (if (<= l 7.2e+263)
             (sqrt (* (* U (- (- t (* (- U U*) (* n t_3))) t_1)) (+ n n)))
             (* l (sqrt (* (/ (* n U) Om) -4.0))))))))))

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 * (l * (l / Om));
	double t_2 = sqrt(((n * (U * (((U - U_42_) * (n / pow(Om, 2.0))) + (2.0 * (1.0 / Om))))) * -2.0));
	double t_3 = pow((l / Om), 2.0);
	double tmp;
	if (l <= -4.1e+171) {
		tmp = l * -t_2;
	} else if (l <= 2.1e-83) {
		tmp = sqrt(((2.0 * (n * U)) * ((t - t_1) - (t_3 * (n * (U - U_42_))))));
	} else if (l <= 7.6e+80) {
		tmp = sqrt(((n * (2.0 * (U * t))) + ((U * pow(l, 2.0)) * ((n / Om) * -4.0))));
	} else if (l <= 2.3e+247) {
		tmp = l * t_2;
	} else if (l <= 7.2e+263) {
		tmp = sqrt(((U * ((t - ((U - U_42_) * (n * t_3))) - t_1)) * (n + n)));
	} else {
		tmp = l * sqrt((((n * U) / Om) * -4.0));
	}
	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 * (l * (l / om))
    t_2 = sqrt(((n * (u * (((u - u_42) * (n / (om ** 2.0d0))) + (2.0d0 * (1.0d0 / om))))) * (-2.0d0)))
    t_3 = (l / om) ** 2.0d0
    if (l <= (-4.1d+171)) then
        tmp = l * -t_2
    else if (l <= 2.1d-83) then
        tmp = sqrt(((2.0d0 * (n * u)) * ((t - t_1) - (t_3 * (n * (u - u_42))))))
    else if (l <= 7.6d+80) then
        tmp = sqrt(((n * (2.0d0 * (u * t))) + ((u * (l ** 2.0d0)) * ((n / om) * (-4.0d0)))))
    else if (l <= 2.3d+247) then
        tmp = l * t_2
    else if (l <= 7.2d+263) then
        tmp = sqrt(((u * ((t - ((u - u_42) * (n * t_3))) - t_1)) * (n + n)))
    else
        tmp = l * sqrt((((n * u) / om) * (-4.0d0)))
    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 * (l * (l / Om));
	double t_2 = Math.sqrt(((n * (U * (((U - U_42_) * (n / Math.pow(Om, 2.0))) + (2.0 * (1.0 / Om))))) * -2.0));
	double t_3 = Math.pow((l / Om), 2.0);
	double tmp;
	if (l <= -4.1e+171) {
		tmp = l * -t_2;
	} else if (l <= 2.1e-83) {
		tmp = Math.sqrt(((2.0 * (n * U)) * ((t - t_1) - (t_3 * (n * (U - U_42_))))));
	} else if (l <= 7.6e+80) {
		tmp = Math.sqrt(((n * (2.0 * (U * t))) + ((U * Math.pow(l, 2.0)) * ((n / Om) * -4.0))));
	} else if (l <= 2.3e+247) {
		tmp = l * t_2;
	} else if (l <= 7.2e+263) {
		tmp = Math.sqrt(((U * ((t - ((U - U_42_) * (n * t_3))) - t_1)) * (n + n)));
	} else {
		tmp = l * Math.sqrt((((n * U) / Om) * -4.0));
	}
	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 * (l * (l / Om))
	t_2 = math.sqrt(((n * (U * (((U - U_42_) * (n / math.pow(Om, 2.0))) + (2.0 * (1.0 / Om))))) * -2.0))
	t_3 = math.pow((l / Om), 2.0)
	tmp = 0
	if l <= -4.1e+171:
		tmp = l * -t_2
	elif l <= 2.1e-83:
		tmp = math.sqrt(((2.0 * (n * U)) * ((t - t_1) - (t_3 * (n * (U - U_42_))))))
	elif l <= 7.6e+80:
		tmp = math.sqrt(((n * (2.0 * (U * t))) + ((U * math.pow(l, 2.0)) * ((n / Om) * -4.0))))
	elif l <= 2.3e+247:
		tmp = l * t_2
	elif l <= 7.2e+263:
		tmp = math.sqrt(((U * ((t - ((U - U_42_) * (n * t_3))) - t_1)) * (n + n)))
	else:
		tmp = l * math.sqrt((((n * U) / Om) * -4.0))
	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(l * Float64(l / Om)))
	t_2 = sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(U - U_42_) * Float64(n / (Om ^ 2.0))) + Float64(2.0 * Float64(1.0 / Om))))) * -2.0))
	t_3 = Float64(l / Om) ^ 2.0
	tmp = 0.0
	if (l <= -4.1e+171)
		tmp = Float64(l * Float64(-t_2));
	elseif (l <= 2.1e-83)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - t_1) - Float64(t_3 * Float64(n * Float64(U - U_42_))))));
	elseif (l <= 7.6e+80)
		tmp = sqrt(Float64(Float64(n * Float64(2.0 * Float64(U * t))) + Float64(Float64(U * (l ^ 2.0)) * Float64(Float64(n / Om) * -4.0))));
	elseif (l <= 2.3e+247)
		tmp = Float64(l * t_2);
	elseif (l <= 7.2e+263)
		tmp = sqrt(Float64(Float64(U * Float64(Float64(t - Float64(Float64(U - U_42_) * Float64(n * t_3))) - t_1)) * Float64(n + n)));
	else
		tmp = Float64(l * sqrt(Float64(Float64(Float64(n * U) / Om) * -4.0)));
	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 * (l * (l / Om));
	t_2 = sqrt(((n * (U * (((U - U_42_) * (n / (Om ^ 2.0))) + (2.0 * (1.0 / Om))))) * -2.0));
	t_3 = (l / Om) ^ 2.0;
	tmp = 0.0;
	if (l <= -4.1e+171)
		tmp = l * -t_2;
	elseif (l <= 2.1e-83)
		tmp = sqrt(((2.0 * (n * U)) * ((t - t_1) - (t_3 * (n * (U - U_42_))))));
	elseif (l <= 7.6e+80)
		tmp = sqrt(((n * (2.0 * (U * t))) + ((U * (l ^ 2.0)) * ((n / Om) * -4.0))));
	elseif (l <= 2.3e+247)
		tmp = l * t_2;
	elseif (l <= 7.2e+263)
		tmp = sqrt(((U * ((t - ((U - U_42_) * (n * t_3))) - t_1)) * (n + n)));
	else
		tmp = l * sqrt((((n * U) / Om) * -4.0));
	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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -4.1e+171], N[(l * (-t$95$2)), $MachinePrecision], If[LessEqual[l, 2.1e-83], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - t$95$1), $MachinePrecision] - N[(t$95$3 * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.6e+80], N[Sqrt[N[(N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.3e+247], N[(l * t$95$2), $MachinePrecision], If[LessEqual[l, 7.2e+263], N[Sqrt[N[(N[(U * N[(N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(n * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -4.0999999999999996e171

