Toniolo and Linder, Equation (13)

Average Error: 47.4% → 59.1%

Time: 1.1min

Precision: binary64

Cost: 37644.00

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_2\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq 10^{+306}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t + t_1 \cdot -2\right) + t_2\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (+ t (* (/ (* l l) Om) -2.0)) t_2))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ t (* t_1 (+ -2.0 (* (/ n Om) (- U* U))))))))
     (if (<= t_4 1e+306)
       (sqrt (* t_3 (+ (+ t (* t_1 -2.0)) t_2)))
       (if (<= t_4 INFINITY)
         (* (sqrt (* n U)) (sqrt (* 2.0 t)))
         (sqrt
          (*
           2.0
           (*
            (* U (* l (* n l)))
            (+ (/ -2.0 Om) (* (/ n Om) (/ (- U* U) Om)))))))))))

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t + (((l * l) / Om) * -2.0)) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 * (-2.0 + ((n / Om) * (U_42_ - U))))))));
	} else if (t_4 <= 1e+306) {
		tmp = sqrt((t_3 * ((t + (t_1 * -2.0)) + t_2)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((n * U)) * sqrt((2.0 * t));
	} else {
		tmp = sqrt((2.0 * ((U * (l * (n * l))) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))));
	}
	return tmp;
}

Java

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

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t + (((l * l) / Om) * -2.0)) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (t_1 * (-2.0 + ((n / Om) * (U_42_ - U))))))));
	} else if (t_4 <= 1e+306) {
		tmp = Math.sqrt((t_3 * ((t + (t_1 * -2.0)) + t_2)));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((n * U)) * Math.sqrt((2.0 * t));
	} else {
		tmp = Math.sqrt((2.0 * ((U * (l * (n * l))) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t + (((l * l) / Om) * -2.0)) + t_2)
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (t_1 * (-2.0 + ((n / Om) * (U_42_ - U))))))))
	elif t_4 <= 1e+306:
		tmp = math.sqrt((t_3 * ((t + (t_1 * -2.0)) + t_2)))
	elif t_4 <= math.inf:
		tmp = math.sqrt((n * U)) * math.sqrt((2.0 * t))
	else:
		tmp = math.sqrt((2.0 * ((U * (l * (n * l))) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))))
	return tmp

Julia

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

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(t_1 * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U))))))));
	elseif (t_4 <= 1e+306)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t + Float64(t_1 * -2.0)) + t_2)));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(n * U)) * sqrt(Float64(2.0 * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * Float64(l * Float64(n * l))) * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t + (((l * l) / Om) * -2.0)) + t_2);
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 * (-2.0 + ((n / Om) * (U_42_ - U))))))));
	elseif (t_4 <= 1e+306)
		tmp = sqrt((t_3 * ((t + (t_1 * -2.0)) + t_2)));
	elseif (t_4 <= Inf)
		tmp = sqrt((n * U)) * sqrt((2.0 * t));
	else
		tmp = sqrt((2.0 * ((U * (l * (n * l))) * ((-2.0 / Om) + ((n / Om) * ((U_42_ - U) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(t$95$1 * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 1e+306], N[Sqrt[N[(t$95$3 * N[(N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 0.0

   1. Initial program 10.5

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

   2. Applied egg-rr 13.1

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

   3. Taylor expanded in n around 0 11.4

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

   4. Simplified 16.9

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

      [Start] 11.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 11.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 11.4	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)} \]
      times-frac [=>] 14.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 14.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*r/ [<=] 16.9	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied egg-rr 15.3

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

   6. Simplified 41.3

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

      [Start] 15.3	\[ e^{\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\right)} - 1 \]
      expm1-def [=>] 41.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 41.3	\[ \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 + \frac{n}{Om} \cdot \left(U - U*\right)\right)\right)\right)}} \]
      *-commutative [=>] 41.3	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(2 + \color{blue}{\left(U - U*\right) \cdot \frac{n}{Om}}\right)\right)\right)} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 1.00000000000000002e306

   1. Initial program 97.2

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

   2. Applied egg-rr 97.2

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

   if 1.00000000000000002e306 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 0.6

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

   2. Simplified 15.5

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

      [Start] 0.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 3.1	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 3.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 3.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 3.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 3.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 3.1	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 14.7	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 15.5	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in t around inf 6.2

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

   4. Simplified 6.1

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
      Proof

      [Start] 6.2	\[ \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]
      associate-*r* [=>] 6.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]
      *-commutative [=>] 6.1	\[ \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied egg-rr 12.1

      \[\leadsto \color{blue}{\sqrt{U \cdot n} \cdot \sqrt{2 \cdot t}} \]

   6. Simplified 12.1

      \[\leadsto \color{blue}{\sqrt{n \cdot U} \cdot \sqrt{t \cdot 2}} \]
      Proof

      [Start] 12.1	\[ \sqrt{U \cdot n} \cdot \sqrt{2 \cdot t} \]
      *-commutative [=>] 12.1	\[ \sqrt{\color{blue}{n \cdot U}} \cdot \sqrt{2 \cdot t} \]
      *-commutative [=>] 12.1	\[ \sqrt{n \cdot U} \cdot \sqrt{\color{blue}{t \cdot 2}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.4

