Toniolo and Linder, Equation (13)

Average Accuracy: 46.1% → 59.3%

Time: 43.1s

Precision: binary64

Cost: 44428.00

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left|n \cdot \frac{\ell}{Om}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

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
         (*
          (* (* 2.0 n) U)
          (+
           (+ t (* (/ (* l l) Om) -2.0))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 1e-311)
     (*
      (sqrt (* U (+ t (* (/ l Om) (+ (* (* l U*) (/ n Om)) (* l -2.0))))))
      (sqrt (* 2.0 n)))
     (if (<= t_1 2e+304)
       (sqrt t_1)
       (if (<= t_1 INFINITY)
         (* (sqrt 2.0) (* (sqrt (* U (- U* U))) (fabs (* n (/ l Om)))))
         (*
          (sqrt 2.0)
          (* l (sqrt (* U (* n (+ (* U* (/ (/ n Om) Om)) (/ -2.0 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 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 1e-311) {
		tmp = sqrt((U * (t + ((l / Om) * (((l * U_42_) * (n / Om)) + (l * -2.0)))))) * sqrt((2.0 * n));
	} else if (t_1 <= 2e+304) {
		tmp = sqrt(t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * (sqrt((U * (U_42_ - U))) * fabs((n * (l / Om))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((U * (n * ((U_42_ * ((n / Om) / Om)) + (-2.0 / 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 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 1e-311) {
		tmp = Math.sqrt((U * (t + ((l / Om) * (((l * U_42_) * (n / Om)) + (l * -2.0)))))) * Math.sqrt((2.0 * n));
	} else if (t_1 <= 2e+304) {
		tmp = Math.sqrt(t_1);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(2.0) * (Math.sqrt((U * (U_42_ - U))) * Math.abs((n * (l / Om))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((U * (n * ((U_42_ * ((n / Om) / Om)) + (-2.0 / 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 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_1 <= 1e-311:
		tmp = math.sqrt((U * (t + ((l / Om) * (((l * U_42_) * (n / Om)) + (l * -2.0)))))) * math.sqrt((2.0 * n))
	elif t_1 <= 2e+304:
		tmp = math.sqrt(t_1)
	elif t_1 <= math.inf:
		tmp = math.sqrt(2.0) * (math.sqrt((U * (U_42_ - U))) * math.fabs((n * (l / Om))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((U * (n * ((U_42_ * ((n / Om) / Om)) + (-2.0 / 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(Float64(Float64(2.0 * n) * U) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_1 <= 1e-311)
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(Float64(Float64(l * U_42_) * Float64(n / Om)) + Float64(l * -2.0)))))) * sqrt(Float64(2.0 * n)));
	elseif (t_1 <= 2e+304)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(U * Float64(U_42_ - U))) * abs(Float64(n * Float64(l / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(Float64(n / Om) / Om)) + Float64(-2.0 / 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 = ((2.0 * n) * U) * ((t + (((l * l) / Om) * -2.0)) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_1 <= 1e-311)
		tmp = sqrt((U * (t + ((l / Om) * (((l * U_42_) * (n / Om)) + (l * -2.0)))))) * sqrt((2.0 * n));
	elseif (t_1 <= 2e+304)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = sqrt(2.0) * (sqrt((U * (U_42_ - U))) * abs((n * (l / Om))));
	else
		tmp = sqrt(2.0) * (l * sqrt((U * (n * ((U_42_ * ((n / Om) / Om)) + (-2.0 / 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[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-311], N[(N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l * U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[Sqrt[t$95$1], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(N[(n / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $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*)))) < 9.99999999999948e-312

   1. Initial program 11.8%

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

   2. Simplified 36.8%

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

      [Start] 11.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 35.8	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 35.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 38.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 36.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 30.9%

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

   4. Simplified 37.3%

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

      [Start] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 30.9	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-*r* [=>] 32.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 32.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 36.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 36.2	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 37.3	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Applied egg-rr 38.4%

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

   6. Applied egg-rr 34.3%

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

   7. Simplified 36.4%

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

      [Start] 34.3	\[ \sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{Om} + \left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)} \]
      *-commutative [=>] 34.3	\[ \color{blue}{\sqrt{U \cdot \left(\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{Om} + \left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}} \]

   if 9.99999999999948e-312 < (*.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*)))) < 1.9999999999999999e304