   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 49.9

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

      [Start] 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 [=>] 49.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Taylor expanded in t around 0 64.0

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

   4. Simplified 64.0

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

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

   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 34.3

      \[\leadsto \color{blue}{\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\right)} \]
      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 [=>] 33.3	\[ \ell \cdot \left(-\color{blue}{\sqrt{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right) \cdot -2}}\right) \]
      rational.json-simplify-43 [=>] 34.4	\[ \ell \cdot \left(-\sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \cdot -2}\right) \]
      rational.json-simplify-49 [=>] 34.3	\[ \ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(\color{blue}{\left(U - U*\right) \cdot \frac{n}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\right) \]

   if -4.0999999999999996e171 < l < 2.0999999999999999e-83

   1. Initial program 28.0

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

   2. Simplified 29.7

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

      [Start] 28.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 28.0	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 28.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 27.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 27.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 27.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 29.7	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   if 2.0999999999999999e-83 < l < 7.59999999999999995e80

   1. Initial program 30.9

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

   2. Simplified 30.9

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

      [Start] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 30.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Taylor expanded in Om around inf 35.6

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Simplified 34.7

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \left(U \cdot {\ell}^{2}\right) \cdot \left(\frac{n}{Om} \cdot -4\right)}} \]
      Proof

      [Start] 35.6	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      rational.json-simplify-43 [=>] 35.6	\[ \sqrt{\color{blue}{n \cdot \left(\left(t \cdot U\right) \cdot 2\right)} + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      rational.json-simplify-2 [=>] 35.6	\[ \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(t \cdot U\right)\right)} + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      rational.json-simplify-2 [=>] 35.6	\[ \sqrt{n \cdot \left(2 \cdot \color{blue}{\left(U \cdot t\right)}\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}} \]
      rational.json-simplify-49 [=>] 34.7	\[ \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + -4 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot U\right) \cdot \frac{n}{Om}\right)}} \]
      rational.json-simplify-43 [=>] 34.7	\[ \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \color{blue}{\left({\ell}^{2} \cdot U\right) \cdot \left(\frac{n}{Om} \cdot -4\right)}} \]
      rational.json-simplify-2 [=>] 34.7	\[ \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \color{blue}{\left(U \cdot {\ell}^{2}\right)} \cdot \left(\frac{n}{Om} \cdot -4\right)} \]

   if 7.59999999999999995e80 < l < 2.29999999999999991e247

   1. Initial program 50.4

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

   2. Simplified 39.6

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

      [Start] 50.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 39.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Taylor expanded in t around 0 64.0