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

   3. Taylor expanded in n around 0 0.1

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

   4. Simplified 0.6

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

      [Start] 0.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{n \cdot {\ell}^{2}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      *-commutative [=>] 0.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 0.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \frac{{\ell}^{2} \cdot n}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)} \]
      times-frac [=>] 0.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n}{Om}\right)} \cdot \left(U - U*\right)\right)} \]
      unpow2 [=>] 0.2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*r/ [<=] 0.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n}{Om}\right) \cdot \left(U - U*\right)\right)} \]

   5. Taylor expanded in l around inf 4.3

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

   6. Simplified 33.2

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

      [Start] 4.3	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      unpow2 [=>] 4.3	\[ \sqrt{-2 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      times-frac [=>] 8.4	\[ \sqrt{-2 \cdot \left(\left(\color{blue}{\frac{n}{Om} \cdot \frac{U - U*}{Om}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r/ [=>] 8.4	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \color{blue}{\frac{2 \cdot 1}{Om}}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      metadata-eval [=>] 8.4	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{\color{blue}{2}}{Om}\right) \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)\right)} \]
      associate-*r* [=>] 9.1	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot U\right)}\right)} \]
      unpow2 [=>] 9.1	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U\right)\right)} \]
      *-commutative [<=] 9.1	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right)\right)} \]
      associate-*l* [=>] 33.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right)\right)} \]
      *-commutative [=>] 33.2	\[ \sqrt{-2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U - U*}{Om} + \frac{2}{Om}\right) \cdot \left(\left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right) \cdot U\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 59.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	60.1%
Cost	51532.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := t_3 \cdot \left(U* - U\right)\\ t_5 := t_2 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_4\right)\\ t_6 := t_5 \leq \infty\\ \mathbf{if}\;t_5 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_6:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + t_1 \cdot -2\right) + t_4\right)}\\ \mathbf{elif}\;t_6:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, t_1, t_3 \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternative 2
Error	60.1%
Cost	50892.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + t_2\right)\\ t_5 := t_4 \leq \infty\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_5:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t + t_1 \cdot -2\right) + t_2\right)}\\ \mathbf{elif}\;t_5:\\ \;\;\;\;\left|\frac{\sqrt{{\left(n \cdot \ell\right)}^{2} \cdot \left(U \cdot \left(U* - U\right)\right)}}{Om} \cdot \sqrt{2}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternative 3
Error	56.3%
Cost	14408.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) + \left(\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{Om}\right)\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\frac{n}{Om}}{Om} \cdot \left(U* - U\right) + \frac{-2}{Om}\right)}\\ \end{array} \]

Alternative 4
Error	53.1%
Cost	14020.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, \frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right), n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + t_1 \cdot -2\right) + \left(\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{Om}\right)\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \]

Alternative 5
Error	53.9%
Cost	13908.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := t + t_1 \cdot -2\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ t_4 := \left(2 \cdot n\right) \cdot U\\ t_5 := \sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{if}\;U \leq -6 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{t_4 \cdot \left(t_2 + \frac{t_1}{\frac{Om}{n}} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;U \leq 5.6 \cdot 10^{-272}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 4 \cdot 10^{-215}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;U \leq 2.15 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;U \leq 5.1 \cdot 10^{-142}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_4 \cdot \left(t_2 + \left(\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{Om}\right)\right) \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 6
Error	53.9%
Cost	13908.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t + t_1 \cdot -2\\ t_4 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{if}\;U \leq -7 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t_3 + \frac{t_1}{\frac{Om}{n}} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-274}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;U \leq 7.8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;U \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t_3 + \left(\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{Om}\right)\right) \cdot \left(U* - U\right)\right)}\\ \end{array} \]

Alternative 7
Error	51.7%
Cost	8652.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}} + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + t_1 \cdot -2\right) + \left(\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{Om}\right)\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	48.9%
Cost	8268.00

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -5 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 9
Error	53.4%
Cost	8136.00

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U \leq -8.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + t_1 \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + t_1 \cdot -2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 10
Error	46.8%
Cost	8008.00

\[\begin{array}{l} \mathbf{if}\;U \leq -8.4 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 7.8 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 11
Error	47.2%
Cost	7888.00

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot -2\right) \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 12
Error	48.2%
Cost	7625.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-43} \lor \neg \left(\ell \leq 1.2 \cdot 10^{-199}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 13
Error	49.9%
Cost	7624.00

\[\begin{array}{l} \mathbf{if}\;U \leq -5.1 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 9.6 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 14
Error	42.3%
Cost	7496.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{2 \cdot \left(\ell \cdot \left(\left(\ell \cdot \left(n \cdot U\right)\right) \cdot \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 16500:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 15
Error	39.5%
Cost	7364.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq 16500:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(U \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 16
Error	38.0%
Cost	6848.00

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

Alternative 17
Error	38.6%
Cost	6848.00

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

Reproduce

herbie shell --seed 2023093 
(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*))))))