   1. Initial program 97.4%

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

   if 1.9999999999999999e304 < (*.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 1.0%

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

   2. Simplified 17.7%

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

      [Start] 1.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 1.1	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 1.1	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 1.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in Om around 0 11.0%

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

   4. Applied egg-rr 10.0%

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

   5. Simplified 11.8%

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

      [Start] 10.0	\[ e^{\mathsf{log1p}\left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right)\right)} - 1 \]
      expm1-def [=>] 10.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right)\right)\right)} \]
      expm1-log1p [=>] 11.0	\[ \color{blue}{\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\frac{n \cdot \ell}{Om} \cdot \sqrt{2}\right)} \]
      associate-*r* [=>] 11.0	\[ \color{blue}{\left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{n \cdot \ell}{Om}\right) \cdot \sqrt{2}} \]
      *-commutative [=>] 11.0	\[ \color{blue}{\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{n \cdot \ell}{Om}\right)} \]
      *-rgt-identity [<=] 11.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\color{blue}{\left(n \cdot \ell\right) \cdot 1}}{Om}\right) \]
      associate-*r/ [<=] 11.0	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \frac{1}{Om}\right)}\right) \]
      associate-*l* [=>] 11.8	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \color{blue}{\left(n \cdot \left(\ell \cdot \frac{1}{Om}\right)\right)}\right) \]
      associate-*r/ [=>] 11.8	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(n \cdot \color{blue}{\frac{\ell \cdot 1}{Om}}\right)\right) \]
      associate-/l* [=>] 11.8	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(n \cdot \color{blue}{\frac{\ell}{\frac{Om}{1}}}\right)\right) \]
      /-rgt-identity [=>] 11.8	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left(n \cdot \frac{\ell}{\color{blue}{Om}}\right)\right) \]

   6. Applied egg-rr 10.2%

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

   7. Simplified 24.1%

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

      [Start] 10.2	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \sqrt{{\left(n \cdot \frac{\ell}{Om}\right)}^{2}}\right) \]
      unpow2 [=>] 10.2	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \sqrt{\color{blue}{\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}}\right) \]
      rem-sqrt-square [=>] 24.1	\[ \sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \color{blue}{\left|n \cdot \frac{\ell}{Om}\right|}\right) \]

   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. Simplified 6.4%

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

      [Start] 0.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 0.0	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      associate--l- [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      sub-neg [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(-\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]
      sub-neg [<=] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]
      cancel-sign-sub [<=] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      cancel-sign-sub [=>] 0.0	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)} \]
      associate-/l* [=>] 6.8	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]
      associate-*l* [=>] 6.4	\[ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Taylor expanded in U around 0 1.4%

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

   4. Simplified 10.6%

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

      [Start] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      *-commutative [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]
      +-commutative [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      mul-1-neg [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)\right)} \]
      unsub-neg [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]
      unpow2 [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-/l* [=>] 1.4	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]
      associate-*r* [=>] 1.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U*}}{{Om}^{2}}\right)\right)\right)\right)} \]
      unpow2 [=>] 1.5	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\left(n \cdot {\ell}^{2}\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)\right)\right)} \]
      times-frac [=>] 5.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \color{blue}{\frac{n \cdot {\ell}^{2}}{Om} \cdot \frac{U*}{Om}}\right)\right)\right)\right)} \]
      unpow2 [=>] 5.1	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{n \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]
      associate-*r* [=>] 10.6	\[ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \ell}}{Om} \cdot \frac{U*}{Om}\right)\right)\right)\right)} \]

   5. Taylor expanded in l around inf 4.8%

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

   6. Simplified 8.5%

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

      [Start] 4.8	\[ \sqrt{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)} \]
      associate-*r* [=>] 6.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)}} \]
      unpow2 [=>] 6.2	\[ \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)} \]
      *-commutative [=>] 6.2	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)} \]
      associate-/l* [=>] 4.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U*}}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      unpow2 [=>] 4.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U*}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      associate-*r/ [<=] 8.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\color{blue}{Om \cdot \frac{Om}{U*}}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]
      associate-*r/ [=>] 8.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om \cdot \frac{Om}{U*}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)} \]
      metadata-eval [=>] 8.5	\[ \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{Om \cdot \frac{Om}{U*}} - \frac{\color{blue}{2}}{Om}\right)\right)\right)} \]