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

   4. Simplified 59.5

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

      [Start] 64.0	\[ \left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \color{blue}{\sqrt{-1 \cdot 2}} \cdot \sqrt{n \cdot \left(U \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]
      metadata-eval [=>] 64.0	\[ \sqrt{\color{blue}{-2}} \cdot \sqrt{n \cdot \left(U \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ \sqrt{-2} \cdot \sqrt{n \cdot \color{blue}{\left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot U\right)}} \]
      exponential.json-simplify-20 [=>] 58.6	\[ \color{blue}{\sqrt{\left(n \cdot \left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot U\right)\right) \cdot -2}} \]
      rational.json-simplify-2 [<=] 58.6	\[ \sqrt{\color{blue}{-2 \cdot \left(n \cdot \left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot U\right)\right)}} \]
      rational.json-simplify-43 [=>] 59.3	\[ \sqrt{-2 \cdot \color{blue}{\left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{n \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot \left(U \cdot n\right)\right)}} \]

   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 36.3

      \[\leadsto \color{blue}{\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}} \]
      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 [=>] 36.2	\[ \ell \cdot \color{blue}{\sqrt{\left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right) \cdot -2}} \]
      rational.json-simplify-43 [=>] 36.4	\[ \ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \cdot -2} \]
      rational.json-simplify-49 [=>] 36.3	\[ \ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\color{blue}{\left(U - U*\right) \cdot \frac{n}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2} \]

   if 2.29999999999999991e247 < l < 7.19999999999999956e263

   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 56.1

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

      [Start] 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-2 [=>] 64.0	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 56.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 56.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 56.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 56.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Applied egg-rr 55.7

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

   4. Simplified 55.7

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

      [Start] 55.7	\[ \sqrt{n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) + n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]
      rational.json-simplify-2 [=>] 55.7	\[ \sqrt{\color{blue}{\left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot n} + n \cdot \left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]
      rational.json-simplify-51 [=>] 55.7	\[ \sqrt{\color{blue}{\left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)}} \]
      rational.json-simplify-2 [=>] 55.7	\[ \sqrt{\left(U \cdot \left(\left(t - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \color{blue}{\left(\left(U - U*\right) \cdot n\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)} \]
      rational.json-simplify-43 [=>] 55.7	\[ \sqrt{\left(U \cdot \left(\left(t - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)} \]

   if 7.19999999999999956e263 < 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 58.9

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

      [Start] 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-2 [=>] 64.0	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-49 [=>] 58.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 58.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 58.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 58.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Taylor expanded in n around 0 64.0

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

   4. Applied egg-rr 64.0

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

   5. Taylor expanded in t around 0 64.0

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

   6. Simplified 36.3

      \[\leadsto \color{blue}{\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}} \]
      Proof

      [Start] 64.0	\[ \left(\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2}\right)\right) \cdot \sqrt{\frac{n \cdot U}{Om}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\sqrt{\frac{n \cdot U}{Om}} \cdot \left(\sqrt{2} \cdot \left(\ell \cdot \sqrt{-2}\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \sqrt{\frac{n \cdot U}{Om}} \cdot \color{blue}{\left(\ell \cdot \left(\sqrt{-2} \cdot \sqrt{2}\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{\ell \cdot \left(\left(\sqrt{-2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot U}{Om}}\right)} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \ell \cdot \left(\color{blue}{\sqrt{2 \cdot -2}} \cdot \sqrt{\frac{n \cdot U}{Om}}\right) \]
      metadata-eval [=>] 64.0	\[ \ell \cdot \left(\sqrt{\color{blue}{-4}} \cdot \sqrt{\frac{n \cdot U}{Om}}\right) \]
      exponential.json-simplify-20 [=>] 36.3	\[ \ell \cdot \color{blue}{\sqrt{\frac{n \cdot U}{Om} \cdot -4}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 31.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+171}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \left(U \cdot {\ell}^{2}\right) \cdot \left(\frac{n}{Om} \cdot -4\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+263}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternatives

Alternative 1
Error	31.7
Cost	14728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{+175}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 2
Error	32.3
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+140}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \left(U \cdot \left(\left(U - U*\right) \cdot \frac{n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot -2}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left({\ell}^{2} \cdot \frac{1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 3
Error	32.5
Cost	14340

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;-\ell \cdot \sqrt{\left(U \cdot \left(\left(U* \cdot \frac{n}{-{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot n\right)\right) \cdot -2}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left({\ell}^{2} \cdot \frac{1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 4
Error	34.1
Cost	14092

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 5
Error	32.9
Cost	14092

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.1 \cdot 10^{+130}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 6
Error	32.8
Cost	14092

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+132}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left({\ell}^{2} \cdot \frac{1}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 7
Error	32.9
Cost	13960

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+126}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 8
Error	33.0
Cost	13960

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+126}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 9
Error	35.8
Cost	7240

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+101}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\ell \leq 3.25 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	35.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+105}:\\ \;\;\;\;\ell \cdot \left(-\sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 11
Error	41.4
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \mathbf{if}\;U \leq -1.5 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	38.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \frac{U}{Om}\right) \cdot -4}\\ \end{array} \]

Alternative 13
Error	38.1
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 14
Error	40.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \end{array} \]

Alternative 15
Error	40.6
Cost	6848

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

Reproduce

herbie shell --seed 2023075 
(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*))))))