   7. Taylor expanded in l around 0 18.8%

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

   8. Simplified 24.4%

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

      [Start] 18.8	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      associate-*l* [=>] 18.8	\[ \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      associate-*r* [=>] 18.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\color{blue}{\left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}}\right) \]
      *-commutative [=>] 18.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(\frac{\color{blue}{U* \cdot n}}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}\right) \]
      unpow2 [=>] 18.9	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(\frac{U* \cdot n}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}\right) \]
      associate-*r/ [<=] 18.8	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(\color{blue}{U* \cdot \frac{n}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}\right) \]
      associate-/r* [=>] 24.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(U* \cdot \color{blue}{\frac{\frac{n}{Om}}{Om}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot U}\right) \]
      associate-*r/ [=>] 24.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right) \cdot U}\right) \]
      metadata-eval [=>] 24.4	\[ \sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} - \frac{\color{blue}{2}}{Om}\right)\right) \cdot U}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \left|n \cdot \frac{\ell}{Om}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.3%
Cost	44428.00

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left|n \cdot \left(\frac{\ell}{Om} \cdot \sqrt{2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 2
Accuracy	55.8%
Cost	38412.00

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(\frac{-1}{U}\right)}^{-0.5} \cdot \sqrt{n \cdot \left(t \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	52.4%
Cost	14800.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -5.7 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \frac{n}{\frac{Om}{\frac{\ell \cdot \left(\ell \cdot U*\right)}{Om}}}\right)}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	53.4%
Cost	14412.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_2 \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

Alternative 5
Accuracy	53.5%
Cost	14412.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om \cdot \frac{Om}{U*}} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 6
Accuracy	53.8%
Cost	14412.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot t_1 - \frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{\frac{n}{Om}}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 7
Accuracy	49.7%
Cost	14040.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(t_2 - t\right)}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{n \cdot U*}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-262}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(t_2 - \frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}}\right) - t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+205}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	51.7%
Cost	13572.00

\[\begin{array}{l} \mathbf{if}\;U \leq -1.8 \cdot 10^{-44}:\\ \;\;\;\;{\left(\frac{-1}{U}\right)}^{-0.5} \cdot \sqrt{n \cdot \left(t \cdot -2\right)}\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 9
Accuracy	52.3%
Cost	13512.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot t_1 - t\right)}\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]

Alternative 10
Accuracy	49.4%
Cost	8524.00

\[\begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\\ t_2 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot t_1\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot t_2 - t\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{U*}{Om} \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{Om} + -2 \cdot t_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}\right)}\\ \end{array} \]

Alternative 11
Accuracy	51.7%
Cost	8393.00

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;U \leq -3 \cdot 10^{+142} \lor \neg \left(U \leq 3.6 \cdot 10^{-55}\right):\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot t_1 - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\frac{\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)}{\frac{Om}{U*}} + -2 \cdot t_1\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	50.2%
Cost	8392.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(\left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \frac{U* \cdot \left(n \cdot \ell\right)}{Om \cdot \frac{Om}{\ell}}\right) - t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	48.7%
Cost	8272.00

\[\begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\ell \cdot t_1\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}\right)}\\ \end{array} \]

Alternative 14
Accuracy	44.5%
Cost	8008.00

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right) \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+215}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	45.7%
Cost	8008.00

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(\ell \cdot U*\right) \cdot \frac{n}{Om} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 16
Accuracy	46.4%
Cost	7756.00

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - t\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Accuracy	46.8%
Cost	7625.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-245} \lor \neg \left(\ell \leq 3.4 \cdot 10^{-114}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	35.9%
Cost	7496.00

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 19
Accuracy	37.6%
Cost	7496.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-197}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{\frac{Om}{\ell}}{U \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	47.0%
Cost	7492.00

\[\begin{array}{l} \mathbf{if}\;U* \leq -6 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)\right)\right)\right)}\\ \end{array} \]

Alternative 21
Accuracy	39.6%
Cost	7112.00

\[\begin{array}{l} \mathbf{if}\;U \leq -1.8 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	37.6%
Cost	6980.00

\[\begin{array}{l} \mathbf{if}\;U* \leq 1.6 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	37.4%
Cost	6848.00

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

Reproduce

herbie shell --seed 2023096 
(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*))))))